Interface and mixed boundary value problems

on n-dimensional polyhedral domains

Constantin Băcuţă^{1}, Anna L. Mazzucato^{2}, Victor Nistor^{3}, and Ludmil Zikatanov^{4}

Received: December 1, 2008

Communicated by Eckhard Meinrenken

Abstract. Let μ ∈ ℤ_{+} be arbitrary. We prove a well-posedness result for mixed boundary value/interface problems of second-order, positive, strongly elliptic operators in weighted Sobolev spaces K_{a}^{μ}(Ω) on a bounded, curvilinear polyhedral domain Ω in a manifold M of dimension n. The typical weight η that we consider is the (smoothed) distance to the set of singular boundary points of ∂Ω. Our model problem is P u := − div(A∇u) = f, in Ω, u = 0 on ∂_{D}Ω, and D_{ν}^{P} u = 0 on ∂_{ν}Ω, where the function A ≥ ϵ > 0 is piece-wise smooth on the polyhedral decomposition Ω̄ = ∪_{j}Ω̄_{j}, and ∂Ω = ∂_{D}Ω ∪ ∂_{N}Ω is a decomposition of the boundary into polyhedral subsets corresponding, respectively, to Dirichlet and Neumann boundary conditions. If there are no interfaces and no adjacent faces with Neumann boundary conditions, our main result gives an isomorphism P : K_{a+1}^{μ+1}(Ω) ∩ {u = 0 on ∂_{D}Ω, D_{ν}^{P} u = 0 on ∂_{N}Ω} → K_{a−1}^{μ−1}(Ω) for μ ≥ 0 and |a| < η, for some η > 0 that depends on Ω and P but not on μ. If interfaces are present, then we only obtain regularity on each subdomain Ω_{j}. Unlike in the case of the usual Sobolev spaces, μ can be arbitrarily large, which is useful in certain applications. An important step in our proof is a regularity result, which holds for general strongly elliptic operators that are not necessarily positive. The regularity result is based, in turn, on a study of the geometry of our polyhedral domain when endowed with the metric (dx/η)^{2}, where η is the weight (the smoothed distance to the singular set). The wellposedness result applies to positive operators, provided the interfaces are smooth and there are no adjacent faces with Neumann boundary conditions.

1The work of C. Bacuta is partially supported by NSF DMS-0713125.

2The work of A. Mazzucato is partially supported by NSF grant DMS-0405803 and DMS-0708902.

3The work of V. Nistor is partially supported by NSF grant DMS-0555831, DMS-0713743, and OCI-0749202.

4The work of L. Zikatanov is partially supported by NSF grant DMS-0810982 and OCI-0749202.

Documenta Mathematica 15 (2010) 687–745

2010 Mathematics Subject Classification: Primary 35J25; Secondary 58J32, 52B70, 51B25. Keywords and Phrases: Polyhedral domain, elliptic equations, mixed boundary conditions, interface, weighted Sobolev spaces, wellposedness, Lie manifold.

Introduction

Let Ω ⊂ ℝ^{n} be an open, bounded set. Consider the boundary value problem

\begin{cases}Δu = f & in Ω \\ u|_{∂Ω} = g, & on Ω, \end{cases}  (1)

where Δ is the Laplace operator. For Ω smooth, this boundary value problem has a unique solution u ∈ H^{s+2}(Ω) depending continuously on f ∈ H^{s}(Ω) and g ∈ H^{s+3/2}(∂Ω), s ≥ 0. See the books of Evans [25], Lions and Magenes [49], or Taylor [72] for proofs of this basic and well known result. It is also well known that this result does not extend to non-smooth domains Ω. For instance, Jerison and Kenig prove in [35] that if g = 0 and Ω ⊂ ℝ^{n}, n ≥ 3, is an open, bounded set such that ∂Ω is Lipschitz, then Equation (1) has a unique solution in W^{s,p}(Ω) depending continuously on f ∈ W^{s−2,p}(Ω) if, and only if, (1/p, s) belongs to a certain explicit hexagon. They also prove a similar result if Ω ⊂ ℝ^{2}. A consequence of this result is that the smoothness of the solution u (measured by the order s of the Sobolev space W^{s,p}(Ω) containing it) will not exceed, in general, a certain bound that depends on the domain Ω and p, even if f is smooth. In addition to the Jerison and Kenig paper mentioned above, a deep analysis of the difficulties that arise for ∂Ω Lipschitz is contained in the papers of Babuška [4], Baouendi and Sjöstrand [9], Băcuţă, Bramble, and Xu [14], Babuška and Guo [31, 30], Brown and Ott [13], Jerison and Kenig [33, 34], Kenig [38], Kenig and Toro [39], Koskela, Koskela and Zhong [43, 44], Mitrea and Taylor [58, 60, 61], Verchota [73], and others (see the references in the aforementioned papers). Results more specific to curvilinear polyhedral domains are contained in the papers of Costabel [17], Dauge [19], Elschner [20, 21], Kondratiev [41, 42], Mazya and Rossmann [54], Rossmann [63] and others. Excellent references are also the monographs of Grisvard [27, 28] as well as the recent books [45, 46, 52, 53, 62], where more references can be found. In this paper, we consider the boundary value problem (1) when Ω is a bounded curvilinear polyhedral domain in ℝ^{n}, or, more generally, in a manifold M of dimension n and, Poisson’s equation Δu = f is replaced by a positive, strongly elliptic scalar equation. We define curvilinear polyhedral domains inductively in Section 2. We allow polyhedral domains to be disconnected for technical reasons, more precisely, for the purpose of defining them inductively. Our results, however, are formulated for connected polyhedral domains. Many polyhedral domains are Lipschitz domains, but not all. This fact is discussed in detail by Vogel and Verchota in [74], where they also prove that the harmonic measure is absolutely continuous with respect to the Lebesgue measure on the boundary as well as the solvability of Equation (1) if f = 0 and g ∈ L^{2−ϵ}(∂Ω), thus generalizing several earlier, classical results. See also the excellent book [50]. The generalized polyhedra we considered are of combinatorial type if no cracks are present. (For a discussion of more general domains, see the references [68, 74, 75].) Instead of working with the usual Sobolev spaces, as in several of the papers mentioned above, we shall work in some weighted analogues of these papers. Let Ω^{(n−2)} ⊂ ∂Ω be the set of singular (or non-smooth) boundary points of Ω, that is, the set of points p ∈ ∂Ω such ∂Ω is not smooth in a neighborhood of p. We shall denote by η_{n−2}(x) the distance from a point x ∈ Ω to the set Ω^{(n−2)}. We agree to take η_{n−2} = 1 if there are no such points, that is, if ∂Ω is smooth. We then consider the weighted Sobolev spaces

K_{a}^{μ}(Ω) = {u ∈ L_{loc}^{2}(Ω), η_{n−2}^{|α|−a}∂^{α}u ∈ L^{2}(Ω), for all |α| ≤ μ}, μ ∈ ℤ_{+}, (2)

which we endow with the induced Hilbert space norm. A similar definition yields the weighted Sobolev spaces K_{a}^{s}(∂Ω), s ∈ ℝ_{+}. By including an extra weight h in the above spaces we obtain the spaces hK_{a}^{μ}(Ω) and hK_{a}^{s}(∂Ω) (where h is required to be an admissible weight, see Definition 3.8 and Subsection 5.1). These spaces are closely related to the weighted Sobolev spaces on non-compact manifolds considered, for example in the references [41, 42, 46, 54, 62, 63] mentioned above, as well as in the works of Erkip and Schrohe [22], Grubb [29], Schrohe [65], as well as the sequence of papers of Schrohe and Schulze [66, 67] concerning related results on boundary value problems on non-compact manifolds and, more generally, on the analysis on non-compact manifolds. The main result of this article, Theorem 1.2 applies to operators with piecewise smooth coefficients, such as div a∇u = f, where a is allowed to have only jumps across the interface. A simplified version of that result, when formulated for the Laplace operator Δ on ℝ^{n} with Dirichlet boundary conditions, reads as follows. In this theorem and throughout this paper, Ω will always denote an open set.

Theorem 0.1. Let Ω ⊂ ℝ^{n} be a bounded, curvilinear polyhedral domain and μ ∈ ℤ_{+}. Then there exists η > 0 such that the boundary value problem (1) has a unique solution u ∈ K_{a+1}^{μ+1}(Ω) for any f ∈ K_{a−1}^{μ−1}(Ω), any g ∈ K_{a+1/2}^{μ+1/2}(∂Ω), and any |a| < η. This solution depends continuously on f and g. If a = μ = 0, this solution is the solution of the associated variational problem.

The case n = 2 of the above theorem is Theorem 6.6.1 in the excellent monograph [45]. Results in higher dimensions related to the ones in our paper can be found, for instance, in [19, 45, 51, 54, 62]. These works also use the framework of the K_{a}^{μ}(Ω) spaces. The spaces hK_{a}^{μ}(Ω), with h an admissible weight are somewhat more general (see Definition 3.8 for a definition of admissible weights). We also take the dimension n of the ambient Euclidean space ℝ^{n} ⊃ Ω to be arbitrary. Furthermore, we impose mixed Dirichlet/Neumann boundary conditions and allow the boundary conditions to change along (n − 2)-dimensional, piecewise smooth hypersurfaces in each hyperface of Ω. To handle this situation, we include in the singular set of Ω all points where the boundary conditions change, giving rise to a polyhedral structure on Ω which is not entirely determined by geometry, but also takes into account the specifics of the boundary value problem. However, we consider only second order, strongly elliptic systems. For n = 3, mixed boundary value problems for such systems in polyhedral domains were studied in weighted L^{p} spaces by Mazya and Rossmann [54] using point estimates for the Green’s function [55]. Since we work in L^{2}-based spaces, we use instead coercive estimates, which directly generalize to arbitrary dimension and to transmission problems. We use manifolds in order to be able to prove estimates inductively. The method of layer potentials seems to give more precise results, but is less elementary (see for example [38, 59, 60, 75]). Solvability of mixed boundary value problems from the point of view of parametrices and pseudodifferential calculus can be found in the papers by Eskin [23, 24], Vishik and Eskin [76, 77, 78], and Boutet de Monvel [10, 11] among others. As we have already pointed out, it is not possible to obtain full regularity in the usual Sobolev spaces, when the smoothness of the solution as measured by μ + 1 in Theorem 0.1 is too large. On the other hand, the weighted Sobolev spaces have proved themselves to be as convenient as the usual Sobolev spaces in applications. Possible applications are to partial differential equations, algebraic geometry, representation theory, and other areas of pure and applied mathematics, as well as to other areas of science, such as continuum mechanics, quantum mechanics, and financial mathematics. See for example [7, 8, 48], where optimal rates of convergence were obtained for the Finite element method on 3D polyhedral domains and for 2D transmission problems using weighted Sobolev spaces. The paper is organized as follows. In Section 1, we introduce the mixed boundary value/interface problem that we study, namely Equation (6), and state the main results of the paper, Theorem 1.1 on the regularity of (6) in weighted spaces of arbitrarily high Sobolev index, and Theorem 1.2 on the solvability of (6) under additional conditions on the operator (positivity) and on the domain (smooth interfaces and no two adjacent faces with Neumann boundary conditions). In Section 2, we give a notion of curvilinear, polyhedral domain in arbitrary dimension using induction, then we specialize to the familiar case of polygonal domains in ℝ^{2} and polyhedral domains in ℝ^{3}, and describe the desingularization Σ(Ω) of the domain Ω in these familiar settings. Before discussing the desingularization in higher dimension, we recall briefly needed notions from the theory of Lie manifolds with boundary in Section 3. Then, in Section 4 we show that Σ(Ω), also defined by induction on the dimension, naturally carries a structure of Lie manifold with boundary. We also discuss the construction of the canonical weight function r_{Ω}, which extends smoothly to Σ(Ω) and is comparable to the distance to the singular set. In turn, the Lie manifold structure on Σ(Ω) allows to identify the spaces K_{a}^{μ}(Ω), μ ∈ ℤ_{+}, with standard Sobolev spaces on Σ(Ω), and hence lead to a definition of the weighted Sobolev spaces on the boundary K_{a}^{s}(∂Ω), s ∈ ℝ. Lastly Section 6 contains the proofs of the main results and most of lemmas of the paper; in particular, it contains a proof of the weighted Hardy-Poincaré inequality used to establish positivity or strict coercivity for the problem of Equation (6). An earlier version of this paper was first circulated as an IMA Preprint in August 2004. We conclude this Introduction with a word on notation. By Ω we always mean an open set in ℝ^{n}. By a diffeomorphisms, we mean a C^{∞} invertible map with C^{∞} inverse. By C we shall denote a generic constant that may change from line to line. We also denote ℤ_{+} = {0, 1, 2, 3, …}.

Acknowledgments

We thank Bernd Ammann, Ivo Babuška, Wolfgang Dahmen, Alexandru Ionescu, and Daniel Tătaru for useful discussions. We also thank Johnny Guzman for pointing the reference [42] to us and Yu Qiao for carefully reading our paper. The second named author would like to thank Institute Henri Poincaré in Paris and the Max Planck Institute for Mathematics in Bonn for their hospitality while parts of this work were being completed.

1 The problem and statement of the main results

We begin by introducing the class of differential operators and the associated mixed Dirichlet-Neumann boundary value/interface problem that will be the object of study. For simplicity, we consider primarily the scalar case, although our results extend to systems as well. Then, we state the main results of this article, namely the regularity and the well-posedness of the mixed boundary value/interface problem (6) in weighted Sobolev spaces for n-dimensional, curvilinear polyhedral domains Ω ⊂ ℝ^{n}. These are stated in Theorems 1.2 and 1.1. Our analysis is general enough to extend to a bounded subdomain Ω ⊂ M of a compact Riemannian manifold M. Initially the reader may assume the polyhedron is straight, that is, informally, that every j-dimensional component of the boundary, j = 1, …, n − 1 is a subset of an affine space. A complete definition of a curvilinear polyhedral domain is given in Section 2.

1.1 The differential operator P and the associated problem

Let us denote by Ω ⊂ ℝ^{n} a bounded, curvilinear stratified polyhedral domain (see Definition 2.1). The domain Ω need not be connected, nor convex. We assume that we are given a decomposition

\overline{Ω} = ∪_{j=1}^{N}\overline{Ω}_{j},  (3)

where Ω_{j} are disjoint polyhedral subdomains. In particular, every face of Ω is also a face of one of the domains Ω_{j}. This is possible since the faces of Ω are not determined only by the geometry of Ω. As discussed in Section 4, a face of codimension 1 of Ω is called a hyperface. For well-posedness results, we shall assume that

Γ = ∪_{j=1}^{N}∂Ω_{j} \ ∂Ω,  (4)

is a finite collection of disjoint, smooth (n−1)-hypersurfaces. We observe that, since each Ω_{j} is a polyhedron, each component of Γ intersects ∂Ω transversely. We refer to Γ as the interface. We also assume that the boundary of Ω is partition into two disjoint subsets

∂Ω = ∂_{D}Ω ∪ ∂_{N}Ω,  (5)

with ∂_{N}Ω consisting of a union of open faces of Ω. For well-posedness results, we shall assume that ∂_{N}Ω does not contain adjacent faces of ∂Ω. We are interested in studying the following mixed boundary value/interface problem for a certain class of elliptic, scalar operators P described below:

\begin{cases}P u = f & on Ω, \\ u|_{∂_{D}Ω} = g_{D} & on ∂_{D}Ω, \\ D_{ν}^{P} u|_{∂_{N} Ω} = g_{N} & on ∂_{N}Ω, \\ u^{+} = u^{−}, D_{ν}^{P +}u = D_{ν}^{P −}u & on Γ. \end{cases}  (6)

Above, ν is the unit outer normal to ∂Ω, which is defined almost everywhere, D_{ν}^{P} is the conormal derivative associated to the operator P (see (10)), and ± refers to one-sided, non-tangential limits at the interface Γ. We observe that D_{ν}^{P ±} is well-defined a.e. on each side of the interface Γ, since each smooth component of Γ is the boundary of exactly two adjacent polyhedral domains Ω_{j}, by (4). The coefficients of P will have in general jump discontinuities along Γ. We next introduce the class of differential operators that we consider. At first, the reader may assume P = −Δ, the Laplace operator. We shall write Re(z) := \frac{1}{2}(z + \overline{z}), or simply Re z for the real part of a complex number z. Let u ∈ H_{loc}^{2}(Ω). We shall study the following scalar, differential operator in divergence form

P u(x) = − ∑_{j,k=1}^{n} ∂_{j}[A_{jk}(x)∂_{k}u(x)] + ∑_{j=1}^{n} B_{j}(x)∂_{j}u(x) + C(x)u(x). (7)

The coefficients A_{jk}, B_{j}, C are real valued with only jump discontinuities on the interface Γ, the operator P is required to be uniformly strongly elliptic and to satisfy another positivity condition. More precisely, the coefficients of P are assumed to satisfy:

A_{jk}, B_{j}, C ∈ ⊕_{j=1}^{N}C^{∞}(\overline{Ω_{j}}) ∩ L^{∞}(\overline{Ω})  (8a)
Re( ∑_{j,k=1}^{n}[A_{jk}(x)]ξ_{j}\overline{ξ}_{k}) ≥ ϵ ∑_{j=1}^{n} |ξ_{j}|^{2}, ∀ξ_{j} ∈ ℂ, ∀x ∈ Ω, and  (8b)

2C(x) − ∑_{j=1}^{n} ∂_{j}B_{j}(x) ≥ 0,  (8c)

for some ϵ > 0. For scalar equations, one may weaken the uniform strong ellipticity condition (8b), but this is not needed for our purposes. Our results extend to systems satisfying the strong Legendre–Hadamard condition, namely

Re( ∑_{j,k=1}^{n} ∑_{p,q=1}^{m}[A_{jk}(x)]_{pq}ξ_{jp}\overline{ξ}_{kq}) ≥ ϵ ∑_{j=1}^{n} ∑_{p=1}^{m} |ξ_{jp}|^{2}, ∀ξ_{jp} ∈ ℂ,  (9)

and a condition on the lower-order terms equivalent to (8c). This condition is not satisfied however by the system of anisotropic elasticity in ℝ^{3}, for which nevertheless the well-posedness result holds if the elasticity tensor is positive definite on symmetric matrices [56]. In (8a), the “regularity condition on the coefficients of P ” means that the coefficients and their derivatives of all orders have well-defined limits from each side of Γ, but as equivalence classes in L^{∞} they may have jump discontinuities along the interface. This condition can be relaxed, but it allows us to state a regularity result of arbitrary order in each subdomains for the solution to the problem (6). The conormal derivative associated to the operator P is formally defined by:

D_{P}^{ν} u(x) = ∑_{i,j=1}^{n} ν_{i} A_{ij}∂_{j}u(x),  (10)

where ν = (ν^{i}) is the unit outer normal vector to the boundary of Ω. We give meaning to (10) in the sense of trace at the boundary. In particular, for u regular enough D_{ν}^{P} u is defined almost everywhere on the boundary as a non-tangential limit. The problem (6) with g_{D} = 0 is interpreted in a weak (or variational) sense, using the bilinear form B(u, v) defined by:

B(u, v) := ∑_{j,k=1}^{n} (A_{jk}∂_{k}u, ∂_{j}v) + ∑_{j=1}^{n} (B_{j}∂_{j}u, v) + (Cu, v),  (11)

which is well-defined for any u, v ∈ H^{1}(Ω). Then, (6) is weakly equivalent to

B(u, v) = (f, v)_{L^{2}(Ω)} + (g_{N}, v)_{∂_{N} Ω},  (12)

where the second parenthesis denotes the pairing between a distribution and a (suitable) function. The jump or transmission conditions, u^{+} = u^{−}, D_{ν}^{P} u^{+} = D_{ν}^{P} u^{−} at the interface Γ follow from the weak formulation and the H^{1}-regularity of weak solutions, and hence justify passing from the strong formulation (6) to the weak one (12). Otherwise, in general, the difference D_{ν}^{P +}u − D_{ν}^{P −}u may be non-zero and may be included as a distributional term in f. Condition (8c) implies the Hardy-Poincaré type inequality

ReB(u, u) > C(η_{n−2}u, η_{n−2}u)_{L^{2}},  (13)

if there are no adjacent faces with Neumann boundary conditions and the interface is smooth. In fact, it is enough to assume that the latter is satisfied instead of (8c). For applications, however, it is more convenient to have the concrete condition (8c). Problems of the form (6) arise in many applications. An important example is given by (linear) elastostatics. In this case, [P u]^{i} = − ∑_{jkl=1}^{3} ∂_{j} C^{ijkl} ∂_{k}u^{l}, i = 1, 2, 3, where C is the fourth-order elasticity tensor, modelling the response of an elastic body under small deformations. Dirichlet or displacement boundary conditions correspond to clamping (parts of) the boundary, while Neumann or traction boundary conditions correspond to loading mechanically (parts of) the boundary. Interfaces arise due to the use of different materials. A careful analysis of mixed Dirichlet/Neumann boundary value problems for linear elastostatics in 3-dimensional curvilinear, polyhedral domains, was carried out by two of the authors in [56]. There, the concept of a “domain with polyhedral structure” is more general than in this paper and includes cracks. In [48], they studied mixed boundary value/interface problems and the implementation of the Finite Element Method on “domains with polygonal structure” with nonsmooth interfaces (see also [15]). The results of this paper can be extended to include domains with cracks, as in [56] and [48], but the topological machinery used there, including the notion of an “unfolded boundary” [19] in arbitrary dimensions is significantly more complex. (See [12] for related results.)

1.1.1 Operators on manifolds

We turn to consider the assumptions on P when the domain Ω is a curvilinear, polyhedral domain in a manifold M of the same dimension. Let then E be a vector bundle on M endowed with a hermitian metric. A coordinate free expression of the conditions in Equations (8a)–(8c) is obtained as follows. We assume that there exist a metric preserving connection ∇ : Γ(E) → Γ(E ⊗T^{∗}M), a smooth endomorphism A ∈ End(E ⊗ T^{∗}M), and a first order differential operator P_{2} : Γ(E) → Γ(E) with smooth coefficients such that

A + A^{∗} ≥ 2ϵI for some ϵ > 0.  (14)

Then we define P_{1} = ∇^{∗}A∇ and P = P_{1}+P_{2}. In particular, the operator P will satisfy the strong Lagrange–Hadamard condition in a neighborhood of Ω in M.

Note that if Ω ⊂ ℝ^{n} and the vector bundle E is trivial, then the condition of (14) reduces to the conditions of (8), by taking ∇ to be the trivial connection. We can allow A to have jump discontinuities as well along polyhedral interfaces.

1.2 The main results

We are ready to state the principal results of this paper. We continue to assume hypotheses (3)–(5) on the domain Ω and its decomposition into disjoint subdomains Ω_{j} separated by the interface Γ. We begin with a regularity results for solutions to the problem (6) in weighted Sobolev spaces hK_{a}^{μ}, μ ∈ ℤ_{+}, a ∈ ℝ, where

K_{a}^{μ}(Ω) := {u ∈ L_{loc}^{2}(Ω), η_{n−2}^{|α|− a}∂^{α}u ∈ L^{2}(Ω), for all |α| ≤ μ} , μ ∈ ℤ_{+},

and

hK_{a}^{μ}(Ω) := {hu, u ∈ K_{a}^{μ}(Ω)}.

(See Section 5 for a detailed discussion and main properties of these spaces.) Above, η_{n−2} is the distance to the singular set in Ω given in Definition 2.5, while h is a so-called admissible weight described in Definition 3.8. Initially, the reader may assume that h = r_{Ω}^{b}, b ∈ ℝ, where r_{Ω} is a function comparable to the distance function η_{n−2} close to the singular set, but with better regularity than η_{n−2} away from the singular set. (We refer again to Subsection 5.1 for more details). The weight h is important in the applications of the theory developed here for numerical methods, where appropriate choices of h yield quasi-optimal rates of convergence for the Finite Element approximation to the weak solution of the problem (6) (see [6, 7, 8, 48]).

Theorem 1.1. Let Ω ⊂ ℝ^{n} be a bounded, curvilinear polyhedral domain of dimension n. Assume that the operator P satisfies conditions (8a) and (8b). Let μ ∈ ℤ_{+}, a ∈ ℝ, and u ∈ hK_{a+1}^{1}(Ω) be such that P u ∈ hK_{a−1}^{μ−1}(Ω_{j}), for all j, u|_{∂_{D}Ω} ∈ hK_{a+1/2}^{μ+1/2}(∂_{D}Ω), D_{ν}^{P} u|_{∂_{N} Ω} ∈ hK_{a−1/2}^{μ−1/2}(∂_{N}Ω). If h is an admissible weight, then u ∈ hK_{a+1}^{μ+1}(Ω_{j}), for all j = 1, …, N, and

\begin{aligned}∥u∥_{hK_{a+1}^{μ+1}(Ω_{j})} & ≤ C( ∑_{k=1}^{N} ∥P u∥_{hK_{a−1}^{μ−1}(Ω_{k})} + ∥u∥_{hK_{a+1}^{0}(Ω)}+ \\ & ∥u|_{∂_{D}Ω}∥_{hK_{a+1/2}^{μ+1/2}(∂_{D}Ω)}) + ∥u|_{∂_{N} Ω}∥_{hK_{a−1/2}^{μ−1/2}(∂_{N} Ω)}) \end{aligned}  (15)

for a constant C = C(Ω, P, μ, a, h) > 0, independent of u.

The proof of the regularity theorem exploits Lie manifolds and their structure to reduce to the classical case of bounded, smooth domains. The proof can be found in Section 6. Note that in this theorem we do not require the interfaces to be smooth and we allow for adjacent faces with Neumann boundary conditions. Under additional conditions on the set Ω and its boundary ensuring strict coercivity of the bilinear form B of equation (11), we obtain a well-posedness result for problem (6). In [48], two of the authors obtained a well-posedness result in an augmented space on polygonal domains with “Neumann-Neumann vertices,” i.e., vertices for which both sides joining at the vertex are given Neumann boundary conditions, and for which the interface Γ is not smooth. Such result is based on specific spectral properties of operator pencils near the vertices and is not easily extendable to higher dimension. Note that K_{a}^{μ}(Ω) = hK_{0}^{μ}(Ω) for a suitable admissible weight h and hence there is no loss of generality to assume a = 0 in Theorem 1.1. We will use the same reasoning to simplify the statements of the following results.

Theorem 1.2. Let Ω ⊂ ℝ^{n} be a bounded, connected curvilinear polyhedral domain of dimension n. Assume that ∂Ω_{N} does not contain any two adjacent hyperfaces, that ∂_{D}Ω is not empty, and that the interface Γ is smooth. In addition, assume that the operator P satisfies conditions (8). Let W_{μ}(Ω), μ ∈ ℤ_{+}, be the set of admissible weights h such that the map P̃(u) := (P u, u|_{∂_{D}Ω}, D_{ν}^{P} u|_{∂_{N} Ω}) establishes an isomorphism

P̃ : {u ∈ ⨁_{j=1}^{N} hK_{1}^{μ+1}(Ω_{j}) ∩ hK_{1}^{1}(Ω), D_{ν}^{P} u^{+} = D_{ν}^{P} u^{−} on Γ}

→ ⨁_{j=1}^{N} hK_{−1}^{μ−1}(Ω_{j}) ⊕ hK_{1/2}^{μ+1/2}(∂_{D}Ω) ⊕ hK_{−1/2}^{μ−1/2}(∂_{N}Ω).

Then the set W_{μ}(Ω) is an open set containing 1.

Theorem 1.2 reduces to a well-known, classical result when Ω is a smooth bounded domain. (See Remark 6.11 for a result on smooth domains that is not classical.) The same is true for the following result, Theorem 1.3, which works for general domains on manifolds. Note however that for manifolds it is more difficult to express the coercive property, so for more complete results we restrict to the case of operators of Laplace type.

Theorem 1.3. Let Ω ⊂ M be a bounded, connected curvilinear polyhedral domain of dimension n. Assume that every connected component of Ω has a non-empty boundary and that the operator P satisfies condition (14). Assume additionally that no two adjacent hyperfaces of ∂Ω are endowed with Neumann boundary conditions and that the interface Γ is smooth. Let c ∈ ℂ and W′_{μ}(Ω) be the set of admissible weights h such that the map P̃_{c}(u) := (P u + cu, u|_{∂Ω}, D_{ν}^{P} u|_{∂Ω}) establishes an isomorphism

P̃_{c} : {u ∈ ⨁_{j=1}^{N} hK_{1}^{μ+1}(Ω_{j}) ∩ hK_{1}^{1}(Ω), u^{+} = u^{−}, D_{ν}^{P} u^{+} = D_{ν}^{P} u^{−} on Γ}

→ ⨁_{j=1}^{N} hK_{−1}^{μ−1}(Ω_{j}) ⊕ hK_{1/2}^{μ+1/2}(∂_{D}Ω) ⊕ hK_{−1/2}^{μ−1/2}(∂_{N}Ω).

Then the set W′_{μ}(Ω) is an open set, which contains 1 if the real part of c is large or if P = ∇^{∗}A∇ with A satisfying (14).

For the rest of this section, Ω and P will be as in Theorem 1.2. We discuss some immediate consequences of Theorem 1.2. Analogous results can be obtained from Theorem 1.3, but we will not state them explicitly. The continuity of the inverse of P̃ is made explicit in the following corollary.

Corollary 1.4. Let A satisfy (14). There exists a constant C = C(Ω, P, μ, a, h) > 0, independent of f, g_{D}, and g_{N}, such that

\begin{aligned}∥u∥_{hK_{1}^{1}(Ω)} + ∥u∥_{hK_{1}^{μ+1}(Ω_{j})} & ≤ C ( ∑_{j=1}^{N} ∥P u∥_{hK_{−1}^{μ−1}(Ω_{j})} \\ & + ∥u|_{∂_{D}Ω}∥_{hK_{1/2}^{μ+1/2}(∂_{D}Ω)} + ∥D_{ν}^{P} u|_{∂_{N} Ω}∥_{hK_{1/2}^{μ+1/2}(∂_{N} Ω)}), \end{aligned}

for any u ∈ hK_{1}^{1}(Ω) and any j.

From the fact that η_{n−2} is equivalent to r_{Ω} by Proposition 4.9 and Corollary 4.11, we obtain the following corollary.

Corollary 1.5. Let A satisfy (14). There exists η > 0 such that

(P, D_{ν}^{P}) : {u ∈ ⨁_{j=1}^{N} K_{a+1}^{μ+1}(Ω_{j}) ∩ K_{a+1}^{1}(Ω), u|_{∂_{D}Ω} = 0,

D_{ν}^{P} u^{+} = D_{ν}^{P} u^{−} on Γ} → ⨁_{j=1}^{N} K_{a−1}^{μ−1}(Ω_{j}) ⊕ K_{a−1/2}^{μ−1/2}(∂_{N}Ω)

is an isomorphism for all μ ∈ ℤ_{+} and all |a| < η.

Note above and in what follows that the interface matching condition u^{+} = u^{−} follows from u ∈ K_{a+1}^{1}(Ω).

Proof. From the results in Sections 5 and 5.1, K_{a+1}^{μ+1} = r_{Ω}^{a}K_{1}^{μ+1} and r_{Ω}^{a} is an admissible weight for any a ∈ ℝ. The result then follows from the fact that W_{μ}(Ω) is an open set containing the weight 1 by Theorem 1.2.

The following corollary gives a characterization of the set W_{μ}(Ω) in the spirit of [15]. There, similar arguments give that for n = 2 the constant η in the previous corollary is η = π/α_{M}, where α_{M} is the largest angle of Ω. See also [42].

Corollary 1.6. Let h = r_{Ω}^{a} and A satisfy (14). Assume that for all λ ∈ [0, 1] the map

(P, D_{ν}^{P}) : {u ∈ h^{λ}K_{1}^{1}(Ω), u|_{∂_{D}Ω} = 0, D_{ν}^{P} u|_{∂_{N} Ω} = 0} → h^{λ}K_{−1}^{−1}(Ω)

is Fredholm. Then h ∈ W_{μ}(Ω).

The corollary holds for more general weights h = ∏_{H} x_{H}^{a_{H}}, where x_{H} is the distance to an hyperface H at infinity (see Section 5.1), as long as all a_{H} ≥ 0 or all a_{H} ≤ 0.

Proof. We proceed as in [15]. The family P_{λ} := h^{−λ}P h^{λ} is continuous for λ ∈ [0, 1], consists of Fredholm operators by hypothesis, and is invertible for λ = 0 by Theorem 1.2. It follows that the family P_{λ} consists of Fredholm operators of index zero. To prove that these operators are isomorphisms, it is hence enough to prove that they are either injective or surjective. Assume first that a ≥ 0 in the definition of h. Then K_{1}^{1+λ}(Ω) = h^{λ}K_{1}^{1}(Ω) ⊂ K_{1}^{1}(Ω). Therefore P is injective on

h^{λ}K_{1}^{1}(Ω) ∩ {u|_{∂_{D}Ω} = 0, D_{P}^{ν} u|_{∂_{N} Ω} = 0}.

This, in turn, gives that P_{λ} is injective. Assume that a ≤ 0 and consider

P_{λ} : h^{λ}K_{1}^{1}(Ω) ∩ {u|_{∂Ω} = 0, D_{P}^{ν} u|_{∂_{N} Ω} = 0} → h^{λ}K_{−1}^{−1}(Ω).  (16)

We have (P_{λ})^{∗} = (P^{∗})_{−λ}. The same argument as above shows that P_{λ}^{∗} is injective, and hence that it is an isomorphism, for all 0 ≤ λ ≤ 1. Hence P_{λ} is an isomorphism for all 0 ≤ λ ≤ 1.

2 Polyhedral domains

In this section we introduce the class of domains to which the results of the previous sections apply. We then specialize to domains in 2 and 3 dimensions and provide ample examples. The reader may at first concentrate on this case. We describe how to desingularize the domain in arbitrary dimension later in the paper, using the theory of Lie manifolds, which we recall in the next section. Let Ω be a proper open set in ℝ^{n} or more generally in a smooth manifold M of dimension n. Our main focus is the analysis of partial differential equations on Ω, specifically the mixed boundary value/interface problem (6). For this reason, we give Ω a structure that is not entirely determined by geometry, rather it takes into account the boundary and interface conditions for the operator P in problem (6). We assume that Ω is given together with a smooth stratification:

Ω^{(0)} ⊂ Ω^{(1)} ⊂ … ⊂ Ω^{(n−2)} ⊂ Ω^{(n−1)} := ∂Ω ⊂ Ω^{(n)} := Ω̄.  (17)

We recall that a smooth stratification S_{0} ⊂ S_{1} ⊂ … ⊂ X of a topological space X is an increasing sequence of closed sets S_{j} = S_{j}(X) such that each point of X has a neighborhood that meets only finitely many of the sets S_{j}, S_{0} is a discrete subset, S_{j+1} ∖ S_{j}, j ≥ 0, is a disjoint union of smooth manifolds of dimension j + 1, and X = ∪S_{j}. Some of the sets S_{j} may be empty for 0 ≤ j ≤ j_{0} < dim(X).

We will always assume that the stratification {Ω^{(j)}} satisfies the condition that Ω^{(j)} ∖ Ω^{(j−1)} has finitely many connected components, for all j. This assumption is automatically satisfied if Ω is bounded, and it is not crucial, but simplifies some of the later constructions. We proceed by induction on the dimension to define a polyhedral structure on Ω. Our definition is very closely related to that of Whitney stratified spaces [79]. We first agree that a curvilinear polyhedral domain of dimension n = 0 is simply a finite set of points. Then, we assume that we have defined curvilinear polyhedral domains in dimension ≤ n − 1, n ≥ 1, and define a curvilinear polyhedral domain in a manifold M of dimension n next. We shall denote by B^{l} the open unit ball in ℝ^{l} and by S^{l−1} := ∂B^{l} its boundary. In particular, we identify B^{0} = {1}, B^{1} = (−1, 1), and S^{0} = {−1, 1}.

Definition 2.1. Let M be a smooth manifold of dimension n ≥ 1. Let Ω ⊂ M be an open subset endowed with the stratification (17). Then Ω ⊂ M is a stratified, curvilinear polyhedral domain if for every point p ∈ ∂Ω, there exist a neighborhood V_{p} in M such that:

(i) if p ∈ Ω^{(l)} \ Ω^{(l−1)}, l = 1, …, n − 1, there is a curvilinear polyhedral domain ω_{p} ⊂ S^{n−l−1}, \overline{ω_{p}} ≠ S^{n−l−1}, and

(ii) a diffeomorphism ϕ_{p} : V_{p} → B^{n−l} × B^{l} such that ϕ_{p}(p) = 0 and

ϕ_{p}(Ω ∩ V_{p}) = {rx′, 0 < r < 1, x′ ∈ ω_{p}} × B^{l},  (18)

inducing a homeomorphism of stratified spaces.

Given any p ∈ ∂Ω, let 0 ≤ ℓ(p) ≤ n − 1 be the smallest integer such that p ∈ Ω^{(ℓ(p))}, but p ∉ Ω^{(ℓ(p)−1)} (by convention we set Ω^{(l)} = ∅ if l < 0). By construction, ℓ(p) is unique given p. Then, the domain ω_{p} ⊂ S^{n−ℓ(p)−1} in the definition above will be called the link of Ω at p. We identify the ”ball” B^{0} = {1} and the “sphere” S^{0} = ∂B^{1} = {−1, 1}. In particular if ℓ(p) = n − 1, then ω_{p} is a point. The notion of a stratified polyhedron is well known in the literature (see for example the monograph [71]). However, our definition is more general, and well suited for applications to partial differential equations. See the papers of Babuška and Guo [5], Mazya and Rossmann [54], and Verchota and Vogel [74, 75] for related definitions. We remark that, according to the above definition, Ω does not need to be bounded, nor connected, nor convex. For applications to the analysis of boundary value/interface problems, however, we will always assume Ω is connected. The boundary ∂Ω need not be connected either, but it does have finitely many connected components. We also stress that polyhedral domains will always be open subsets. The condition \overline{ω_{p}} ≠ S^{n−l−1} can be relaxed to ω_{p} ≠ S^{n−l−1}, thus allowing for cracks and slits, but not punctured domains of the form M ∖ {p}. We will not pursue this generality in the paper, given also that submanifolds of codimension greater than 2 consists of irregular boundary points for elliptic equations and may lead to ill-posedness in boundary value problems. We refer to the articles [48, 56] for a detailed analysis of polyhedral domains with cracks in 2 and 3 dimensions. We continue with some comments on Definition 2.1 before providing several concrete examples in dimension n = 1, 2, 3. We denote by tB^{l} the ball of radius t in ℝ^{l}, l ∈ ℕ, centered at the origin. We also let tB^{0} to be a point independent of t. Sometimes it is convenient to replace Condition (18) with the equivalent condition that there exist t > 0 such that

ϕ_{p}(Ω ∩ V_{p}) = {rx′, 0 < r < t, x′ ∈ ω_{p}} × tB^{l}.  (19)

We shall interchange conditions (18) and (19) at will from now on. For a cone or an infinite wedge, t = +∞, so cones and wedges are particular examples of polyhedral domain. We have the following simple result that is an immediate consequence of the definitions.

Proposition 2.2. Let ψ : M → M′ be a diffeomorphism and let Ω ⊂ M be a curvilinear polyhedral domain. Then ψ(Ω) is also a curvilinear polyhedral domain.

Next, we introduce the singular set of Ω, Ω_{sing} := Ω^{(n−2)}. A point p ∈ Ω^{(n−2)} will be called a singular point for Ω. We recall that a point x ∈ ∂Ω is called a smooth boundary point of Ω if the intersection of ∂Ω with a small neighborhood of p is a smooth manifold of dimension n − 1. In view of Definition 2.1, the point p is smooth if ϕ_{p} satisfies

ϕ_{p}(Ω ∩ V_{p}) = (0, t) × B^{n−1}.  (20)

This observation is consistent with ω_{p} being a point in this case, since it is a polyhedral domain of dimension 0. Any point p ∈ ∂Ω that is not a smooth boundary point in this sense is a singular point. But the singular set may include other points as well, in particular the points where the boundary conditions change, i.e., the points of the boundary of ∂_{D}Ω in ∂Ω, and the points where the interface Γ meets ∂Ω. It is known [36, 37] that the solution to the problem (6) near such points behaves in a similar way as in the neighborhood of non-smooth boundary points. We call the non-smooth points in ∂Ω the true or geometric singular points, while we call all the other singular points artificial singular points. The true singular points can be characterized by the condition that the domain ω_{p} of Definition 2.1 be an “irreducible” subset of the sphere S^{n−l−1}, in the sense of the following definition.

Definition 2.3. A subset ω ⊂ S^{n−1} := ∂B^{n}, the unit sphere in ℝ^{n} will be called irreducible if ℝ_{+}ω := {rx′, r > 0, x′ ∈ ω} cannot be written as V + V′ for a linear subspace V ⊂ ℝ^{k} of dimension ≥ 1 and V′ an arbitrary subset of ℝ^{n−k}. (The sum does not have to be a direct sum and, in fact, V′ is not assumed to be an affine subspace.)

For example, (0, α) ⊂ S^{1} is irreducible if, and only if, α ≠ π. A subset ω ⊂ S^{n−1} strictly contained in an open half-space is irreducible, but the intersection of S^{n−1}, n ≥ 2, with an open half-space is not irreducible. If p ∈ Ω^{(0)}, then we shall call p a vertex of Ω and we shall interpret the condition (18) as saying that ϕ_{p} defines a diffeomorphism such that

ϕ_{p}(Ω ∩ V_{p}) = {rx′, 0 < r < t, x′ ∈ ω_{p}}.  (21)

This interpretation is consistent with our convention that the set B^{0} (the zero dimensional unit ball) consists of a single point. We shall call any open, connected component of Ω^{(1)} ∖ Ω^{(0)} an (open) edge of Ω, necessarily a smooth curve in M. Similarly, any open, connected component of Ω^{(j)} ∖ Ω^{(j−1)} shall be called a (open) j–face if 2 ≤ j ≤ n − 1. A n − 1-face will be called a hyperface. A j–face H is a smooth manifold of dimension j, but in general it is not a curvilinear polyhedral domain (except if n = 2), because there might not exist a j-manifold containing the closure of H in ∂Ω. This point will be addressed in terms of the desingularization Σ(Ω) of Ω constructed in Section 4.

Notations 2.4. From now on, Ω will denote a curvilinear polyhedral domain in a manifold M of dimension n with given stratification Ω^{(0)} ⊂ Ω^{(1)} ⊂ … ⊂ Ω^{(n)} := \overline{Ω}.

Some or all of the sets Ω^{(j)}, j = 0, …, n − 2, in the stratification of Ω may be empty. In fact, Ω^{(n−2)} is empty if, and only if, \overline{Ω} is a smooth manifold, possibly with boundary, a particular case of a curvilinear, stratified polyhedron. Finally we introduce the notion of distance to the singular set Ω^{(n−2)} of Ω (if not empty) on which the constructions of the Sobolev spaces K_{a}^{μ}(Ω) given in Section 5 is based. If Ω^{(n−2)} = ∅, we let η_{n−2} ≡ 1.

Definition 2.5. Let Ω be a curvilinear, stratified polyhedral domain of dimension n. The distance function η_{n−2}(x) from x to the singular set Ω^{(n−2)} is

η_{n−2}(x) := inf_{γ} ℓ(γ),  (22)

where ℓ(γ) is the length of the curve γ, and γ ranges through all smooth curves γ : [0, 1] → \overline{Ω}, γ(0) = x, p := γ(1) ∈ Ω^{(n−2)}.

If Ω is not bounded, for example Ω is an infinite cone, then we modify the definition of the distance function as follows:

η_{n−2}(x) := χ(inf_{γ} ℓ(γ)), where

χ ∈ C^{∞}([0, +∞)), χ(s) = \begin{cases}s, & 0 ≤ s ≤ 1 \\ ≥ 1, & s ≥ 1 \\ 2, & s ≥ 3, \end{cases}

(23)

which has the effect of making η_{n−2} a bounded function.

2.1 Curvilinear polyhedral domains in 1, 2, and 3 dimensions

In this subsection we give some examples of curvilinear polyhedral domains Ω in ℝ^{2}, in S^{2}, or in ℝ^{3}. These examples are crucial in understanding Definition 2.1, which we specialize here for n = 2, n = 3. The desingularization Σ(Ω) and the function r_{Ω} will be introduced in the next subsection in these special cases. We have already defined a polyhedron in dimension 0 as a finite collection of points. Accordingly, a subset Ω ⊂ ℝ or Ω ⊂ S^{1} is a curvilinear polyhedral domain if, and only if, it is a finite union of open intervals. Let M be a smooth 2-manifold or ℝ^{2}. Definition 2.1 can be more explicitly stated as follows.

Definition 2.6. A subset Ω ⊂ M together with smooth stratification Ω^{(0)} ⊂ Ω^{(1)} ≡ ∂Ω ⊂ Ω^{(2)} ≡ Ω will be called a curvilinear, stratified polygonal domain if, for every point of the boundary p ∈ ∂Ω, there exists a neighborhood V_{p} ⊂ M of p and a diffeomorphism ϕ_{p} : V_{p} → B^{2}, ϕ_{p}(p) = 0, such that:

(a) ϕ_{p}(V_{p} ∩ Ω) = { (r cos θ, r sin θ), 0 < r < 1, θ ∈ ω_{p} }, where ω_{p} is a union of open intervals of the unit circle such that \overline{ω_{p}} ≠ S^{1};

(b) if p ∈ Ω^{(1)} ∖ Ω^{(0)}, then ω_{p} is exactly an interval of length π.

Any point p ∈ Ω^{(0)} is a vertex of Ω, and p is a true vertex precisely when ω_{p} is not an interval of length π. The open, connected components of ∂Ω ∖ Ω^{(0)} are the (open) sides of Ω. In view of condition (b) above, sides are smooth curves γ_{j} : [0, 1] → M, j = 1, …, N, with no common interior points. Recall that by hypothesis, there are finitely many vertices and sides. The condition that \overline{ω_{p}} ≠ S^{1} implies that either a side γ_{j} has a vertex in common with another side γ_{k} or γ_{j} is a closed smooth curve or an unbounded smooth curve. In the special case Ω^{(1)} ∖ Ω^{(0)} = ∅, Ω has only isolated conical points (see Example 2.11 in the next subsection), while if Ω^{(0)} = ∅, Ω has smooth boundary.

Notations 2.7. Any curvilinear, stratified polygon in ℝ^{2} will be denoted by ℙ and its stratification by ℙ^{(0)} ⊂ ℙ^{(1)} = ∂ℙ ⊂ ℙ^{(2)} = \overline{ℙ}.

Let now M be a smooth 3-manifold or ℝ^{3}. Definition 2.1 can also be stated more explicitly.

Definition 2.8. A subset Ω ⊂ M together with a smooth stratification Ω^{(0)} ⊂ Ω^{(1)} ⊂ Ω^{(2)} ≡ ∂Ω ⊂ Ω^{(3)} ≡ \overline{Ω} will be called a curvilinear, stratified polyhedral domain if, for every point of the boundary p ∈ ∂Ω, there exists a neighborhood V_{p} ⊂ M of p and a diffeomorphism ϕ_{p} : V_{p} → B^{l} × B^{3−l}, ϕ_{p}(p) = 0, such that:

(a) ϕ_{p}(V_{p} ∩ Ω) = { (y, rx′), y ∈ B^{l}, 0 < r < t, x′ ∈ ω_{p} }, where t ∈ (0, +∞] and ω_{p} ⊂ S^{2−l} is such that \overline{ω_{p}} ≠ S^{2−l};

(b) if l = 0 (i.e., if p ∈ Ω^{(0)}), then ω_{p} ⊂ S^{2} is a stratified, curvilinear polygonal domain; (c) if l = 1 (i.e., if p ∈ Ω^{(1)} ∖Ω^{(0)}), then ω_{p} is a finite, disjoint union of finitely

many open intervals in S^{1} of total length less than 2π.

(d) if l = 2 then p is a smooth boundary point;

(e) ϕ_{p} preserves the stratifications.

Each point p ∈ Ω^{(0)} is a vertex of Ω and p is a true vertex precisely when ω_{p} is an irreducible subset of S^{2} (according to Definition 2.3). The open, connected components of Ω^{(1)} ∖ Ω^{(0)} are the edges of Ω, smooth curves with no interior common points by condition (c) above. The open, connected components of Ω^{(2)} ∖ Ω^{(1)}, smooth surfaces with no common interior points, are the faces of Ω. Recall that by hypothesis, there are only finitely many vertices, edges, and faces in Ω. The condition that \overline{ω_{p}} be not the whole sphere S^{2−l} (l = 1, 0) implies that either an edge γ_{j} has a vertex in common with another edge γ_{k} or γ_{j} is a closed smooth curve or an unbounded smooth curve (such as in a wedge), and similarly for faces. Again, in the the case Ω^{(1)} = Ω^{(0)}, Ω has only isolated conical points, in the case Ω^{(0)} = ∅, Ω has only edge singularities, and in the case Ω^{(1)} = Ω^{(0)} = ∅, Ω is smooth. The following subsection contains several examples.

2.2 Definition of Σ(Ω) and of r_{Ω} if n = 2 or n = 3

We now introduce the desingularization Σ(Ω) for some of the typical examples of curvilinear polyhedral domains in n = 2 or n = 3. The desingularization of a domain Ω ⊂ M depends in general on M, but we do not explicitly show this dependence in the notation, and generally ignore it in order to streamline the presentation, given that the manifold M will be mostly implicit. Associated to the singularization is the function r_{Ω}, which is comparable with the distance to the singular set η_{n−2} but is more regular. We also frame these definitions as examples. The general case (of which the examples considered here are particular cases) is in Section 4. The reader can skip this part at first reading. The case n = 2 of a polygonal domain ℙ in ℝ^{2} is particularly simple. We use the notation in Definition 2.6. Example 2.9. The desingularization Σ(ℙ) of ℙ will replace each of the vertices A_{j}, j = 1, …, k, of ℙ with a segment of length α_{j} > 0, where α_{j} is the magnitude of the angle at A_{j} (if A_{j} is an artificial vertex, then α_{j} = π). We can realize Σ(ℙ) in three dimensions as follows. Let θ_{j} be the angle in a polar coordinates system (r_{j}, θ_{j}) centered at A_{j}. Let ϕ_{j} be a smooth function on ℙ that is equal to 1 on {r_{j} < ϵ} and vanishes outside V_{j} := {r_{j} < 2ϵ}. By choosing ϵ > 0 small enough, we can arrange that the sets V_{j} do not intersect. We define then

Φ : \overline{ℙ} ∖ \{A_{1}, A_{2}, …, A_{k}\} → ℙ × ℝ ⊂ ℝ^{3}

by Φ(p) = (p, ∑ ϕ_{j}(p)θ_{j}(p)). Then Σ(ℙ) is (up to a diffeomorphism) the closure in ℝ^{3} of Φ(ℙ). The desingularization map is κ(p, z) = p. The structural Lie algebra of vector fields V(ℙ) on Σ(ℙ) is given by (the lifts of) the smooth vector fields X on \overline{ℙ} ∖ \{A_{1}, A_{2}, …, A_{k}\} that on V_{j} = \{r_{j} < 2ϵ\} can be written as

X = a_{r}(r_{j}, θ_{j})r_{j}∂_{r_{j}} + a_{θ}(r_{j}, θ_{j})∂_{θ_{j}},  (24)

with a_{r} and a_{θ} smooth functions of (r_{j}, θ_{j}) on [0, 2ϵ) × [0, α_{j}]. We can take r_{Ω}(x) := ψ(x) ∏_{j=1}^{k} r_{j}(x), where ψ is a smooth, nowhere vanishing function on Σ(Ω). (Such a factor ψ can always be introduced, and the function r_{Ω} is determined only up to this factor. We shall omit this factor in the examples below.) The examples of a domain with a single edge or of a domain with a single vertex are among of the most instructive. Example 2.10. Let first Ω be the wedge

𝕎 := {(r cos θ, r sin θ, z), 0 < r, 0 < θ < α, z ∈ ℝ },  (25)

where 0 < α < 2π, and x = r cos θ and y = r sin θ define the usual cylindrical coordinates (r, θ, z), with (r, θ, z) ∈ [0, ∞) × [0, 2π) × ℝ. Then the manifold of generalized cylindrical coordinates is, in this case, just the domain of the cylindrical coordinates on \overline{𝕎}:

Σ(𝕎) = [0, ∞) × [0, α] × ℝ.

The desingularization map is κ(r, θ, z) = (r cos θ, r sin θ, z) and the structural Lie algebra of vector fields of Σ(𝕎) is

a_{r}(r, θ, z)r∂_{r} + a_{θ}(r, θ, z)∂_{θ} + a_{z}(r, θ, z)r∂_{z},

where a_{r}, a_{z}, and a_{θ} are smooth functions on Σ(𝕎). Note that the vector fields in V(𝕎) may not extend to the closure \overline{𝕎}. We can take r_{Ω} = r, the distance to the Oz-axis. At this stage, we can describe a domain with one conical point and its desingularization in any dimension. Example 2.11. Let next Ω be a domain with one conical point, that is, Ω is a curvilinear, stratified polyhedron in ℝ^{n} such that Ω^{(j)} = Ω^{(0)} for all 1 ≤ j ≤ n − 2. We assume Ω is bounded for simplicity. Let p ∈ Ω^{(0)} denote the single vertex of Ω. There exists a neighborhood V_{p} of p such that, up to a local change of coordinates,

V_{p} ∩ Ω = {rx′, 0 ≤ r < ϵ, x′ ∈ ω},  (26)

for some smooth, connected domain ω ⊂ S^{n−1} := ∂B^{n}. Then we can realize Σ(Ω) in ℝ^{2n} as follows. Assume p = 0, the origin, for simplicity. We define Φ(x) = (x, |x|^{−1}x) for x ≠ p, where |x| is the distance from x to the origin (i.e., to p). The set Σ(Ω) is defined to be the closure of the range of Φ. The map κ is the projection onto the first n components. The map κ is one-to-one, except that κ^{−1}(p) = {p} × ω. We can take r_{Ω}(x) = |x|. The Lie algebra of vector fields V(Ω) consists of the vector fields on Σ(Ω) that are tangent to κ^{−1}(p). This example is due to Melrose [57].

Example 2.12. Let Ω ⊂ ℝ^{3} be a convex polyhedral domain, such that all edges are straight segments. To construct Σ(Ω), we combine the ideas used in the previous examples. First, for each edge e we define (r_{e}, θ_{e}, z_{e}) to be a coordinate system aligned to that edge and such that θ_{e} ∈ (0, α_{e}), as in Example 2.10. Let v_{1}, v_{2}, …, v_{b} be the set of vertices of Ω and e_{1}, …, e_{a} be the set of edges. Then, for x not on any edge of Ω, we define Φ(x) ∈ ℝ^{3+a+b} by

Φ(x) = (x, θ_{e_{1}}, θ_{e_{2}}, …, θ_{e_{a}}, |x − v_{1}|^{−1}(x − v_{1}), …, |x − v_{b}|^{−1}(x − v_{b})) .

The desingularization Σ(Ω) ⊂ ℝ^{3+a+b} is defined as the closure of the range of Φ. The resulting set will be a manifold with corners with several different types of hyperfaces. Namely, the manifold Σ(Ω) will have a hyperface for each face of Ω, a hyperface for each edge of Ω, and, finally, a hyperface for each vertex of Ω. The last two types of hyperfaces are the so-called hyperfaces at infinity of Σ(Ω). Let x_{H} be the distance to the hyperface H. We can take then r_{Ω} = ∏_{H} x_{H}, where H ranges through the hyperfaces at infinity of Σ(Ω). We can imagine Σ(Ω) as follows. Let ϵ > 0. Remove the sets {x ∈ Ω, |x − v_{j}| ≤ ϵ} and {x ∈ Ω, |x − e_{k}| ≤ ϵ^{2}}. Call the resulting set Ω_{ϵ}. Then, for ϵ small enough, the closure of Ω_{ϵ} is diffeomorphic to Σ(Ω).

The example above can be generalized to a curvilinear, stratified polyhedron, using local change of coordinates as in Example 2.9 in 2 dimensions. A detailed construction will be given in Section 4. A nonstandard example of a curvilinear polyhedral domain is given below.

Example 2.13. We start with a connected polygonal domain ℙ with connected boundary and we deform it, within the class of connected polygonal domains, until one, and exactly one of the vertices, say A, touches the interior of another edge, say [B, C]. (It is clear that such a deformation exists since we allow each side to have arbitrary finite curvature and length.) Let Ω be the resulting connected open set. Then Ω will be a curvilinear polyhedral domain. We define the set Σ(Ω) as for the polygonal domain ℙ, but by introducing polar coordinates in the whole neighborhood of the point A. If we deform ℙ to Ω, Σ(ℙ) will deform continuously to a space Σ′(Ω), different from Σ(Ω). For certain purposes, the desingularization Σ′(Ω) is better suited than Σ(Ω).

3 Lie manifolds with boundary

The construction of the desingularization Σ(Ω) of a general, curvilinear, stratified polyhedron Ω in n dimensions will be discussed in Section 4. Σ(Ω) will be used both in the definition of weighted Sobolev spaces on the boundary and the proof of a weighted Hardy-Poincaré inequality in Subsection 6.2, which in turn is crucial in establishing coercive estimates for the mixed boundary value/interface problem (6). Since the construction of the desingularization Σ(Ω) relies on properties of manifolds with a Lie structure at infinity, we now recall the definition of a Lie manifold from [2] and of a Lie manifold with boundary from [1], in order to make this paper as self-contained as possible. We also recall a few other needed definitions and results from those papers.

3.1 Definition

We recall that a topological space 𝔐 is, by definition, a manifold with corners if every point p ∈ 𝔐 has a coordinate neighborhood diffeomorphic to [0, 1)^{k} × (−1, 1)^{n−k}, k = 0, 1, …, n, such that the transition functions are smooth (including at the boundary). Given p ∈ 𝔐, the least integer k with the above property is called the depth of p. Since the transition functions are smooth, it therefore makes sense to talk about smooth functions on 𝔐, these being the functions that correspond to smooth functions on [0, 1)^{k} ×(−1, 1)^{n−k}. We denote by C^{∞}(𝔐) the set of smooth functions on a manifold with corners 𝔐. Throughout this paper, 𝔐 will denote a manifold with corners, not necessarily compact. We shall denote by 𝔐_{0} the interior of 𝔐 and by ∂𝔐 = 𝔐 ∖ 𝔐_{0} the boundary of 𝔐. The set 𝔐_{0} consists of the set of points of depth zero of 𝔐. It is usually no loss of generality to assume that 𝔐_{0} is connected. Let 𝔐_{k} denote the set of points of 𝔐 of depth k and F_{0} be a connected component of 𝔐_{k}. We shall call F_{0} an open face of codimension k of 𝔐 and F := \overline{F}_{0} a face of codimension k of 𝔐. A face of codimension 1 will be called a hyperface of 𝔐, so that ∂𝔐 is the union of all hyperfaces of 𝔐. In general, a face of 𝔐 need not be a smooth manifold (with or without corners). A face F ⊂ 𝔐 which is a submanifold with corners of 𝔐 will be called an embedded face. Anticipating, a Lie manifold with boundary 𝔐_{0} is the interior of a manifold with corners 𝔐 together with a Lie algebra of vector fields V on 𝔐 satisfying certain conditions. To state these conditions, it will be convenient first to introduce a few other concepts.

Definition 3.1. Let 𝔐 be a manifold with corners and V be a space of vector fields on 𝔐. Let U ⊂ 𝔐 be an open set and Y_{1}, Y_{2}, …, Y_{k} be vector fields on U ∩ 𝔐_{0}. We shall say that Y_{1}, Y_{2}, …, Y_{k} form a local basis of V on U if the following three conditions are satisfied:

(i) there exist vector fields X_{1}, X_{2}, …, X_{k} ∈ V, Y_{j} = X_{j} on U ∩ 𝔐_{0};

(ii) V is closed under products with smooth functions in C^{∞}(𝔐) (i.e., V = C^{∞}(𝔐)V) and for any X ∈ V, there exist smooth functions ϕ_{1}, ϕ_{2}, …, ϕ_{k} ∈ C^{∞}(𝔐_{0}) such that

X = ϕ_{1}X_{1} + ϕ_{2}X_{2} + … + ϕ_{k}X_{k} on U ∩ 𝔐_{0} ;  (27)

and

(iii) if ϕ_{1}, ϕ_{2}, …, ϕ_{k} ∈ C^{∞}(𝔐) and ϕ_{1}X_{1} +ϕ_{2}X_{2} +…+ϕ_{k}X_{k} = 0 on U ∩𝔐_{0}, then ϕ_{1} = ϕ_{2} = … = ϕ_{k} = 0 on U.

We now recall structural Lie algebras of vector fields from [2].

Definition 3.2. A subspace V ⊆ Γ(𝔐, T 𝔐) of the Lie algebra of all smooth vector fields on 𝔐 is said to be a structural Lie algebra of vector fields on 𝔐 provided that the following conditions are satisfied:

(i) V is closed under the Lie bracket of vector fields;

(ii) every vector field X ∈ V is tangent to all hyperfaces of 𝔐;

(iii) C^{∞}(𝔐)V = V; and

(iv) for each point p ∈ 𝔐 there exist a neighborhood U_{p} of p in 𝔐 and a local basis of V on U_{p}.

The concept of Lie structure at infinity, defined next, is also taken from [2].

Definition 3.3. A Lie structure at infinity on a smooth manifold 𝔐_{0} is a pair (𝔐, V), where 𝔐 is a compact manifold, possibly with corners, and V ⊂ Γ(𝔐, T 𝔐) is a structural Lie algebra of vector fields on 𝔐 with the following properties:

(i) 𝔐_{0} = 𝔐 ∖ ∂𝔐, the interior of 𝔐, and

(ii) If p ∈ 𝔐_{0}, then any local basis of V in a neighborhood of p is also a local basis of the tangent space to 𝔐_{0}. (In particular, the constant k of Equation (27) equals n, the dimension of 𝔐_{0}.)

A manifold with a Lie structure at infinity (or, simply, a Lie manifold) is a manifold 𝔐_{0} together with a Lie structure at infinity (𝔐, V) on 𝔐_{0}. We shall sometimes denote a Lie manifold as above by (𝔐_{0}, 𝔐, V), or, simply, by (𝔐, V), because 𝔐_{0} is determined as the interior of 𝔐. Let V_{b} be the set of vector fields on 𝔐 that are tangent to all faces of 𝔐. Then (𝔐, V_{b}) is a Lie manifold [57]. See [1, 2, 47] for more examples.

3.2 Riemannian metric

Let (𝔐, V) be a Lie manifold and g a Riemannian metric on 𝔐_{0} := 𝔐 ∖ ∂𝔐. We shall say that g is compatible (with the Lie structure at infinity (𝔐, V)) if, for any p ∈ 𝔐, there exist a neighborhood U_{p} of p in 𝔐 and a local basis X_{1}, X_{2}, …, X_{n} of V on U_{p} that is orthonormal with respect to g on U_{p}. It was proved in [2] that (𝔐_{0}, g_{0}) is necessarily of infinite volume and complete. Moreover, all the covariant derivatives of the Riemannian curvature tensor of g are bounded. We also know that the injectivity radius is bounded from below by a positive constant, i.e., (𝔐_{0}, g_{0}) is of bounded geometry [18]. (A manifold with bounded geometry is a Riemannian manifold with positive injectivity radius and with bounded covariant derivatives of the curvature tensor, see for example [16] or [69] and references therein).

3.3 V-differential operators

We are especially interested in the analysis of the differential operators generated using only derivatives in V. Let Diff_{V}^{∗}(𝔐) be the algebra of differential operators on 𝔐 generated by multiplication with functions in C^{∞}(𝔐) and by differentiation with vector fields X ∈ V. The space of order m differential operators in Diff_{V}^{∗}(𝔐) will be denoted Diff_{V}^{m}(𝔐). A differential operator in Diff_{V}^{∗}(𝔐) will be called a V-differential operator. We define the set Diff_{V}^{∗}(𝔐; E, F) of V-differential operators acting between sections of smooth vector bundles E, F → 𝔐 in the usual way [1, 2]. A simple but useful property of the differential operator in Diff_{V}^{∗}(𝔐) is that

x^{s}P x^{−s} ∈ Diff_{V}^{∗}(𝔐)  (28)

for any P ∈ Diff_{V}^{∗}(𝔐) and any defining function x of some hyperface of 𝔐 [3]. This property is easily proved using the fact that X is tangent to the hyperface defined by x, for any X ∈ V (a proof of a slightly more general fact is included in Corollary 6.3).

3.4 Lie manifolds with boundary

A subset 𝔑 ⊂ 𝔐 is called a submanifold with corners of 𝔐 if 𝔑 is a closed submanifold of 𝔐 such that 𝔑 is transverse to all faces of 𝔐 and any face of 𝔑 is a component of 𝔑 ∩ F for some face F of 𝔐. The following definition is a reformulation of a definition of [1].

Definition 3.4. Let (𝔑, W) and (𝔐, V) be Lie manifolds, where 𝔑 ⊂ 𝔐 is a submanifold with corners and

W = {X|_{𝔑}, X ∈ V, X|_{𝔑} tangent to 𝔑}.

We shall say that (𝔑, W) is a tame submanifold of (𝔐, V) if, for any p ∈ ∂𝔑 and any X ∈ T_{p}𝔐, there exist Y ∈ V and Z ∈ T_{p}𝔑 such that X = Y (p) + Z.

Let 𝔑 ⊂ 𝔐 be a submanifold with corners. We assume that 𝔐 and 𝔑 are endowed with the Lie structures (𝔑, W) and (𝔐, V). We shall say that 𝔑 is a regular submanifold of (𝔐, V) if we can choose a tubular neighborhood V of 𝔑_{0} := 𝔑 ∖ ∂𝔑 = 𝔑 ∩ 𝔐_{0} in 𝔐_{0}, a compatible metric g_{1} on 𝔑_{0}, a product-type metric g_{1} on V that reduces to g_{1} on 𝔑_{0}, and a compatible metric on 𝔐_{0} that restricts to g_{1} on V. Theorem 5.8 of [2] states that every tame submanifold is regular. The point of this result is that it is much easier to check that a submanifold is tame than to check that it is regular. In the case when 𝔑 is of codimension one in 𝔐, the condition that 𝔑 be tame is equivalent to the fact that there exists a vector field X ∈ V that restricts to a normal vector of 𝔑 in 𝔐. The neighborhood V will then be of the form V ≃ (∂𝔑_{0}) × (−ε_{0}, ε_{0}). Moreover, there will exist a compatible metric on 𝔐_{0} that restricts to the product metric g_{1} + dt^{2} on V, where g_{1} is a compatible metric on 𝔑_{0}.

Let A be a subset of 𝔐. We denote by ∂_{𝔐}A := \overline{A} ∖ A the boundary of A computed within 𝔐, which should not be confused with ∂A = A ∖ A_{0}, where A_{0} is the interior of A, as a manifold with corners. Let 𝔻 ⊂ 𝔐 be an open subset. We say that 𝔻 is a Lie domain in 𝔐 if, and only if,

∂_{𝔐}𝔻 = ∂_{𝔐}\overline{𝔻}  (29)

and ∂_{𝔐}𝔻 is a regular submanifold of 𝔐. The condition (29) is included in order to make sure that 𝔻 is on only one side of its boundary. A typical example of a Lie domain 𝔻 ⊂ 𝔐 is obtained by considering a regular submanifold with corners 𝔑 ⊂ 𝔐 of codimension one with the property that 𝔐 ∖ 𝔑 consists of two connected components. Any of these two components will be a Lie domain.

Definition 3.5. A Lie manifold with boundary is a triple (𝔒_{0}, 𝔒, V′), where 𝔒_{0} is a smooth manifold with boundary, 𝔒 is a compact manifold with corners containing 𝔒_{0} as an open subset, and V′ is a Lie algebra of vector fields on 𝔒 with the property that there exists a Lie manifold (𝔐_{0}, 𝔐, V), a Lie domain 𝔻 in 𝔐 and a diffeomorphism ϕ : 𝔒 → \overline{𝔻} such that ϕ(𝔒_{0}) = \overline{𝔻} ∩ 𝔐_{0} and Dϕ(V|_{𝔻}) = V′.

We continue with some simple observations. First note that if (𝔒_{0}, 𝔒, V) is a Lie manifold with boundary, then 𝔒_{0} is determined by (𝔒, V). Indeed, if we remove from 𝔒 the hyperfaces H with the property that V consists only of vectors tangent to H, then the resulting set is 𝔒_{0}. Therefore, we can denote the Lie manifold with boundary (𝔒_{0}, 𝔒, V) simply by (𝔒, V). Another observation is that ∂𝔒_{0}, the boundary of 𝔒_{0} (as a smooth manifold with boundary), has a canonical structure of Lie manifold (∂𝔒_{0}, D = ∂_{𝔐}𝔻, W), where W = {X|_{D}, X ∈ V, X|_{D} is tangent to D}. The compactification D is the closure of ∂𝔒_{0} in 𝔒.

3.5 Sobolev spaces

The main reason for considering Lie manifolds (with or without boundary) in our setting is that they carry some naturally defined Sobolev spaces and these Sobolev spaces behave almost exactly like the Sobolev spaces on a compact manifold with a smooth boundary. Let us recall one of the equivalent definitions in [1]. See also [16, 32, 57, 64, 70] for results on Sobolev spaces on non-compact manifolds.

Definition 3.6. Fix a Lie manifold (𝔐, V). The spaces L^{2}(𝔐_{0}) = L^{2}(𝔐_{0}) are defined using the natural volume form on 𝔐_{0} given by an arbitrary compatible metric g on 𝔐_{0} (i.e., compatible with the Lie structure at infinity). All such volume forms are known to define the same space L^{2}(𝔐), but with possibly different norms. Let k ∈ ℤ_{+}. Choose a finite set of vector fields X ⊂ V such that C^{∞}(𝔐)X = V. The system X gives rise to the norm

∥u∥_{X ,Ω}^{2} := ∑ ∥X_{1}X_{2} … X_{μ}u∥_{L^{2}(Ω)}^{2} , 1 ≤ p < ∞,  (30)

the sum being over all possible choices of 0 ≤ l ≤ k and all possible choices of vector fields X_{1}, X_{2}, …, X_{l} ∈ X, not necessarily distinct. We then set

H^{k}(𝔐_{0}) = H^{k}(𝔐) := {u ∈ L^{2}(𝔐), ∥u∥_{X ,𝔐} < ∞}.

The spaces H^{s}(𝔐_{0}) = H^{s}(𝔐) are defined by duality (with pivot L^{2}(𝔐_{0})) when −s ∈ ℤ_{+}, and then by interpolation, as above. Let (𝔒_{0}, 𝔒, V) be a Lie manifold with boundary. We shall assume that 𝔒 is the closure of a Lie domain 𝔻 of the Lie manifold 𝔐. The Sobolev spaces H^{k}(𝔒_{0}) are defined as the set of restrictions to 𝔒_{0} of distributions u ∈ H^{k}(𝔐_{0}), using the notation of Definition 3.5, k ∈ ℤ. In particular, we obtain the following description of H^{k}(𝔒_{0}) from [1].

Lemma 3.7. We have, for k ≥ 0,

H^{k}(𝔒_{0}) = {u ∈ L^{2}(𝔒_{0}), ∥u∥_{X} < ∞},

and

H^{−k}(𝔒_{0}) = H_{0}^{k}(𝔒_{0})^{∗},

where H_{0}^{k}(𝔒_{0}) is the closure of C_{c}^{∞}(𝔒_{0}) in H^{k}(𝔐_{0}).

Definition 3.8. The hyperfaces of 𝔒 that do not intersect the boundary ∂𝔒_{0} of the manifold with boundary 𝔒_{0} will be called hyperfaces at infinity. Let x_{H} be a defining function of the hyperface H of 𝔒. Any function of the form h = ∏ x_{H}^{a_{H}}, where H ranges through the set of hyperfaces at infinity of 𝔒 and a_{H} ∈ ℝ, will be called an admissible weight. If h is an admissible weight, we set

hH^{μ}(𝔒_{0}) = {h u; u ∈ H^{μ}(𝔒_{0})}

with the induced norm.

Later in the paper, we will identify the weighted Sobolev spaces K_{a}^{s}(Ω) with suitable spaces hH^{s}(𝔒_{0}) in Proposition 5.7 and utilize the spaces hH^{s}(∂𝔒_{0}) to define the spaces K_{a}^{s}(∂Ω) on the boundary in Definition 5.8, for Ω a curvilinear, stratified polyhedral domain in dimension n. The following proposition, which summarizes the relevant results from Theorem 3.4 and 3.7 from [1], will then imply Theorem 5.9.

Proposition 3.9. The restriction to the boundary extends to a continuous, surjective map hH^{μ}(𝔒_{0}) → hH^{μ−1/2}(∂𝔒_{0}), for any μ ≥ 1 and any admissible weight h. The kernel of this map, for μ = 1, consists of the closure of C_{c}^{∞}(𝔒_{0}) in hH^{1}(𝔒_{0}).

For 𝔻, 𝔒, 𝔒_{0} as in the proposition above, hH^{s}(𝔻), hH^{s}(𝔒), and hH^{s}(𝔒_{0}) will all denote the same space.

4 Desingularization of polyhedra

In this section, we introduce a desingularization procedure that we shall use for studying curvilinear polyhedral domains. The desingularization will carry a natural structure of Lie manifold with boundary. This construction will allow us to study curvilinear polyhedral domains using Lie manifolds with boundary. The desingularization of a domain Ω ⊂ M depends on M in general (since \overline{Ω} depends on M), but we do not include the dependence on M in the notation, and generally ignore this issue in order to streamline the presentation, since the manifold M will be clear from the context in most cases. As before, Ω ⊂ M denotes a curvilinear, stratified polyhedral domain in an n-dimensional manifold M. We shall construct by induction on n a canonical manifold with corners Σ(Ω) and a differentiable map κ : Σ(Ω) → \overline{Ω} that is a diffeomorphism from the interior of Σ(Ω) to Ω. In particular, the map κ allows us to identify Ω with a subset of Σ(Ω). We shall also construct a canonical Lie algebra of vector fields V(Ω) on Σ(Ω). The manifold Σ(Ω) will be called the desingularization of Ω, the map κ will be called the desingularization map, and the Lie algebra of vector fields will be called the structural Lie algebra of vector fields of Σ(Ω). We shall also introduce in this section a smooth weight function r_{Ω} equivalent to η_{n−2}. The space Σ(Ω) that we construct is not optimal if the links ω_{p} are not connected. A better desingularization would be obtained if one considers a diffeomorphism ϕ_{pC} for each connected component C of V_{p} ∩ Ω that maps C to a conic set of the form ω_{p,C} ×B^{λ}, with λ largest possible. The difference between these two constructions is seen by looking at the Example 2.13.

Notations 4.1. From now on V_{p} and ϕ_{p} : V_{p} → tB^{n−l} × tB^{l}, l = ℓ(p), will denote a neighborhood of p ∈ ∂Ω in M ⊃ Ω and ϕ_{p} will be a diffeomorphism satisfying the conditions of Definition (2.1). In addition, ω_{p} ⊂ S^{n−l−1} will be the curvilinear, stratified polyhedron such that

ϕ_{p}(V_{p} ∩ Ω) = {(rx′, x′′)}, r ∈ (0, t), x′ ∈ ω_{p} and x′′ ∈ tB^{l}},

i.e., ω_{p} is the link of Ω at p. This notation will remain fixed throughout the paper.

Recall that 0 ≤ ℓ(p) ≤ n − 1 is defined to be the smallest integer such that p ∈ Ω^{(ℓ(p))}, but p ∉ Ω^{(ℓ(p)−1)}. If ℓ(p) = 0, then B^{l} is reduced to a point, and we just drop x′′ from the notation above. We will assume that ϕ_{p} extends to the closure of V_{p}, if necessary.

4.1 The desingularization Σ(Ω)

We now define the canonical desingularization of a curvilinear polyhedral domain Ω ⊂ M, M an n-dimensional smooth manifold. For n = 0, Ω consists of finitely many points. Then we define Σ(Ω) = Ω and κ = id. To define Σ(Ω) for general Ω, we shall proceed by induction.

We need first to make the important observation that the set ω_{p}, p ∈ ∂Ω, of Definition 2.1 is determined up to a linear isomorphism of ℝ^{n−l−1}. Indeed, let S_{p} ⊂ ∂Ω be the maximal connected manifold of dimension l = ℓ(p) passing through p that is, the connected component of Ω^{(l)} ∖ Ω^{(l−1)} containing p. Let (T_{p}S_{p})^{⊥} = T_{p}M/T_{p}S_{p}. The differential Dϕ_{p} : T_{p}M → ℝ^{n} = T_{0}ℝ^{n} of the map ϕ_{p} at p has then the property that Dϕ_{p}(T_{p}S_{p}) = T_{0}ℝ^{l}, Dϕ_{p}((T_{p}S_{p})^{⊥}) = T_{0}ℝ^{n}/T_{0}ℝ^{l} = ℝ^{n−l}. We will define a canonical set N_{p} ⊂ (T_{p}S_{p})^{⊥} such that

Dϕ_{p}(N_{p}) ≃ ℝ_{+}ω_{p}.

Since the definition of N_{p}, which we give next, is independent of any choices used in the definition of a polyhedral domain, it follows that ω_{p} is unique, up to a linear isomorphism of ℝ^{n−l−1}. It remains to define the set N_{p} with the desired independence property. It is enough to define the complement of N_{p}. This complement is the projection onto (T_{p}S_{p})^{⊥} = T_{p}M/T_{p}S_{p} of the set γ′(0) ∈ T_{p}M, where γ ranges through the set of smooth curves γ : [0, 1] → M, with γ(t) ∉ Ω for t > 0, and γ(0) = p. We let then σ_{p} := N_{p}/ℝ_{+}, the set of rays in N_{p}, for p ∈ ∂Ω. Any choice of a metric on T_{p}M/T_{p}S_{p} ⊃ N_{p} will identify σ_{p} with a subset of the unit sphere of T_{p}M/T_{p}S_{p}, which depends however on the metric. In particular, Dϕ_{p} : σ_{p} → ω_{p} is a diffeomorphism. If p is not in the singular set Ω^{(n−2)} of Ω, then σ_{p} consists exactly of one point. The map κ is the projection onto the second component and is one-to-one above Ω and above Ω^{(n−1)} ∖Ω^{(n−2)} ⊂ ∂Ω. We now proceed with the induction step. Assume Σ(ω) and κ : Σ(ω) → \overline{ω} have been constructed for all curvilinear, stratified polyhedral domains ω of dimension at most n − 1. If p ∈ Ω, we then set σ_{p} = {0} = Σ(σ_{p}). Let Ω be an arbitrary curvilinear, stratified polyhedral domain of dimension n. We define

Σ(Ω) := ⋃_{p∈Ω}{p} × Σ(σ_{p}) = Ω ∪ ⋃_{p∈∂Ω}{p} × Σ(σ_{p}).  (31)

In particular, if Ω is a bounded domain with smooth boundary, then Σ(Ω) ≃ Ω. This definition is consistent as ω_{p} is a curvilinear polyhedral domain of dimension at most n−1. Below, an open embedding will mean a diffeomorphism onto an open subset of the codomain.

Proposition 4.2. Let Ω ⊂ M and Ω′ ⊂ M′ be curvilinear, stratified polyhedral domains and Φ : M → M′ be an open embedding such that Φ(Ω) is a union of connected components of Ω′ ∩Φ(M). Then the embedding Φ defines a canonical map Σ(Φ) : Σ(Ω) → Σ(Ω′) such that

Σ(Φ ∘ Φ′) = Σ(Φ) ∘ Σ(Φ′),

for all open embeddings Φ and Φ′ for which Σ(Φ ∘ Φ′), Σ(Φ) ∘ Σ(Φ′) are welldefined.

Proof. The proof is by induction. There is nothing to prove for n = 0. Let p ∈ \overline{Ω}. We have that Φ(\overline{Ω}) ⊂ \overline{Ω′}, and hence Φ(p) ∈ \overline{Ω′}, as well. Let V′_{p} be an open neighborhood of Φ(p) in M′ such that there exists a diffeomorphism ϕ′_{p} : V′_{p} → B^{n−l} × B^{l} satisfying the condition (18) of the definition of a polyhedral domain (i.e., ϕ′_{p}(Ω′ ∩V′_{p}) is ℝ_{+}ω′_{p} ×B^{l}, for some curvilinear polyhedral domain ω′_{p} ⊂ S^{n−l−1}). By decreasing V′_{p}, if necessary, we can assume that V′_{p} ⊂ Φ(M). Then V′_{p} ∩ Φ(Ω) is a union of connected components of V′_{p} ∩ Ω′. Therefore ω′_{p} is a union of connected components of Φ(ω_{p}), where ω_{p} ⊂ S^{n−l−1} is associated to p ∈ \overline{Ω} in the same way as ω′_{p} was associated to Φ(p) ∈ \overline{Ω′}. The induction hypothesis then gives rise to a canonical, injective map Σ(ω_{p}) → Σ(ω′_{p}). The map Σ(Φ) is obtained by combining these different maps. The functoriality (i.e., the relation Σ(Φ∘Φ′) = Σ(Φ)∘Σ(Φ′)) is proved similarly by induction.

Here is a corollary of the above proof.

Corollary 4.3. If Ω = Ω′∪Ω′′ is the disjoint union of two open sets, then the inclusions Σ(Ω′) ⊂ Σ(Ω) and Σ(Ω′) ⊂ Σ(Ω) defined in Proposition 4.2 realize Σ(Ω) = Σ(Ω′) ∪ Σ(Ω′′), where the union is a disjoint union.

Proof. We use the same argument as in the proof of Proposition 4.2.

The desingularization has a simple behavior with respect to products.

Lemma 4.4. We have a canonical identification

Σ(M′ × Ω) = M′ × Σ(Ω),

for any smooth manifolds M and M′ and any curvilinear polyhedral domain Ω ⊂ M.

Proof. Since M′ is smooth, we can choose the structural local diffeomorphism ϕ_{(p,q)} in M′ × Ω to be given by ϕ_{p} × ψ_{q}, where ψ_{p} is a local coordinate chart defined in a neighborhood of p ∈ M′ and ϕ_{q} is the local diffeomorphism of a neighborhood of q in Ω. Indeed, then

Σ(M′×Ω) := ∪_{p,q}{(p, q)}×Σ(σ_{(p,q)}) = ∪_{p,q}{(p, q)}×Σ(σ_{q}) = M′×Σ(Ω), (32)

where q ∈ \overline{Ω} and p ∈ M′. Consequently, there is a canonical bijection σ_{(p,q)} ≃ σ_{q} for any q ∈ \overline{Ω} and any p ∈ M′ (so (p, q) is in the closure of M′ × Ω in M′ × M).

It remains to define the topology and differentiable structure on Σ(Ω). These definitions will again be canonical if we require that the map of the above lemma, as well as the maps κ and Σ(ϕ), be differentiable, for any open embedding ϕ. Let V_{p} ⊂ M and ϕ_{p} be as in Equation (19). By Proposition 4.2, we may assume that ϕ_{p} is the identity, so that p = 0, V_{p} = B^{n−l} × B^{l}, and V_{p} ∩ Ω = Iω_{p} × B^{l}, with I = (0, 1). Let (Σ(ω_{p}), κ′_{p}) be the canonical desingularization of ω_{p} in S^{n−l−1}. We shall need the following lemma.

Lemma 4.5. We have a canonical identification

Σ(V_{p} ∩ Ω) = [0, 1) × Σ(ω_{p}) × B^{l}

such that the desingularization map

κ_{p} : [0, 1) × Σ(ω_{p}) × B^{l} → \overline{V_{p} ∩ Ω} ⊂ B^{n−l} × B^{l}

is given by κ_{p}(r, x′, y) = (rκ′_{p}(x′), y).

Proof. We may assume p = 0. Let I = (0, 1). The closure of V_{0} ∩ Ω in V_{0} = V_{p} is the disjoint union \{0\} × B^{l} ∪ I\overline{ω}_{0} × B^{l}. Accordingly, we decompose Σ(V_{0}∩Ω, M) into two disjoint sets, corresponding to this splitting of the closure of V_{0}∩Ω. Recall that by definition Σ(V_{0}∩Ω) is the union ∪_{p∈\overline{V_{0}∩Ω}} \{p\}×Σ(σ_{p}). Using also Lemma 4.4, we then obtain

Σ(V_{0} ∩ Ω) = Σ(V_{0} ∩ Ω) ∪ ⋃_{q∈B^{l}}{(0, q)} × Σ(ω_{0})

= Σ((0, 1) × ω_{0} × B^{l}) ∪ ⋃_{q∈B^{l}}{(0, q)} × Σ(ω_{0})

= (0, 1) × Σ(ω_{0}) × B^{l} ∪ {0} × Σ(ω_{0}) × B^{l} = [0, 1) × Σ(ω_{0}) × B^{l}.

The formula for κ_{0} follows from the definition.

Since Σ(Ω) is the union of all the sets Σ(V_{p} ∩ Ω), with V_{p} in the covering above, we can define the topology and smooth structure on Σ(Ω) by induction as follows (there is nothing to define in the case Ω has dimension zero, since then Σ(Ω) = Ω).

Definition 4.6. Let ϕ_{p} : V_{p} → B^{n−l} × B^{l} and ω_{p} be as in Definition 2.1. The topology and smooth structure on Σ(Ω) are such that the induced structure on Σ(V_{p} ∩ Ω) is the same as the one obtained from the canonical identification Σ(V_{p} ∩ Ω) = [0, 1) × Σ(ω_{p}) × B^{l} of Lemma 4.5.

The smooth structure on Σ(Ω) is therefore defined using a covering with sets of the form Σ(V_{p} ∩ Ω) (this desingularization is with respect to V_{p} and not M ⊂ Ω!). We need to prove that the transition functions are smooth. This property follows from the fact that the maps ϕ_{p} are diffeomorphisms and from Lemma 4.5. We have therefore completed the definition of the desingularization Σ(Ω) and of its smooth structure.

4.2 The distance to singular boundary points

We continue with a study of the geometric and, especially, metric properties of Σ(Ω). We first argue that Σ(Ω) has embedded faces and hence that every hyperface of Σ(Ω) has a defining function.

Let F_{0} be an open hyperface of a manifold with corners 𝔐. Then F_{0} is a manifold of dimension n − 1. Its closure F, in general, will not necessarily be a manifold, because it may have self-intersections. (A typical example is the boundary of a curvilinear polygonal domain with only one vertex, the “tear drop domain.”) By induction, however, it follows that F_{0} ∩ V_{p} will be a manifold with corners, for any p. In particular, we obtain that all (closed) faces of Σ(Ω) are embedded submanifolds of Σ(Ω). Let H be a hyperface of Σ(Ω), since H is an embedded submanifold of codimension 1, there will exist a function x_{H} > 0 on Ω, H = {x_{H} = 0}, and dx_{H} ≠ 0 on H. A function x_{H} with this property is called a defining function of H [57]. One of the main reasons for introducing the desingularization space Σ(Ω) is the following result.

Proposition 4.7. Let Ω be a bounded, curvilinear, stratified polyhedral domain and g_{1} and g_{2} be two smooth Riemannian metrics on M. Let us fix k and assume Ω^{(k)} ≠ ∅. Let f_{j}(x) be the modified distance from x ∈ \overline{Ω} to the set Ω^{(k)} in the metric g_{j}, computed within \overline{Ω}. Then the quotient f_{2}/f_{1} extends to a continuous function on Σ(Ω).

Proof. It is enough to prove the given property in the neighborhood of every point p ∈ \overline{Ω}. So let us fix p ∈ \overline{Ω}. By replacing V_{p} with a smaller neighborhood of p, if necessary, we can also assume that g_{2}(ξ) ≤ Cg_{1}(ξ), which implies that f_{2} ≤ Cf_{1}, and hence that f_{2}/f_{1} is bounded. We shall prove the statement by induction on n. In the case n = 1, the only possibility is that k = 0, or otherwise Ω^{(k)} = ∅. Then f(x) is the distance to the vertices of Ω. Recall that Ω is a disjoint union of open intervals in this case, so that we can reduce to consider a single interval. If say Ω = [a, b], then close to a, f_{j}(x) = a_{j}(x)(x − a), with a_{j} smooth near a and a_{j}(a) ≠ 0. The same situation holds at b. This proves our result in the case n = 1. We now proceed with the induction step. The function f_{1}/f_{2} is clearly continuous on the open set Ω. Fix p ∈ ∂Ω. We shall construct an open neighborhood U_{p} of p in \overline{Ω} such that f_{1}/f_{2} extends to a continuous function on κ^{−1}(U_{p}). Let V_{p} be as in the definition of polyhedral domains (Definition 2.1). We shall identify V_{p} ∩ Ω with Iω_{p} × B^{l} using the diffeomorphism ϕ_{p} of Equation (19). If l > k, that is, p ∈ Ω^{(l)} ∖ Ω^{(k)}, then both f_{1} and f_{2} extend to continuous, non-vanishing functions on V_{p} ∩ \overline{Ω}, which can be lifted to continuous, non-vanishing functions on κ^{−1}(V_{p} ∩ Ω). We shall assume hence that k ≥ l. On a smaller neighborhood V′ ⊂ V_{p}, if necessary, we can arrange that the function f_{1} gives the distance to V_{p}^{(k)}, that is, that the point of Ω^{(k)} closest to x ∈ V′ ∩ Ω is, in fact, in V_{p}. By decreasing V′ even further, we can further arrange that the same holds for f_{2}. Then we shall take U_{p} := V′. To prove that f_{2}/f_{1} extends to a continuous function on κ^{−1}(U_{p}), it is enough to do that in the case Ω = V_{p} ∩ Ω, because the quotient f_{2}/f_{1} does not change on U_{p} ∩Ω if we replace Ω with V_{p} ∩Ω, as explained in the paragraph above. We can also assume that g_{2} is the standard Euclidean metric, but then we have to prove that f_{1}/f_{2} extends to a nowhere vanishing continuous function on Σ(Ω). (Using also Proposition 4.2, we have reduced to the case Ω = Iω_{p} × B^{l} ⊂ ℝ^{n}, I = (0, t).) The scaling property of the Euclidean metric and our assumption that k ≥ l imply that

f_{2}(rx′, x′′) = rf_{2}(x′, x′′),

for any r ∈ [0, 1]. Let g_{0} be a constant metric on ℝ^{n} that coincides with g_{1} at the origin. Let f_{0} be associated to g_{0} in the same way as f_{j} is associated to g_{j}, for j = 1, 2, i.e., f_{0}(x) = dist(x, Ω^{(k)}) using the metric g_{0}. We then have similarly f_{0}(rx′, x′′) = rf_{0}(x′, x′′), so that the quotient f_{0}(rx′, x′′)/f_{1}(rx′, x′′) does not depend on r. We can therefore fix r = 1. Consequently, we can work with the lower dimensional polyhedral domain ω := ω_{p} × B^{l} instead of Ω = Iω_{p} × B^{l}, and prove that f_{0}/f_{1} extends by continuity to Σ(ω). It remains to see that we can use induction to prove the existence of this extension. Since ω is by construction a stratified polyhedron, we denote by ω^{(k)} = ω^{(k)} × B^{l} k < n, its associated stratification, where we set ω_{p}^{(k−l−1)} = ∅ if k − l − 1 < 0 as before. Let f′_{1} be the distance function to ω^{(k−1)} on ω (i.e., computed within \overline{ω}, with respect to the metric induced by g_{1}, as in the statement of Proposition 4.7). We let f′_{1} = 1 if ω_{p}^{(k−l−1)} = ∅. We define f′_{0} similarly. The inductive hypothesis guarantees that f′_{0}/f′_{1} extends to a continuous function on Σ(ω) = Σ(ω_{p}) × B^{l}. On the other hand, it is easy to see that both f_{1}/f′_{1} and f′_{1}/f_{1} extend to continuous functions on \overline{ω} if we set them to be equal to 1 on ω^{(k−1)}. The same is true of f_{0}/f′_{0} and f′_{0}/f_{0}. Putting all these estimates together, it follows that

f_{0}/f_{1} = (f_{0}/f′_{0})(f′_{0}/f′_{1})(f′_{1}/f_{1})

extends to a continuous, nowhere vanishing function on Σ(ω). Let us tackle now the case g_{2} arbitrary. Let f_{0} be defined as before. We then have that f_{2}(rx′, x′′) = rf_{0}(x′, x′′) + r^{2}h(rx′, x′′), with h a continuous function on Σ(V_{p} × Ω) that vanishes on Ω^{(k)}. Then

\frac{f_{2}}{f_{1}} = \frac{f_{0}}{f_{1}} + r \frac{h(rx′, x′′)}{f_{1}(x′, x′′)}.

The function f_{0}/f_{1} was already shown to extend by continuity to Σ(Ω). The same argument as above shows that h/f_{1} extends by continuity to a nowhere vanishing function on

[ϵ, 1) × Σ(ω_{p}) × B^{l} ⊂ [0, 1) × Σ(ω_{p}) × B^{l} =: Σ(Ω).

The continuity of f_{2}/f_{1} then follows from the boundedness of f_{2}/f_{1}. The resulting function does not vanish at r = 0, because it is equal to f_{0}/f_{1} there. It was already proved that it does not vanish for ϵ > 0. The proof is complete.

We shall need also the following corollary of the above proof.

Corollary 4.8. Identify V_{p} ∩Ω = Ω with Iω_{p} ×B^{l}, I = (0, a), l = ℓ(p), using the diffeomorphism ϕ_{p} given in Definition 2.1. Let g be a smooth metric on V_{p}, and let f(x) be the distance from x to Ω^{(k)}, k ≥ l, f′(x′, x′′) be the distance from (x′, x′′) ∈ ω := ω_{p} × B^{l} to ω^{(k−1)} (within \overline{ω}, as in Proposition 4.7) if ω^{(k−1)} ≠ ∅, and f′(x′, x′′) = 1 otherwise. Assume ω_{p} is connected. Then

f(rx′, x′′)/rf′(x′, x′′)

extends to a continuous, nowhere vanishing function on Σ(Ω) = [0, a)×Σ(ω_{p})× B^{l}.

Proof. Assume first that ω^{(k−1)} ≠ ∅, where ω^{(k)} is defined as in Proposition 4.7. Let f_{0} and f′_{0} be defined in the same way f and f′ were defined, but replacing g with a constant metric g_{0}. Then the proof of Proposition 4.7 gives that f_{0}(rx′, x′′)/rf′_{0}(x′, x′′) is independent of r. Hence f_{0}(rx′, x′′)/rf′_{0}(x′, x′′) extends to a continuous, nowhere vanishing function on Σ(Ω), as it was shown in the proof of Proposition 4.7. Then

\frac{f(rx′, x′′)}{rf′(x′, x′′)} = \frac{f(rx′, x′′)}{f_{0}(rx′, x′′)} × \frac{f_{0}(rx′, x′′)}{rf′_{0}(rx′, x′′)} × \frac{f′_{0}(x′, x′′)}{f′(x′, x′′)} .

We have just argued that the middle quotient in the above product extends to a continuous function on Σ(Ω). The other two quotients also extend to continuous functions on Σ(Ω), by Proposition 4.7 applied to Ω and ω. Let us assume now that ω^{(k−1)} = ∅. Then the same proof applies, given that f′_{0}/f′ = 1 clearly extends to a continuous function on Σ(Ω).

4.3 The weight function r_{Ω}

Recall that η_{n−2}(x), given in Definition 2.5, denotes the distance from x ∈ \overline{Ω} to the singular set Ω^{(n−2)}. The main goal of this subsection is to define on any curvilinear polyhedral domain Ω a function

r_{Ω} : \overline{Ω} → [0, ∞)

that lifts to a smooth function on Σ(Ω) and is equivalent to η_{n−2}. (Additional properties of r_{Ω} will be established later on.) This will lead to a definition of the Sobolev spaces K_{a}^{m}(Ω) as weighted Sobolev spaces on Lie manifolds with boundary, Proposition 5.7. We again proceed by induction on n. We define r_{Ω} = 1 if n ≤ 1 (recall Ω^{(n−2)} = ∅ if n < 2) or if Ω^{(n−2)} = ∅, that is, Ω is a smooth manifold, possibly with boundary. Assume now that a function r_{ω} was defined for all curvilinear polyhedral domains ω of dimension < n and let us define it for a given bounded n-dimensional curvilinear polyhedral domain Ω. We localize first to a neighborhood of a generic point p ∈ ∂Ω and then use a partition of unity argument. We recall that by definition there exists a neighborhood V_{p} of p in M, a diffeomorphism ϕ_{p} : V_{p} → B^{n−l} × B^{l}, for some 0 ≤ l = ℓ(p) ≤ n − 1, and a polyhedral domain ω_{p} ⊂ S^{n−l−1} such that ϕ_{p}(V_{p} ∩ Ω) = I ω_{p} × B^{l}, I = (0, ϵ), see Condition (18)). Therefore, we can assume that ϕ_{p} is the identity map and replace in what follows V_{p} with ϕ_{p}^{−1}(\frac{1}{2}B^{n−l}× \frac{1}{2}B^{l}). Since r_{Ω} is already defined equal to 1 if p ∈ Ω^{(n−1)}∖Ω^{(n−2)}, we restrict n − l − 1 ≥ 1 above. Let r_{ω_{p}} be the function associated to the curvilinear polyhedral domain ω_{p}. Then we define

r_{V_{P}} (rx′, x′′) := rr_{ω_{p}}(x′), (rx′, x′′) ∈ Ω ⊂ V_{p},  (33)

if x′ ∈ ω_{p}, x′′ ∈ B^{l}, and 1 ≤ l = ℓ(p) ≤ n − 2. Following our usual procedures, we set r_{V_{p}}(rx′) = rr_{ω_{p}}(x′) if l = 0. We consider next a locally finite covering of \overline{Ω} with open sets U_{α} of one of the three following forms

(i) U_{α} ⊂ \overline{U_{α}} ⊂ Ω with ∂U_{α} smooth;

(ii) U_{α} = V_{p} with ℓ(p) = n − 1 (i.e., p is not in the singular set of \overline{Ω}); or

(iii) such that for any x ∈ U_{α} ∩ Ω, the point of Ω^{(n−2)} closest to x is in V_{p} with ℓ(p) ≤ n − 2, and

p ∈ U_{α} ⊂ \overline{U}_{α} ⊂ V_{p}.  (34)

A condition similar to (iii) above was already used in the proof of Proposition 4.7. The conditions (i) and (ii) above correspond exactly to the case when (∂U_{α} ∩ ∂Ω) is smooth (this includes the case when (∂U_{α} ∩ ∂Ω) is empty). We then set

r_{α} = \{ \begin{matrix}1 & if (∂U_{α} ∩ ∂Ω) is smooth \\ r_{V_{p}} & if U_{α} is as in (34). \end{matrix}  (35)

and define

r_{Ω} = ∑_{α} φ_{α}r_{α},  (36)

where φ_{α} is a smooth partition of unity subordinated to U_{α}. If Ω is not bounded, we define instead:

r_{Ω} = χ( ∑_{α} φ_{α}r_{α}),  (37)

where χ is defined as in (23). We notice that the definition of r_{Ω} is not canonical, because it depends on a choice of a covering \{U_{α}\} of \overline{Ω} as above and a choice of a subordinated partition of unity.

Proposition 4.9. Let Ω be a curvilinear, stratified polyhedral domain of dimension n ≥ 2. Then r_{Ω} defined in Equation (36) (or (37)) is continuous on \overline{Ω} and r_{Ω} ∘ κ is smooth on Σ(Ω). Moreover, η_{n−2}/r_{Ω} extends to a continuous, nowhere vanishing function on Σ(Ω) and r_{α}/r_{Ω} extends to a smooth function on Σ(V_{p} ∩ Ω).

Proof. Let η_{−1} := 1 for the inductive step. We shall prove the statement on η_{n−2}/r_{Ω} by induction on n ≥ 1. Since r_{Ω} = 1 for polyhedral domains of dimension n = 0, the result is obviously true for n = 1. We now proceed with the inductive step. We shall use the above results, in particular, Proposition 4.7, for k = n−2 ≥ 0. Thus f = η_{n−2} in the notation of Proposition 4.7. Let f_{α}(x) be the distance from x ∈ V_{p} to V_{p} ∩ Ω^{(n−2)}, if U_{α} ⊂ V_{p} is as in Equation (34) (so ℓ(p) ≤ n − 2 in this case). Thus f_{α} = f on U_{α} ∩ Ω, by the construction of U_{α}. We identify once again V_{p} ∩ Ω with (0, ϵ) ω_{p} × B^{l}, l = ℓ(p), using the diffeomorphism ϕ_{p}, and set again ω := ω_{p} × B^{l}. Also, for any x ∈ ω, let f′_{α}(x) be the distance from x to the singular set ω^{(n−l−2)} of ω if it is not empty, f′_{α}(x) = 1 otherwise. Let r_{α} be as in the definition of r_{Ω}, Equation (36). Then

\frac{f_{α}(rx′, x′′)}{r_{α}(rx′, x′′)} = \frac{f_{α}(rx′, x′′)}{rf′_{α}(x′, x′′)} \frac{f′_{α}(x′, x′′)}{r_{ω_{p}}(x′, x′′)} , for (rx′, x′′) ∈ V_{p} ∩ Ω.

The quotient f′_{α}(rx′, x′′)/rf′_{α}(x′, x′′) extends to a continuous, nowhere vanishing function on Σ(V_{p} ∩ Ω), by Corollary 4.8. By the induction hypothesis, the quotient f′_{α}(x′, x′′)/r_{ω_{p}}(x′, x′′) also extends to a continuous, nowhere vanishing function on Σ(ω) = Σ(ω_{p}) × B^{l}. Since this quotient is independent of r, it also extends to a continuous, nowhere vanishing function on Σ(V_{p}∩Ω). Hence f_{α}/r_{α} extends to a continuous, nowhere vanishing function on Σ(V_{p}). Therefore

r/f = ∑_{α} φ_{α}r_{α}/f = ∑_{α} φ_{α}r_{α}/f_{α}

extends to a continuous function on Σ(Ω). The quotient r/f is immediately seen to be non-zero everywhere, from the definition. Hence f/r also extends to a continuous function on Σ(Ω). We have already noticed that r_{α}/f extends to a continuous, nowhere vanishing function on Σ(V_{p}). Hence r_{α}/r_{Ω} = (r_{α}/f)(f/r_{Ω}) extends to a continuous, nowhere vanishing function on Σ(V_{p} ∩ Ω). Since both r_{α} and r_{Ω} are smooth on Σ(V_{p} ∩ Ω) and the set of zeroes of r_{Ω} is the union of transversal manifolds on which r_{Ω} has simple zeroes, it follows that r_{α}/r_{Ω} extends to a smooth function on Σ(V_{p}). Since \overline{U}_{α} ⊂ V_{p} is compact, it follows from a compactness argument that r_{α} and r are equivalent on U_{α}. The proof is complete.

We can now prove the following result, which will be used in the proof of Theorem 6.4.

Proposition 4.10. Let Ω be a bounded, curvilinear, stratified polyhedral domain. Suppose r_{Ω}, r′_{Ω} are two functions on \overline{Ω} defined by formula (36) (or (37)) with possibly different choices of open covering {U_{α}}, subordinate partition {φ_{α}}, and diffeomorphisms ϕ_{p}. Then r′_{Ω}/r_{Ω} extends to a smooth, nowhere vanishing function on Σ(Ω).

Proof. We know from Proposition 4.9, that f/r′_{Ω} and f/r_{Ω} extend to continuous, nowhere vanishing functions on Σ(Ω). Hence r′_{Ω}/r_{Ω} extends to a continuous, nowhere vanishing function on Σ(Ω). Since both r′_{Ω} and r_{Ω} are smooth functions on Σ(Ω) and the set of zeroes of r_{Ω} is a union of transverse manifolds, each a set of simple zeroes of r_{Ω}, it follows that the quotient r′_{Ω}/r_{Ω} is smooth on Σ(Ω).

We obtain the following corollary. Let H ⊂ Σ(Ω) be a hyperface (i.e., face of maximal dimension) of Σ(Ω). Recall that a defining function of H is a smooth function x_{H} ≥ 0 defined on Σ(Ω), such that H = {x = 0} and dx_{H} ≠ 0 on H. All the faces of Σ(Ω) are closed subsets of Σ(Ω), by definition. We have already noticed that any face of Σ(Ω) has a defining function. We then have the following corollary.

Corollary 4.11. Let η = ∏_{H} x_{H}, where H ranges through the set of hyperfaces of Σ(Ω) that do not intersect ∂Ω ∖ Ω^{(n−2)}. Then η/r_{Ω} extends to a smooth, nowhere vanishing function on Σ(Ω).

Proof. This is a local statement that can be checked by induction, as in the previous proofs.

In particular, since the function r_{Ω} is anyway determined only up to a factor of h ∈ C^{∞}(Σ(Ω)), h ≠ 0, we obtain that we could take r_{Ω} = ∏_{H} x_{H}, where H ranges through the set of hyperfaces of Σ(Ω) that do not intersect ∂Ω∖Ω^{(n−2)}. The function r_{Ω}, for various versions of the set Ω, will play an important role in the inductive definition of the structural Lie algebra of vector fields V(Ω) on Σ(Ω), which we address next. The faces considered in the above corollary are the hyperfaces at infinity of Σ(Ω). See Definition 3.8.

4.4 The structural Lie algebra of vector fields

We now proceed to define by induction a canonical Lie algebra of vector fields V(Ω) on Σ(Ω), for Ω a curvilinear, stratified polyhedral domain of dimension n ≥ 1. In view of Corollary 4.3, we can assume that Ω is connected. We denote by

X (M) := Γ(M; T M)

the space of vector fields on a manifold M. We let

V(Ω) = X (\overline{Ω}) = X (Σ(Ω)), if n = 1.  (38)

In other words, there is no restriction on the vector fields X ∈ V(Ω), if Ω has dimension one. Assume now that the Lie algebra of vector fields V(ω) has been defined on Σ(ω) for all curvilinear polyhedral domains ω of dimension 1 ≤ k ≤ n − 1 and let us define V(Ω) for a curvilinear polyhedral domain of dimension n. We fix p ∈ ∂Ω and let V_{p} and ϕ_{p} be as in Definition 2.1, as usual. We identify V_{p} ∩ Ω with (0, 1)ω_{p} × B^{l} using ϕ_{p}. Assume 1 ≤ ℓ(p) ≤ n − 2, so that in particular ω_{p} is a curvilinear polyhedral domain of dimension ≥ 1. Let M_{1} := [0, 1)×Σ(ω_{p})×B^{l}. We notice that

T M_{1} = T ([0, 1) × Σ(ω_{p}) × B^{l}) = T ([0, 1)) × T Σ(ω_{p}) × T B^{l}

and hence

X (M_{1}) = Γ(M_{1}; T [0, 1)) ×_{M_{1}} Γ(M_{1}; T Σ(ω_{p})) ×_{M_{1}} Γ(M_{1}; T B^{l})

⊂ Γ(M_{1}; T [0, 1)) × Γ(M_{1}; T Σ(ω_{p})) × Γ(M_{1}; T B^{l}).

Then we define

V(V_{p} ∩ Ω) = {X = (X_{1}, X_{2}, X_{3}) ∈ X (M_{1})

X_{1} ∈ Γ(M_{1}; T [0, 1)), X_{2} ∈ Γ(M_{1}; T Σ(ω_{p})), X_{3} ∈ Γ(M_{1}; T B^{l})
Y_{1} := r_{Ω}^{−1}X_{1} and Y_{3} := r_{Ω}^{−1}X_{3} are smooth, and

X_{2}(t, x′, x′′) ∈ V({t} × ω_{p} × {x′′}) = V(ω_{p}), for any fixed t, x′′}. (39)

In Equation (39) above, “smooth” means, “smooth including at r = 0.” If ℓ(p) = 0, then we just drop the component X_{3}, but keep the same conditions on X_{1} and X_{2}. By Proposition 4.10, the definition of V(V_{p} ∩ Ω) is independent of the choice of r_{Ω}. All vector fields are assumed to be smooth. Finally, we define V(Ω) to consist of the vector fields X ∈ X (Σ(Ω)) such that X|_{V_{p}∩Ω} ∈ V(V_{p} ∩ Ω) for all p ∈ Ω^{(n−2)}. In particular, only the smoothness condition is imposed on our vector fields at the smooth points of ∂Ω. Note that the vector fields in V(Ω) may not extend to the closure \overline{Ω}, in general. This was seen in Example 2.10.

4.5 Lie manifolds with boundary

We now proceed to show that the pair (Σ(Ω), V(Ω)) defines a Lie manifold with boundary, introduced in [1], and the construction of which was recalled in Definition 3.5. We first establish some lemmata.

Lemma 4.12. Let X ∈ X (Σ(Ω)) be such that X = 0 in a neighborhood of the boundary of Σ(Ω). Then X ∈ V(Ω).

Proof. The result follows immediately by induction from the definition of V(Ω).

We also get the following simple fact.

Lemma 4.13. If f : Σ(Ω) → ℂ is a smooth function and X ∈ V(Ω), then X(f) is a smooth function on Σ(Ω) and fX ∈ V.

Proof. The vector field X is smooth on Σ(Ω), hence X(f) is smooth on Σ(Ω). The second statement is local, so it is enough to check it on Ω and on each V_{p}, on which it is as a direct consequence of the definition and induction.

Lemma 4.14. For any X ∈ V(Ω) and any continuous function f : \overline{Ω} → ℂ such that f ∘ κ is smooth on Σ(Ω), we have

X(f) = f̃r_{Ω},

where f̃ is a smooth function on Σ(Ω). In particular, X(r_{Ω}) = f_{X}r_{Ω}, where f_{X} is a smooth function on Σ(Ω).

Proof. This is a local statement that can be checked by induction in any neighborhood V_{p}, using the definition, as follows. Let us use the notation of Equation (39) and write

X = (X_{1}, 0, 0) + (0, X_{2}, 0) + (0, 0, X_{3}).

We shall write, with abuse of notation, X_{1} = (X_{1}, 0, 0). Define X_{2} and X_{3} similarly. It is enough to check that X_{j}f(rx′, x′′) is of the indicated form, for j = 1, 2, 3. We have X_{1} = r_{Ω}Y_{1} and X_{3} = r_{Ω}Y_{3}, where Y_{1} and Y_{3} are smooth (in appropriate spaces), by Equation (39). This observation proves our lemma if X = X_{1} or X = X_{3}. If X = X_{2}, then we have

(Xf)(r, x′, x′′) = X_{2}(f(rx′, x′′)) = r_{ω_{p}}f_{1}(r, x′, x′′),  (40)

with f_{1} a smooth function on Σ(V_{p} ∩ Ω) = [0, ϵ) × Σ(ω_{p}) × ℝ^{l}, by the induction hypothesis. Moreover, given that by assumption (39) κ_{∗}X is a vector field tangent to the sphere S^{n−l−1}, we see that Xf(0, x′, x′′) = 0. Therefore Xf = rr_{ω_{p}} f̃, for some smooth function f̃ on Σ(V_{p} ∩ Ω). Let us denote r_{α} = rr_{ω_{p}}, as in Equation (35) and in Proposition 4.9. Proposition 4.9 gives that r_{α}/r_{Ω} is smooth on its domain of definition. Hence Xf = r_{α}f_{1} = r_{Ω}(r_{α}/r_{Ω})f_{1} = r_{Ω}f̃, with f̃ smooth on each Σ(V_{p} ∩ Ω). Hence f̃ is smooth on Σ(Ω).

We next characterize which vector fields on Ω are restrictions of vector fields on V(Ω). We begin by showing that the restriction property is local.

Lemma 4.15. Let Y be a vector field on Ω with the property that every point p ∈ \overline{Ω} has a neighborhood U_{p} in M such that Y = X_{U} on U ∩ Ω, for some X_{U} ∈ V(Ω). Then there exists X ∈ V(Ω) such that Y is the restriction of X to Ω.

Proof. Let us cover \overline{Ω} with a locally finite family of sets U_{p}, p ∈ B ⊂ \overline{Ω}. Let ψ_{p}, p ∈ B, be a subordinated partition of unity. We claim that X = ∑_{p∈B} ψ_{p}X_{U_{p}} ∈ V(Ω) (by Lemma 4.13) satisfies X(x) = Y (x), x ∈ Ω. Indeed, X(x) = ∑_{p∈B} ψ_{p}(x)X_{U_{p}}(x) = ( ∑ ψ_{p}(x))Y (x) = Y (x).

We can now prove the following lemma.

Lemma 4.16. Let Y be a smooth vector field on \overline{Ω}. Then r_{Ω}Y is the restriction to Ω ⊂ Σ(Ω) of a vector field X in V(Ω).

Proof. By Lemma 4.15, it is enough to check this statement on a neighborhood V_{p} of some p ∈ ∂Ω. We shall proceed by induction. Since the desingularization and the definition of r_{Ω} are covariant with respect to diffeomorphism (that respect the stratification of Ω), we can assume that V_{p} = B^{n−l} × B^{l} and that V_{p} ∩ Ω ≃ (0, 1)ω_{p} × B^{l}. Assume first that Y = ∂_{j} is a constant vector field on V_{p}. Let α_{t}(x′, x′′) = (tx′, x′′). Then Dα_{t}(∂_{j}) = t∂_{j}. Therefore,

Dα_{t}(X) = X,  (41)

where X = r_{Ω}∂_{j}, where r_{Ω} can be taken, on V_{p}, to be given by rr_{ω_{p}}. Let us decompose ∂_{j} = (Y_{1}, Y_{2}, Y_{3}) on V_{p} using the notation of Equation (39). Then Y_{3} is constant. In fact, either Y_{3} = ∂_{j} or Y_{3} = 0. In any instance, if we write X = (X_{1}, X_{2}, X_{3}), then X_{3} = r_{Ω}Y_{3} satisfies the condition of Equation (39). The relation (41) gives that Y_{1}(r, x′, x′′) = a_{1}(x′)∂_{r} and Y_{2}(r, x′, x′′) = r^{−1}Z(x′), with a_{1} a smooth function and Z a smooth vector field on \overline{ω}_{p}. Clearly X_{1} = r_{Ω}Y_{1} will satisfy the conditions of Equation (39). The induction hypothesis then gives that X_{2}(r, x′, x′′) = r_{Ω}Y_{2}(r, x′, x′′) = r_{ω_{p}}(x′)Z(x′) is the restriction to V_{p} ∩ Ω of a smooth vector field in V(V_{p} ∩ Ω). (This vector field depends only on the second factor in Σ(V_{p} ∩ Ω) = [0, 1) × ω_{p} × B^{l}.)

We now identify a canonical metric on the vector fields V. Recall that the concept of local basis of a space of vector fields was defined in Definition 3.1.

Proposition 4.17. Let us fix a metric h on M ⊃ Ω. Let q ∈ Σ(Ω) be arbitrary. Then there exists a neighborhood U of q in Σ(Ω) and X_{1}, X_{2}, …, X_{n} ∈ V(Ω) that form a local basis of V(Ω) on U and satisfy

h(X_{j}, X_{k}) = r_{Ω}^{2}δ_{jk}.

In other words, the vectors X_{1}, X_{2}, …, X_{n} form an orthonormal system on Ω∩U for the metric r_{Ω}^{−2}h. A local basis X_{1}, X_{2}, …, X_{n} with this property will be called a local orthonormal basis of V(Ω) over U.

Proof. If q ∈ Ω ⊂ Σ(Ω), the result follows from Lemma 4.12. Let p = κ(q). We shall hence assume that p ∈ ∂Ω. This is again a local statement in p ∈ ∂Ω. We can therefore proceed by induction. If the dimension n of Ω is 1, then there is nothing to prove because r_{Ω} = 1 in this case. Once again, we let ϕ_{p} : V_{p} → B^{n−l} × B^{l} and ω_{p} be as in Definition 2.1. We can assume that ϕ_{p} is the identity map. If we can prove the result for the function r = r_{Ω}, then we can prove it for the function r′ = f′r, where f′, 1/f′ ∈ C^{∞}(Σ(Ω)), simply by replacing X_{j} with f′X_{j}. By Proposition 4.9, we can therefore assume that r_{Ω} = rr_{ω_{p}} on V_{p} ∩ Ω. Let q = (0, x′, x′′) ∈ [0, 1) × Σ(ω_{p}) × B^{l}. Let h_{0} be the standard metric on V_{p}. For the induction hypothesis, we shall need that the metric h_{0} is given by

h_{0}(r, x′, x′′) = (dr)^{2} + r^{2}(dx′)^{2} + (dx′′)^{2}  (42)

on Ω∩V_{p} = (0, 1)ω_{p} ×B^{l}. Here (dx′)^{2} denotes the metric on ω_{p} induced by the Euclidean metric on the sphere S^{n−l−1}. In other words, if X = (X_{1}, X_{2}, X_{3}) is a vector field on V_{p} ∩ Ω, written using the product decomposition explained above (or as in the Equation (39)), then

h_{0}(X) = ∥X_{1}∥^{2} + r^{2}∥X_{2}∥^{2} + ∥X_{3}∥^{2}

where the norms come from the standard metrics, respectively, on T [0, 1), on T S^{n−l−1} ⊃ T ω_{p}, and on T ℝ^{l}. Let us assume first that h = h_{0}, the standard metric on ℝ^{n}. By the induction hypothesis, we can construct Y_{2}, …, Y_{n−l} ∈ V(ω_{p}) forming a local orthonormal basis of V over some small neighborhood U′ of x′ in Σ(ω_{p}) (i.e., {Y_{2}, …, Y_{n−l}} ⊂ V(ω_{p}) is orthonormal with respect to the metric r_{ω_{p}}^{−2}(dx′)^{2}). Here (dx′)^{2} denotes the metric on ω_{p} induced by the Euclidean metric on the sphere S^{n−l−1}, as above. Let Y_{1} = r_{Ω}∂_{r} and Y_{j} = r_{Ω}∂_{j}, j = n − l + 1, …, n, where ∂_{j} forms the standard basis of ℝ^{l−1}. Then we claim that we can take U = [0, 1) × U′ × B^{l} and

{X_{1}, X_{2}, …, X_{n}} = {Y_{1}} ∪ {Y_{2}, …, Y_{n−l}} ∪ {Y_{n−l+1}, …, Y_{n}}.  (43)

(If n−l = 1, then the second set in the above union is empty. If l = 0, then the third set in the above union is empty.) Indeed, {X_{1}, …, X_{n}} is a local basis by construction and by the local definition of V(Ω) in Equation (39). Let us check that this is an orthonormal local basis. To this end, we shall use the form of the standard metric h_{0} given in Equation (42), to obtain

h_{0}(X_{1}) = r_{Ω}^{2}∥∂_{r}∥^{2} = r_{Ω}^{2} , h_{0}(X_{n−l+1}) = … = h_{0}(X_{n}) = r_{Ω}^{2}

and h_{0}(X_{2}) = … = h_{0}(X_{n−l}) = r^{2}∥X_{2}∥^{2} = r^{2}r_{ω_{p}}^{2} = r_{Ω}^{2} .

It is also clear that {X_{1}, X_{2}, …, X_{n}} is an orthogonal system. This completes the induction step if h = h_{0}, the standard metric on ℝ^{n}. If h is not the standard metric on V_{l}, we can nevertheless chose a matrix valued function T defined on a neighborhood of q in U such that h(T ξ, T η) = h_{0}(ξ, η). We then let X_{j} = T Y_{j} and replace U with this smaller neighborhood.

This lemma gives the following corollary.

Corollary 4.18. Let X, Y ∈ V(Ω) and h be a fixed metric on M. Then the function r_{Ω}^{−2}h(X, Y), defined first on Ω, extends to a smooth function on Σ(Ω).

Proof. This is a local statement in the neighborhood of each point q ∈ Σ(Ω). Let X_{1}, X_{2}, …, X_{n} be a local basis of V on a neighborhood U of q in Σ(Ω) satisfying the conditions of Proposition 4.17 (i.e., orthogonal with respect to r_{Ω}^{−2}h). Let X = ∑ ϕ_{j}X_{j} and Y = ∑ ψ_{j}X_{j} on U ∩ Ω, where ϕ_{j}, ψ_{j} are smooth functions on Σ(Ω). Then r_{Ω}^{−2}h(X, Y) = ∑ ϕ_{j}\overline{ψ}_{j} is smooth on U.

Lemma 4.19. Let p ∈ ∂Ω and X_{1}, X_{2}, …, X_{n} be vector fields on \overline{Ω} that define a local basis of T M on \overline{U}, for some neighborhood U of p. Then r_{Ω}X_{1}, r_{Ω}X_{2}, …, r_{Ω}X_{n} is a local basis of V(Ω) on U, that is, for any Y ∈ V(Ω), there exist unique smooth function ϕ_{1}, ϕ_{2}, …, ϕ_{n} on Σ(Ω) satisfying

Y = ϕ_{1}r_{Ω}X_{1} + ϕ_{2}r_{Ω}X_{2} + … + ϕ_{n}r_{Ω}X_{n} on U ∩ Ω ⊂ Σ(Ω). (44)

Conversely, if a vector field Y on Ω satisfies Condition (44) for any p and any local basis X_{1}, …, X_{n} of T M at p, then Y is the restriction to Ω of a vector field in V(Ω).

Proof. The converse part is easier, so we prove it first. Let Y be a vector field on Ω that satisfies Condition (44) for any p and any local basis X_{1}, …, X_{n} of T M at p. Fix an arbitrary p ∈ Ω. Lemmata 4.13 and 4.16 give that ϕ_{j}r_{Ω}X_{j} is the restriction to Ω of a vector field in V(Ω). Hence on each U ∩ Ω, Y is the restriction of a vector field Y_{U} ∈ V(Ω). Lemma 4.15 then gives the converse part of our lemma. We now prove the direct part of the lemma. We can assume that the vector fields X_{1}, …, X_{n} form an orthonormal system on U with respect to some fixed metric h on M. We know from Lemma 4.16 that r_{Ω}X_{j} ∈ V(Ω). Let then Y ∈ V(Ω) and note that ϕ_{j} = r_{Ω}^{−1}h(Y, X_{j}) = r_{Ω}^{−2}h(Y, r_{Ω}X_{j}) ∈ C^{∞}(Σ(Ω)), by Corollary 4.18. Then Y = ∑_{j=1}^{n} ϕ_{j}r_{Ω}X_{j} on U ∩ Ω. The local uniqueness of the functions ϕ_{j} follows from the fact that r_{Ω}X_{1}, r_{Ω}X_{2}, …, r_{Ω}X_{n} also form a local basis of T Ω on U ∩ Ω.

We are now ready to prove the following characterizations of V(Ω). We notice that the restriction map V(Ω) ∋ X → X|_{Ω} is injective, so we may identify V(Ω) with a subspace of the space Γ(Ω, T M) of vector fields on Ω.

Proposition 4.20. Let Ω ⊂ M be a curvilinear, stratified polyhedral domain of dimension n and let X be a smooth vector field on Ω. Fix an arbitrary metric h on M. Then X ∈ V(Ω) if, and only if, r_{Ω}^{−1}h(X, Y) extends to a smooth function on Σ(Ω) for any smooth vector field Y on \overline{Ω}.

Proof. In one direction the result follows from Lemma 4.16 and Corollary 4.18. Indeed, let X ∈ V(Ω) and Y be a smooth vector field on \overline{Ω}. Then r_{Ω}Y ∈ V(Ω) by Lemma 4.16 and hence r_{Ω}^{−1}h(X, Y) = r_{Ω}^{−2}h(X, r_{Ω}Y) extends to a smooth function on Σ(Ω) by Corollary 4.18. (We have already used this argument in the proof of the previous lemma.) Conversely, assume that r_{Ω}^{−1}h(X, Y) extends to a smooth function on Σ(Ω) for any smooth vector field on \overline{Ω}. The statement that X ∈ V(Ω) is a local statement, by Lemma 4.15. So let p ∈ \overline{Ω} and let U be an arbitrary neighborhood of p. Choose smooth vector fields defined in a neighborhood of \overline{Ω} in M such that X_{1}, X_{2}, …, X_{n} is a local orthonormal basis on U (orthonormal with respect to h). Let

ϕ_{j} = r_{Ω}^{−1}h(Y, X_{j}),

by assumption ϕ_{j} ∈ C^{∞}(Σ(Ω). Then Y = ∑_{j=1}^{n} ϕ_{j}X_{j} on U ∩ Ω and ∑_{j=1}^{n} ϕ_{j}X_{j} ∈ V(Ω). Lemma 4.15 then shows that X ∈ V(Ω).

We now prove the main characterization of the structural Lie algebra of vector fields V(Ω).

Theorem 4.21. Let Ω ⊂ M be a bounded curvilinear, stratified polyhedral domain of dimension n. Then V(Ω) is generated as a vector space by the vector fields of the form ϕ r_{Ω}X, where ϕ ∈ C^{∞}(Σ(Ω)) and X is a smooth vector field on \overline{Ω}.

Proof. We know that ϕr_{Ω}X ∈ V(Ω) whenever X is a smooth vector field on \overline{Ω}, by Lemmata 4.13 and 4.16. This remark shows that the linear span of vectors of the form ϕ r_{Ω}X, where ϕ ∈ C^{∞}(Σ(Ω)) and X is a smooth vector field in a neighborhood of Σ, is contained in V(Ω). Conversely, let Y ∈ V(Ω). Then Lemma 4.19 shows that we can find, in the neighborhood U_{p} of any point p ∈ \overline{Ω} vector fields X_{1p}, X_{2p}, …, X_{np} and smooth functions ϕ_{jp} such that Y = ∑ ϕ_{jp}r_{Ω}X_{jp} on U_{p}. The result then follows using a finite partition of unity on Σ(Ω) subordinated to the covering U_{p}.

If we drop the condition that Ω be bounded, we obtain the following result, which was established in the first half of the above proof.

Proposition 4.22. Let Ω ⊂ M be a curvilinear polyhedral domain of dimension n. Then V(Ω) consists of the set of vector fields that locally can be written as linear combinations of vector fields of the form ϕr_{Ω}X, where ϕ ∈ C^{∞}(Σ(Ω)) and X is a smooth vector field on \overline{Ω}.

We are finally in the position to endow Σ(Ω) with a structure of Lie manifold, which we will exploit in the following sections to study the mixed boundary value/interface problem (6). We set ∂′′Σ(Ω) to be the union of all hyperfaces (i.e., faces of maximal dimension) H of Σ(Ω) such that κ(H) ⊂ \overline{Ω} lies in the singular set Ω^{(n−2)}, and let ∂′Σ(Ω) = ∂Σ(Ω) ∖ ∂′′Σ(Ω). In particular, ∂′′Σ(Ω) is the union of the hyperfaces at infinity of Σ(Ω), see Definition 3.8. The next theorem is the main result of this subsection.

Theorem 4.23. Let Ω be a bounded curvilinear, stratified polyhedral domain and let

𝔒_{0} := Σ(Ω) ∖ ∂′′Σ(Ω) = Ω ∪ ∂′Σ(Ω) = κ^{−1}(\overline{Ω} ∖ Ω^{(n−2)}).

Then (𝔒_{0}, Σ(Ω), V(Ω)) is a Lie manifold with boundary ∂′Σ(Ω). The projection map κ : 𝔒_{0} → \overline{Ω} ∖ Ω^{(n−2)} is such that κ^{−1}(p) consists of exactly one point if p ∈ \overline{Ω} ∖ Ω^{(n−2)}.

Proof. The last statement (on the number of elements in κ^{−1}(p), p ∈ \overline{Ω} ∖ Ω^{(n−2)}) follows from the definition. Therefore, to prove the proposition, we need, using the notation of Definition 3.5, to construct a compactification 𝔒 of 𝔒_{0} that identifies with the closure of a Lie domain in a Lie manifold 𝔐. We shall choose then 𝔒 = Σ(Ω). Then we shall let 𝔐 be the “double” of Σ(Ω), also denoted ^{d}Σ(Ω). More precisely, 𝔐 is obtained from the disjoint union of two copies of Σ(Ω) by identifying the hyperfaces that are not at infinity. We let V to be the set of smooth vector fields on 𝔐 such that the restriction to either copy of Σ(Ω) is in V(Ω). Let 𝔻 be obtained from the closure of Ω in 𝔐 by removing the closure of ∂′Σ(Ω). Then 𝔻 is an open subset of 𝔐 whose closure is Σ(Ω). Moreover, ∂_{𝔐}𝔻 (the boundary of 𝔻 regarded as a subset of 𝔐) is the closure of ∂′Σ(Ω). To prove our theorem, we shall check that 𝔐 is a manifold with corners, that (𝔐, V) is a Lie manifold, and that ∂𝔻 is a regular submanifold of 𝔐. Each of these properties is local, so it can be checked in the neighborhood of a point of 𝔐. Fix V_{p} = (0, ϵ) × ω_{p} × B^{l}. Then the union of the two copies of Σ(V_{p}) is the double ^{d}Σ(V_{p}) of Σ(V_{p}). Denote by ^{d}ω_{p} the double of ω_{p}. Then

dΣ(V_{p}) = [0, ϵ) ×^{d}ω_{p} × B^{l}.

An inductive argument then shows that ^{d}Σ(Ω) is a manifold with corners and that ∂𝔻 is a regular submanifold of 𝔐. Let us check that V satisfies the conditions defining a Lie manifold structure on 𝔐. It follows from Theorem 4.21 that V is a C^{∞}(𝔐)–module (this checks condition (iii) of Definition 3.2). Theorem 4.21 and Lemmata 4.14, 4.13 show that V is closed under Lie brackets (this checks condition (i) of Definition 3.2). Condition (ii) of that definition follows from the definition of V(Ω). Condition (iv) of Definition 3.2 as well as Condition (ii) of Definition 3.3 were proved in Lemma 4.19. This shows that (𝔐, V) is a Lie manifold.

An immediate consequence of the above Proposition is that the boundary ∂𝔒_{0} = ∂′Σ(Ω) of 𝔒_{0} = Σ(Ω) ∖ ∂′′Σ(Ω) will acquire the structure of a Lie manifold, as explained after the definition of a Lie manifold with boundary, Definition 3.5. Let D be the closure of ∂𝔒_{0} in 𝔒. Then the Lie structure at infinity is (∂𝔒_{0}, D, W), where

W = {X|_{D}, X ∈ V, X|_{D} is tangent to D}.  (45)

As always, X ∈ W is completely determined by its restriction to 𝔒_{0}.

5 Weighted Sobolev spaces

One of the main goals of this work, as mentioned already, is the study of mixed boundary value/interface problems for second-order elliptic operators on ndimensional curvilinear, stratified polyhedral domains Ω in the framework of certain weighted Sobolev spaces. This framework is adapted to the singular geometry of polyhedral domains and allows to obtain optimal regularity, which does not hold in general in the standard (unweighted) spaces. We begin by giving a rigorous definition of the weighted spaces. Let f be a continuous function on Ω, f > 0 on the interior of Ω. We define the μ-th Sobolev space with weight f and index a by

K_{a,f}^{μ}(Ω) = {u ∈ L_{loc}^{2}(Ω), f^{|α|− a}∂^{α}u ∈ L^{2}(Ω), for all |α| ≤ μ} , μ ∈ ℤ_{+}.

(46)

The norm on K_{a,f}^{μ}(Ω) is given by

∥u∥_{K_{a,f}^{μ} (Ω)}^{2} := ∑_{|α|≤μ} ∥f^{|α|− a}∂^{α}u∥_{L^{2}(Ω)}^{2} .  (47)

Definition 5.1. Let f, g be two continuous, non-negative functions on Ω. We shall say that f and g are equivalent (written f ∼ g) if there exists a constant C > 0 such that

C^{−1}f(x) ≤ g(x) ≤ Cf(x),

for all x ∈ Ω.

Clearly, if f ∼ g, then the norms ∥u∥_{K_{a,f}^{μ} (Ω)} and ∥u∥_{K_{a,g}^{μ}(Ω)} are equivalent, and hence we have K_{a,f}^{μ}(Ω) = K_{a,g}^{μ}(Ω) as Banach spaces.

Definition 5.2. Let f = η_{n−2} be the distance to Ω^{(n−2)}, as before. Then we define K_{a}^{μ}(Ω) = K_{a,f}^{μ}(Ω) and ∥u∥_{K_{a}^{μ}(Ω)} = ∥u∥_{K_{a,f}^{μ} (Ω)}.

For example, K_{0}^{0}(Ω) = L^{2}(Ω). For Ω a polygon in the plane, η_{n−2}(x) = η_{0}(x) is the distance from x to the vertices of Ω and the resulting spaces K_{a}^{μ}(Ω) are the spaces considered in Kondratiev’s paper [41]. Above in Definition 5.2, we can and will replace η_{n−2} with the equivalent function r_{Ω} by Proposition 4.9. If μ ∈ ℕ = ℤ_{+} ∖ {0}, we define K_{−a}^{−μ}(Ω) to be the dual of

K_{a}^{μ}^{∘} (Ω) := K_{a}^{μ}(Ω) ∩ {∂_{ν}^{j}u|_{∂Ω} = 0, j = 0, 1, …, μ − 1}  (48)

with pivot K_{0}^{0}(Ω). Later in this Section, we will identify K_{a}^{m}(Ω) with a suitable space hH^{μ}(Σ(Ω)) using the Lie structure on Σ(Ω). Then, Theorem 3.4 of [1]

(see also Lemma 3.7 and Proposition 3.9) gives that C_{c}^{∞}(Ω) is dense in K_{a}^{μ}^{∘} (Ω) and consequently K_{−a}^{−μ}(Ω) is the completion of the space of smooth functions u on Ω satisfying

∥u∥_{K_{−a}^{−μ}(Ω)} = sup_{0≠v∈C_{c}^{∞}(Ω)} \frac{|(u, v)|}{∥v∥_{K_{a}^{μ}(Ω)}} < +∞.  (49)

In order to make the identification K_{a}^{μ}(Ω) ≈ hH^{μ}(Σ(Ω)), for suitable h, we introduce next a class of “admissible weights” h.

5.1 The set of weights

If h > 0 on Ω, we denote

hK_{a}^{μ}(Ω) := {hu, u ∈ K_{a}^{μ}(Ω)},  (50)

with induced norm, that is ∥hu∥_{hK_{a}^{μ}(Ω)} = ∥u∥_{K_{a}^{μ}(Ω)}. A weight h on Ω will be called admissible if it is admissible on Σ(Ω). One of the main examples of an admissible weight is η_{n−2}^{a}, for a ∈ ℝ. We recall that an admissible weight on Σ(Ω) is a function h of the form h = ∏_{H} x_{H}^{a_{H}}, where H ranges through the set of hyperfaces at infinity of Σ(Ω) and a_{H} ∈ ℝ. The topology is induced from the topology on the set {(a_{H})} of exponents. As discussed after Corollary 4.11, we can always assume r_{Ω} := ∏_{H} x_{H}. In particular, r_{Ω}^{a}, a ∈ ℝ, is the most important example of an admissible weight. It is more suitable to use this weight, which is intimately related to the structure of Lie manifold on Σ(Ω) (∖∂”Σ(Ω)) described in Theorem 4.23, instead of η_{n−2}^{a}. We also have that

r_{Ω}^{t}K_{a}^{μ}(Ω) = K_{a+t}^{μ}(Ω),  (51)

so in a statement about the spaces hK_{a}^{μ}(Ω), where h is an admissible weight, we can usually assume that a = 0, without loss of generality. These spaces are weighted Sobolev spaces in the sense of the following definition. (These spaces are sometimes called Babuška–Kondratiev spaces.)

Definition 5.3. Let h be an admissible weight on Ω. The weighted Sobolev space of order μ ∈ ℤ and weight h on Ω is the space hK_{0}^{μ}(Ω).

5.2 Sobolev spaces and Lie manifolds

We now identify the weighted Sobolev space K_{a}^{μ}(Ω) with hH^{μ}(Σ(Ω)), for a suitable admissible weight h; more precisely, h = r_{Ω}^{a−n/2}. The following description of V(Ω) for Ω a curvilinear polyhedral domain in ℝ^{n} will be useful. It follows readily from Theorem 4.21.

Corollary 5.4. Let Ω ⊂ ℝ^{n} be a bounded curvilinear, polyhedral domain. Then

V(Ω) = {ϕ_{1}r_{Ω}∂_{1} + ϕ_{2}r_{Ω}∂_{2} + … + ϕ_{n}r_{Ω}∂_{n}, where ϕ_{j} ∈ C^{∞}(Σ(Ω))}.

We shall denote by

Diff_{Ω}^{k} := Diff_{V(Ω)}(Σ(Ω))  (52)

the space of differential operators with coefficients in C^{∞}(Σ(Ω)) of order ≤ k on Σ(Ω) generated by V(Ω). The algebra of differential operators Diff_{Ω}^{∞} is an example of the algebra of differential operators considered in 3.3. From the last corollary, we obtain directly the following lemma.

Lemma 5.5. Let X_{1}, X_{2}, …, X_{k} be smooth vector fields on M. Then

P := r_{Ω}^{k}X_{1}X_{2} … X_{k} ∈ Diff_{Ω}^{k}.

Moreover, Diff_{Ω}^{k} is generated linearly by ϕP, with P as above and ϕ ∈ C^{∞}(Σ(Ω)).

Proof. For k = 1, this follows from Lemma 4.16. Next, we have

r_{Ω}^{k+1}X_{0}X_{1} … X_{k} = r_{Ω}X_{0}r_{Ω}^{k}X_{1} … X_{k} − kX_{0}(r_{Ω})r_{Ω}^{k}X_{1} … X_{k}.

The fact that P ∈ Diff_{Ω}^{k} then follows by induction, since X_{0}(r_{Ω}) ∈ C^{∞}(Σ(Ω)), by Lemma 4.14. Conversely, we can similarly check by induction (using the same identity above) that the product r_{Ω}X_{1}r_{Ω}X_{2} … r_{Ω}X_{k} can be written as a linearly combination of differential operators of the form ϕP, with ϕ ∈ C^{∞}(Σ(Ω)) and P as above. Since r_{Ω}X generate V(Ω) as a C^{∞}(Σ(Ω))–module (see Corollary 5.4 or the second part of Theorem 4.21), the result follows.

We next provide a different description of the weighted Sobolev spaces K_{a}^{μ}(Ω), μ ∈ ℤ_{+}. For a multiindex α, we denote

(r_{Ω}∂)^{α} := (r_{Ω}∂_{1})^{α_{1}}(r_{Ω}∂_{2})^{α_{2}} … (r_{Ω}∂_{n})^{α_{n}}.  (53)

Theorem 5.6. Let Ω ⊂ ℝ^{n} be a bounded curvilinear, stratified polyhedral domain and

∥u∥_{μ,a}^{2} := ∑_{|α|≤μ} ∥r_{Ω}^{−a}(r_{Ω}∂)^{α} u∥_{L^{2}(Ω)}^{2}.

Then ∥u∥_{μ,a} is equivalent to ∥u∥_{K_{a}^{μ}(Ω)} of Definition 5.2. In particular,

K_{a}^{μ}(Ω) = {u, ∥u∥_{μ,a} < ∞}.

Proof. We have that

u ∈ K_{a}^{μ}(Ω) ⇔ r_{Ω}^{|α|−a}∂^{α}u ∈ L^{2}(Ω) for all |α| ≤ μ by Proposition 4.9

⇔ r_{Ω}^{−a}(r_{Ω}∂)^{α}u ∈ L^{2}(Ω) for all |α| ≤ μ by Proposition 5.5.

Above the corresponding square integrability conditions define the topology on the indicated spaces. Therefore ⇔ also means that the topologies are the same.

We are in position to identify the spaces K_{a}^{μ} with Sobolev spaces on Lie manifolds. If Ω is a bounded curvilinear, stratified polyhedral domain, we let

𝔒_{0} := Σ(Ω) ∖ ∂′′Σ(Ω) = Ω ∪ ∂′Σ(Ω) = κ^{−1}(\overline{Ω} ∖ Ω^{(n−2)}),

as in Theorem 4.23. Since (𝔒_{0}, 𝔒 := Σ(Ω), V(Ω)) is a Lie manifold with boundary by the same theorem, the definitions of Sobolev spaces on Lie manifolds (with or without boundary) of Subsection 3.5 provide us with natural spaces H^{s}(Σ(Ω)) = H^{s}(𝔒) = H^{s}(𝔒_{0}) and H^{s}(∂′Σ(Ω)) = H^{s}(∂𝔒_{0}). For the last equality we used that the boundary of 𝔒_{0} is ∂′Σ(Ω).

Proposition 5.7. Let Ω be an n-dimensional, bounded curvilinear, stratified polyhedral domain and let h be an admissible weight on Ω. We have an equality

hK_{a}^{μ}(Ω) = hr_{Ω}^{a−n/2}H^{μ}(Σ(Ω)), μ ∈ ℤ.

Proof. This is again a local statement. We can therefore assume that Ω ⊂ ℝ^{n}. Furthermore, it is enough to prove the statement in the case h = 1, since the weight h does not enter into the condition on derivatives in the definition 50 of weighted spaces. Equation (51) and Proposition 4.9 show that we can also assume a = 0. Recall from Lemma 3.7 that the spaces H^{k}(Σ(Ω)) are defined using L^{2}(Σ(Ω)). In turn, L^{2}(Σ(Ω)) is defined using the volume element of a compatible metric. A typical compatible metric is r_{Ω}^{−2}g_{e}, where g_{e} is the Euclidean metric. Therefore the volume element on Σ(Ω) is r_{Ω}^{−n}dx, where dx is the Euclidean volume element. In particular, v ∈ L^{2}(Ω) ⇔ v ∈ r_{Ω}^{−n/2}L^{2}(Σ(Ω)). We notice next that r_{Ω}^{−t}(r_{Ω}∂)^{α}r_{Ω}^{t} − (r_{Ω}∂)^{α} is a linear combination with C^{∞}(Σ(Ω))–coefficients of monomials (r_{Ω}∂)^{β}, with |β| < |α|, by the second part of Lemma 4.14. From this observation we obtain

u ∈ K_{0}^{μ}(Ω) ⇔ (r_{Ω}∂)^{α}u ∈ L^{2}(Ω) for all |α| ≤ μ by Theorem 5.6
⇔ (r_{Ω}∂)^{α}u ∈ r_{Ω}^{−n/2}L^{2}(Σ(Ω)) for all |α| ≤ μ
⇔ (r_{Ω}∂)^{α}r_{Ω}^{n/2}u ∈ L^{2}(Σ(Ω)) for all |α| ≤ μ
⇔ u ∈ r_{Ω}^{−n/2}H^{μ}(Σ(Ω)).

This proves that K_{a}^{μ}(Ω) = r_{Ω}^{a−n/2}H^{μ}(Σ(Ω)) for μ ∈ ℤ_{+}. For μ ∈ ℤ_{−}, we observe that, for (𝔒, 𝔒_{0}, V) a Lie manifold with boundary in a manifold with corner 𝔐, the set of restrictions of distributions u ∈ H^{−μ}(𝔐) to 𝔒_{0} is the dual of the closure of C_{c}^{∞}(𝔒_{0}) in H^{−μ}(𝔐). Hence

K_{0}^{−μ}(Ω) :=K_{a}^{μ}^{∘} (Ω)^{∗} = (r_{Ω}^{−n/2}H^{∘}^{μ}(Σ(Ω)))^{∗} = r_{Ω}^{−n/2}H^{−μ}(Σ(Ω)).

The proof is concluded.

The identification given in Proposition 5.7 above allows to define weighted spaces on the boundary hK_{a}^{m}(∂Ω). We recall that the closure of a hyperface of a curvilinear, stratified polyhedral domain Ω need not be contained in any smooth n − 1 manifold. Consequently, we utilize the desingularization Σ(Ω). In the special case that Ω ⊂ ℝ^{n} is a (bounded) convex, stratified polyhedron that in addition has straight faces (i.e., each connected component D_{j}^{(l)} of Ω^{(l)} ∖ Ω^{(l−1)}, l = 1, …, n − 1 is contained in an affine space V_{j}^{(l)} of dimension l), for example an n-simplex, we can more simply define the spaces on the boundary as follows:

K_{a}^{μ}(D_{j}^{(n−1)}) = {u ∈ L_{loc}^{2}(D_{j}^{(n−1)}), r_{Ω}^{k− a}X_{1} … X_{k}u ∈ L^{2}(D_{j}^{(n−1)}), 0 ≤ k ≤ l} , for all choices of vector fields X_{j} in a basis of the linear space containing D_{j}^{(n−1)}. Then for any admissible weight h,

hK_{a}^{μ}(∂Ω) = {hu, u ∈ L_{loc}^{2}(∂Ω), u|_{D_{j}^{(n−1)}} ∈ K_{a}^{μ}(D_{j}^{(n−1)}), for all j}. (54)

In the general case of a curvilinear, stratified polyhedral domain, we exploit the structure of Lie manifold on Σ(Ω), following the notation of Proposition 5.7.

Definition 5.8. Let Ω be a bounded, curvilinear, stratified polyhedral domain. Then we define

hK_{a}^{μ}(∂Ω) := hr_{Ω}^{a−(n−1)/2}H^{μ}(∂′Σ(Ω)),

for any admissible weight h.

Note that on each hyperface, the natural weight is the distance to the boundary of that face, not the distance to the set of singular boundary points of that face. The spaces K_{−a}^{−μ}(∂Ω) are defined to be the duals of K_{a}^{μ}(∂Ω) with pivot L^{2}(∂Ω). For reasons that will be explained later, we do not have to restrict to functions with vanishing trace when studying weighted Sobolev spaces on the boundary. In particular, the usual difficulties that appear in the treatment of Sobolev spaces of fractional order on smooth, bounded domains [49], do not arise when studying the weighted Sobolev spaces on ∂Ω, and we can define the spaces K_{a}^{s}(∂Ω), with s ∉ ℤ, by complex interpolation. A similar attempt at defining K_{a}^{s}(Ω), with s ∈ ℤ + 1/2, would lead to the usual difficulties encountered in the case of smooth domain [49]. We next prove a trace theorem, generalizing the corresponding well-known result for smooth domains. Let C_{c}^{∞}(Ω) be the space of compactly supported functions on the open set Ω.

Theorem 5.9. The restriction C_{c}^{∞}(\overline{Ω}∖Ω^{(n−2)}) ∋ u → u|_{∂Ω} ∈ C_{c}^{∞}(∂Ω∖Ω^{(n−2)}) extends to a continuous, surjective map

T r : K_{a}^{μ}(Ω) → K_{a−1/2}^{μ−1/2}(∂Ω), μ ≥ 1.

Moreover, C_{c}^{∞}(Ω) is dense in the kernel of this map if μ = 1. Similarly, the normal derivative ∂_{ν} extends to a continuous, surjective map

∂_{ν} : K_{a}^{μ}(Ω) → K_{a−3/2}^{μ−3/2}(∂Ω), μ ≥ 1.

The result is a consequence of similar results for Lie manifolds contained in Theorems 3.4 and 3.7 of [1] recalled here in Proposition 3.9. For the normal derivative, we also use the fact that the rescaled normal vector r_{Ω}ν extends to a smooth vector field, first on the boundary of Σ(Ω), and then on the whole of Σ(Ω). The rescaled normal vector is then a unit normal vector for the boundary of Σ(Ω).

Proof. The map H^{μ}(Σ(Ω)) → H^{μ−1/2}(∂′Σ(Ω)), where we follow the notation of Proposition 5.7, is well defined, continuous, and surjective by Proposition 3.9. Proposition 5.7 then shows that the map

hK_{a}^{μ}(Ω) = hr_{Ω}^{a−n/2}H^{μ}(Σ(Ω)) → hr_{Ω}^{a−n/2}H^{μ−1/2}(∂Σ(Ω)) = hK_{a−1/2}^{μ−1/2}(∂Ω)

is also well defined, continuous, and surjective. The density of C_{c}^{∞}(Ω) in the subspace of elements in hK_{a}^{1}(Ω) with trace zero also follows from Proposition 3.9 and Proposition 5.7.

6 Proofs

In this section, we establish the main results of the paper, Theorems 1.1, 1.2, 1.3, using material from previous sections. We first discuss some results on the behavior of differential operators on the spaces hK_{a}^{m}(Ω).

6.1 Differential operators

We recall that the algebra Diff_{Ω}^{∞} is the natural algebra of differential operators on Ω associated to the Lie algebra of vector fields V(Ω), namely, it is generated as an algebra by X ∈ V(Ω) and ϕ ∈ C^{∞}(Σ(Ω)). (This algebra was used also in Equation (52) and in Subsection 3.3.)

Proposition 6.1. Let P be a differential operator of order m on a manifold M with smooth coefficients. Let Ω ⊂ M be a curvilinear, stratified polyhedral domain. Then P maps hK_{a}^{μ}(Ω) to hK_{a−m}^{μ−m}(Ω) continuously, for any admissible weight h and any μ ∈ ℤ. Moreover, the resulting family h^{−λ}P h^{λ} : K_{a}^{μ}(Ω) → K_{a−m}^{μ−m}(Ω) of bounded operators depends continuously on λ.

Before proceeding with the proof, we discuss a corollary, which will be relevant in showing that Theorems 1.2 and 1.3 hold. Following the notation of those theorems, below W_{μ}(Ω) represents the set of admissible weights h such that

the map P̃(u) := (P u, u|_{∂_{D}Ω}, D_{ν}^{P} u|_{∂_{N} Ω}) is an isomorphism {⨁_{j=1}^{N} hK_{1}^{μ+1}(Ω_{j}) ∩

hK_{1}^{1}(Ω); u^{+} = u^{−}, D_{ν}^{P} u^{+} = D_{ν}^{P} u^{−} on Γ} ≃ ⨁_{j=1}^{N} hK_{−1}^{μ−1}(Ω_{j})⊕hK_{1/2}^{μ+1/2}(∂_{D}Ω)⊕

hK_{−1/2}^{μ−1/2}(∂_{N}Ω).

Proposition 6.2. The set W_{μ}(Ω) is open.

Proof. This follows directly from Proposition 6.1. Indeed, the family P : ⨁_{j=1}^{N} hK_{1}^{μ+1}(Ω_{j})∩hK_{1}^{1}(Ω) → hK_{−1}^{μ−1}(Ω) is unitarily equivalent to h^{−1}P h : ⨁_{j=1}^{N} K_{1}^{μ+1}(Ω_{j}) ∩ K_{1}^{1}(Ω) → K_{−1}^{μ−1}(Ω). The result then follows since the set of invertible operators is open.

For the proof of Proposition 6.1, we observe that if Ω ⊂ ℝ^{n}, the principal symbol of (r_{Ω}∂)^{α} is (ıξ)^{α}. This result follows from the definition of the principal symbol in [2, 1] and from Corollary 5.4. (The reader can just assume σ((r_{Ω}∂)^{α}) = (ıξ)^{α} by definition.)

Corollary 6.3. Let P be a differential operator of order m on M with smooth coefficients. Then

(i) P_{0} := r_{Ω}^{m}P ∈ Diff_{Ω};

(ii) P is uniformly strongly elliptic if, and only if, r_{Ω}^{m}P is uniformly strongly elliptic in Diff_{Ω}^{m};

(iii) h^{λ}P h^{−λ} depends continuously on λ;

(iv) P maps hK_{a}^{μ}(Ω) → hK_{a−m}^{μ−m}(Ω) continuously.

Proof. The relation r_{Ω}^{m}P ∈ Diff_{Ω}^{m} was proved as part of Lemma 5.5. Strong ellipticity is a local property, so we can assume Ω ⊂ ℝ^{n}. The proof of Lemma 5.5 shows that P and r_{Ω}^{m}P have the same principal symbol. Therefore they are elliptic (or strongly elliptic) at the same time. For any X ∈ V and any defining function x of some hyperface at infinity of Σ(Ω), we have that x^{λ}Xx^{−λ} = X − λx^{−1}X(x). Since x^{−1}X(x) is a smooth function (as X is tangent to the face defined by x), we see that x^{λ}Xx^{−λ} ∈ Diff_{Ω}^{1} and depends continuously on λ, establishing (iii). It also shows, in particular, that Diff_{Ω}^{k} is conjugation invariant with respect to defining functions of hyperfaces at infinity (Equation (28)). We can therefore assume that h = 1. Since (Σ(Ω), V(Ω)) is a Lie manifold with boundary (Theorem 4.23) any P_{0} ∈ Diff_{Ω}^{k} maps H^{μ}(Σ(Ω)) → H^{μ−k}(Σ(Ω)) continuously. (This simple property, proved in [1], is an immediate consequence of the definitions.) The continuity of P = r_{Ω}^{−m}P_{0} : K_{a}^{μ}(Ω) → K_{a−m}^{μ−m}(Ω) then follows using also the fact that multiplication by r_{Ω}^{−m} defines an isometry K_{a}^{μ−m}(Ω) ≃ K_{a−m}^{μ−m}(Ω).

6.2 A weighted Hardy–Poincaré’s inequality

The stepping stones in the proof of our main result on the solvability of the mixed boundary value/interface problem (6), Theorem 1.2, consist of

(i) a Hardy–Poincaré type inequality (Theorem 6.4);

(ii) the regularity result for polyhedra (Theorem 1.1).

We address the Hardy-Poincaré inequality first and turn to the proof of the regularity result, which is more general and of independent interest in the next subsection. Let dx = dx_{1}dx_{2} … dx_{n} denote the standard volume element in ℝ^{n}. We continue to denote by Ω a curvilinear, stratified polyhedral domain satisfying hypotheses (3)–(5).

Theorem 6.4. Let Ω be a bounded, connected, curvilinear, stratified polyhedral domain Ω ⊂ M. Assume that ∂_{D}Ω ≠ ∅ and ∂_{N}Ω does not contain any two adjacent hyperfaces. Then there exists a constant κ_{Ω} > 0, depending only on the polyhedral structure of Ω, such that

∥u∥_{K_{1}^{0}(Ω)}^{2} := ∫_{Ω} \frac{|u(x)|^{2}}{η_{n−2}(x)^{2}} dx ≤ κ_{Ω} ∫_{Ω} |∇u(x)|^{2} dx,  (55)

for any function u ∈ H_{loc}^{1}(Ω) such that u|_{∂_{D}Ω} = 0.

Above, if u/η_{n−2} is not square integrable, the statement of the theorem is understood to mean that ∇u is not square integrable either. By Propositions 4.9 and 4.10, we can replace the distance to the singular set η_{n−2} with the more regular weight r_{Ω}. The proof proceeds by induction on the dimension n. We discuss first the case n = 2, 3. The case n = 2. In view of the local nature of the definition of a curvilinear, stratified polygonal domain, Definition 2.6, it will be sufficient to have the Hardy-Poincaré inequality in a sector. By abuse of notation, we shall write u(r, θ) := u(r cos θ, r sin θ) for a function u(x_{1}, x_{2}) expressed in polar coordinates. The proof of the following elementary lemma can be found in e.g. [62][Subsection 2.3.1]. See also [40].

Lemma 6.5. Let C = C_{R}(α, β) := {(r cos θ, r sin θ) ∈ ℝ^{2}, 0 < r < R, β < θ < α}, 0 < α − β < 2π. Then

∫_{C} \frac{|u|^{2}}{r^{2}} dx ≤ \frac{π^{2}}{(α − β)^{2}} ∫_{C} |∇u|^{2}dx

for any u ∈ H_{loc}^{1}(C) satisfying u(r, θ) = 0 if θ = β or θ = α. The same result holds if C is the disjoint union of domains C_{R}(α, β), for different values of R, α, and β.

From the Lemma above, we obtain the case n = 2 in Theorem 6.4, the first step in our induction proof. A detailed proof can be found e.g. in the papers [15, 56].

Lemma 6.6. Let Ω be a connected, curvilinear, stratified polygonal domain in a two dimensional manifold M. Assume that ∂_{D}Ω ≠ ∅ and ∂_{N}Ω does not contain any two adjacent sides of Ω. Fix an arbitrary metric g on M and let η_{0}(z) be the distance from z to the vertices of Ω. Then there exists a constant κ_{Ω} > 0 such that

∫_{Ω} \frac{|u(w)|^{2}}{η_{0}(w)^{2}} dz ≤ κ_{Ω} ∫_{Ω} |∇u(w)|^{2}dz

for any u ∈ H_{loc}^{1}(Ω) satisfying u = 0 on ∂_{D}Ω.

The case n = 3. The proof of Theorem 6.4 for n = 3 combines the methods used in the previous two Lemmata and the inequality for the case n = 2. We give a self-contained proof again, especially because the induction step in the general case is very similar. The general case n > 3 will be completed using Proposition 4.10.

Proof. Let us fix, for any p ∈ ∂Ω, a neighborhood V_{p} of p in M and a diffeomorphism ϕ_{p} : V_{p} → U = B^{3−l} × B^{l} as in Definition 2.8, where l = ℓ(p). We denote C := ϕ_{p}(V_{p} ∩ Ω). We shall use the notation ω_{p} introduced in that definition. By decreasing V_{p}, if necessary, we may assume that ϕ_{p} extends to a diffeomorphism defined in a neighborhood of \overline{V}_{p} in ℝ^{3}. Since η_{n−2} = η_{1} is the distance to the singular set Ω^{(1)} of Ω, we need only discuss two cases:

(a) l = ℓ(p) = 0, i.e., p is a true or artificial vertex;

(b) l = ℓ(p) = 1, i.e., p belongs to a true or artificial edge.

If l = 0, we denote by ψ_{0}(x′) the distance from a point x′ ∈ ω_{p} ⊂ S^{2} to the vertices of ω_{p} and let r_{p}(w) = ρψ_{0}(x′), if ϕ_{p}(w) = ρx′, where 0 < ρ and x′ ∈ ω_{p}. If l = 1, we let r_{p}(w) = r if ϕ_{p}(w) = (r cos θ, r sin θ, z), where 0 < r, 0 < θ < α, and z ∈ ℝ. (These definitions agree with the general definition of r_{Ω} given in (36) with r_{p} = r_{α} given in (35).) As before, the function η_{1}(x)/r_{p}(x) is bounded for any p, provided that we choose the neighborhoods V_{p} small enough, uniformly in p. Below, we shall write u(x) instead of u(ϕ_{p}^{−1}(x)), by abuse of notation. If l = 1, C = C′ × (−1, 1), so that we exploit the Hardy-Poincaré inequality in a sector of Lemma 6.5. In fact

∫_{V_{p}∩Ω} \frac{|u(w)|^{2}}{η_{1}(w)^{2}} dw = C ∫_{Ω∩V_{p}} \frac{|u(x)|^{2}}{r^{2}} |\frac{∂w}{∂x} | dx ≤ C ∫_{C} \frac{|u(x)|^{2}}{r^{2}} dx.  (56)

so that we obtain

∫_{C} \frac{|u(x)|^{2}}{r^{2}} dx = ∫_{−1}^{1} (∫_{C′} \frac{|u(x)|^{2}}{r^{2}} dx_{1}dx_{2}) dx_{3}

≤ ∫_{−1}^{1} (∫_{C′} |∇_{(x_{1},x_{2})}u(x)|^{2} dx_{1}dx_{2}) dx_{3} ≤ ∫_{−1}^{1} (∫_{C′} |∇u(x)|^{2} dx_{1}dx_{2}) dx_{3}

≤ C ∫_{V_{p}∩Ω} |∇u(w)|^{2} dw. (57)

We perform a similar calculation on V_{p} ∩ Ω when l = 0, using spherical coordinates instead. Recall that C = ϕ_{p}(V_{p} ∩ Ω) = {ρx′, 0 < ρ < 1, x′ ∈ ω_{p}}, hence following (56) and using that Cη_{1}(x) ≥ ρψ_{0}(x) The inequality

∫_{V_{p}∩Ω} \frac{|u(w)|^{2}}{η_{1}(w)^{2}} dw ≤ C ∫_{C} \frac{|u(x)|^{2}}{ρ^{2}ψ_{0}(x′)^{2}} dx, x = ρx′, |x′| = 1,  (58)

Next, we observe that ∇u(ρx′) = ρ^{−1}∇′u(ρx′)+∂_{ρ}u(ρx′), with ∇′ the gradient of a function defined on ω_{p}, so that |∇′u(ρx′)|^{2} ≤ ρ^{2}|∇u(ρx′)|, which gives

∫_{C} \frac{|u(x)|^{2}}{ρ^{2}ψ_{0}(x)^{2}} dx = ∫_{0}^{1} (∫_{ω_{p}} \frac{|u(ρx′)|^{2}}{ψ_{0}^{2}} dx′) dρ

≤ C ∫_{0}^{1} (∫_{ω_{p}} |∇′u(ρx′)|^{2}dx′) dρ ≤ C ∫_{0}^{1} (∫_{ω_{p}} ρ^{2}|∇u(ρx′)|^{2}dx′) dρ

≤ C ∫_{V_{p}∩Ω} |∇u(w)|^{2} dw. (59)

We can rewrite the above inequalities simply as

∫_{V_{p}∩Ω} \frac{|u(w)|^{2}}{η_{1}(w)^{2}} dz ≤ C_{p} ∫_{V_{p}∩Ω} |∇u(w)|^{2} dz ≤ C_{p} ∫_{Ω} |∇u(w)|^{2} dw,  (60)

where the constant C_{p} depends on the point p ∈ Ω^{(1)} but not on u. To conclude the proof, as before we cover the singular set Ω^{(1)} with finitely many sets V_{p} = V_{p_{k}}. Let C_{0} > η_{1}^{−2} outside the union of the sets V_{p_{k}}. Let κ_{Ω} = C_{0}C_{Ω} + ∑ C_{p_{k}}, where C_{Ω} is the standard Poincaré inequality constant for the domain Ω. We add all inequalities (60) for p = p_{k} and combine it with the Poincaré inequality to get

∫_{Ω} \frac{|u(w)|^{2}}{η_{1}^{2}(w)} dw ≤ κ_{Ω} ∫_{Ω} |∇u(w)|^{2}dw.  (61)

The proof of Theorem 6.4 is now complete for n = 3.

The general case n > 3. To conclude the proof of theorem 6.4, we need only establish the induction step. The induction step follows very closely the proof of the case n = 3. The only delicate point is showing that the ratio η_{n−2}(x)/r_{α}(x) is bounded on \overline{Ω}, where η_{n−2} is the distance to the singular set Ω^{(n−2)} of Ω and r_{α} is as in Equation 35. This fact was established in Proposition 4.9. We conclude with an immediate corollary of Theorem 6.4, which will be used in the proof of Theorem 1.2

Corollary 6.7. There exists a constant κ′_{Ω} > 0, depending only on Ω, such that

\frac{1}{κ′_{Ω}} ∥u∥_{K_{1}^{1}(Ω)}^{2} ≤ ∫_{Ω} |∇u(x)|^{2} dx ,

for any function u ∈ H_{loc}^{1}(Ω) such that u|_{∂_{D}Ω} = 0, if ∂_{D}Ω ≠ ∅ and ∂_{N}Ω does not contain any two adjacent hyperfaces.

6.3 Proofs of the main results

In this subsection, we finally tackle the proofs of the main results of the paper. We first show how the proof of the regularity property for the mixed boundary value/interface problem (6), Theorem 1.1 can be obtained from the results of [1] and the theory developed in Section 4. The following result was proved in [1].

Theorem 6.8. Let (𝔐, V) be a Lie manifold with boundary and P_{0} := r_{Ω}^{2}P ∈ Diff^{2}(𝔐) be a second order, uniformly strongly elliptic operator. Let h be an admissible weight and u ∈ hH^{1}(𝔐) be such that P u ∈ hH^{μ−1}(𝔐) and u|_{∂𝔐} ∈ hH^{μ+1/2}(∂𝔐), μ ∈ ℤ_{+}. Then u ∈ hH^{μ+1}(𝔐) and

∥u∥_{hH^{μ+1}(𝔐)} ≤ C(∥P u∥_{hH^{μ−1}(𝔐)} + ∥u∥_{hH^{0}(𝔐)} + ∥u|_{∂Ω}∥_{hH^{μ+1/2}(∂𝔐)}). (62)

For mixed boundary value/interface problems we need the following extension of this theorem, which is proved exactly in the same way.

Theorem 6.9. Let (𝔐, V) be a Lie manifold with boundary and P_{0} = r_{Ω}^{2}P ∈ Diff^{2}(𝔐) be a second order, uniformly strongly elliptic operator with jump discontinuities on sub Lie manifolds of 𝔐 that partition it into subsets 𝔐_{j}. Assume that ∂𝔐 = ∂_{D}𝔐 ∪ ∂_{N}𝔐 is a disjoint decomposition into open, disjoint subsets. Let h be an admissible weight and u ∈ hH^{1}(𝔐) be such that P u ∈ hH^{μ−1}(𝔐_{j}) and u|_{∂𝔐} ∈ hH^{μ+1/2}(∂𝔐), μ ∈ ℤ_{+}. Then u ∈ hH^{μ+1}(𝔐_{j}) and

∥u∥_{hH^{μ+1}(𝔐_{j})} + ∥u∥_{hH^{1}(𝔐)} ≤ C( ∑_{k} ∥P u∥_{hH^{μ−1}(𝔐_{k})} + ∥u∥_{hH^{0}(𝔐)}

+ ∥u|_{∂Ω}∥_{hH^{μ+1/2}(∂_{D}𝔐)} + ∥D_{ν}^{P} u|_{∂Ω}∥_{hH^{μ−1/2}(∂_{N} 𝔐)}).

Theorem 1.1 then follows by applying the above theorem to P_{0} := r_{Ω}^{2}P, which is strongly elliptic by Corollary 6.3(ii), and using the identifications of Proposition 5.7 and Definition 5.8. We now prove Theorem 1.2 assuming the results stated in the previous subsection. The proof of Theorem 1.3 is completely similar.

Remark 6.10. In the statement of Theorems 1.2 and 1.3, the spaces K_{1}^{μ+1}(Ω_{j}) are defined intrinsically, without reference to Ω. However, the interface Γ is assumed smooth for well-posedness in this paper (more general conditions on Γ were for example considered in [48]) and the points where Γ intersects ∂Ω, necessarily transversely, are included in the singular sets Ω^{(n−2)} of Ω; consequently, r_{Ω} is equivalent to r_{Ω_{j}} in each Ω_{j}.

Proof. We first notice that Theorem 5.9 allows us to reduce the proof to the case g_{D} = 0.

We continue to denote with W_{μ}(Ω) the set of weights such that the operator P̃, defined below, is an isomorphism

P̃ := (P u, u|_{∂_{D}Ω}, D_{ν}^{P} u|_{∂_{N} Ω}) :

{u ∈ ⨁_{j=1}^{N} hK_{1}^{μ+1}(Ω_{j}) ∩ hK_{1}^{1}(Ω); u|_{∂_{D}Ω} = 0, u^{+} = u^{−}, D_{ν}^{P} u^{+} = D_{ν}^{P} u^{−} on Γ}

→ ⨁_{j=1}^{N} hK_{−1}^{μ−1}(Ω_{j}) ⊕ hK_{−1/2}^{μ−1/2}(∂_{N}Ω), (63)

which is an open set by Proposition 6.2. Therefore, it is enough to show that 1 ∈ W_{μ}(Ω) to complete the proof. For solvability, we consider the case μ = 0. For μ = 0, the problem (6) is interpreted in the weak sense (11), using that K_{1}^{1}(Ω) ⊂ H^{1}(Ω). More precisely, we let

H := {u ∈ K_{1}^{1}(Ω), u = 0 on ∂_{D}Ω},  (64)

and we define the weak solution u of Equation (11) with g_{D} = 0 as the unique u ∈ K_{1}^{1}(Ω) satisfying u = 0 on ∂_{D}Ω in trace sense and

B_{P} (u, v) = Φ(v) for all v ∈ H,  (65)

where the element Φ ∈ H^{∗} is defined by Φ(u) = ∫_{Ω} fu dx + ∫_{∂_{N} Ω} g_{N}u dS(x),

this last integral being the pairing between K_{1/2}^{1/2}(∂Ω) and K_{−1/2}^{−1/2}(∂Ω). Here, we have employed the trace property, Theorem 5.9. We will establish the existence and uniqueness of u by using the Lax-Milgram Lemma and coercive estimates for P in weighted Sobolev spaces, which in turn follow from the (uniform) strong ellipticity of P and the Hardy-Poincaré inequality of Theorem 6.4. This result gives the first step of the proof, that is, 1 ∈ W_{0}(Ω). We refer to [26] for the version of the Lax–Milgram lemma needed in this proof, where P contains lower-order terms. Indeed, the sesquilinear form B is continuous on H × H by Proposition 6.1. Furthermore, assumptions 8 on the coefficients A_{jk}, B_{j}, and C of the operator P, together with Corollary 6.7 imply the following inequality for the real part of B(u, v):

Re(P u, u) = ∫_{Ω} (Re ∑_{j,k=1}^{n} A_{jk}∂_{k}u\overline{∂_{j}u}) dx + ((2C − ∑_{j} ∂_{j}B_{j})u, u)/2

≥ ϵ ∑_{j=1}^{n} ∥∂_{j}u∥^{2} ≥ ϵ∥u∥_{K_{1}^{1}(Ω)}^{2} =: ϵ∥u∥_{K_{1}^{1}(Ω)}^{2}, (66)

which shows that B is strictly coercive on H. The assumptions of the Lax-Milgram lemma are therefore satisfied, Hence P : H → H^{∗} is an isomorphism (i.e., P is continuous with continuous inverse), proving that 1 ∈ W_{0}(Ω).

We next consider the case μ ≥ 1 and prove that W_{0}(Ω) ⊂ W_{μ}(Ω) for any μ ∈ ℤ_{+}, so that, in particular, 1 ∈ W_{μ}(Ω). We pick h ∈ W_{0}(Ω) and observe that by the regularity theorem, Theorem 1.1, the map P̃ of Equation 63 above is surjective. Since this map is also continuous (Proposition 6.1) and injective (because h ∈ W_{0}(Ω)), it is an isomorphism by the open mapping theorem. This observation shows that W_{0}(Ω) ⊂ W_{μ}(Ω), for any μ ∈ ℤ_{+}. Since we have already proved that W_{μ}(Ω) is open, the proof is complete.

Remark 6.11. It seems that it would be more natural to work in the framework of stratified spaces than in the framework of polyhedral domains. For example, if we consider a smooth, bounded domain Ω ⊂ ℝ^{n} and a submanifold X ⊂ ∂Ω of lower dimension, then we can consider η_{n−2}(x) to be the distance from x to X. Then Theorem 1.2 remains true, with essentially the same proof, by taking Ω^{(n−2)} := X in this case.

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Constantin Băcuţă University of Delaware Department of Mathematical Sciences 501 Ewing Hall Newark, DE 19716-2553 USA bacuta@math.udel.edu Anna L. Mazzucato Pennsylvania State University Department of Mathematics University Park, PA 16802 USA alm24@psu.edu

Victor Nistor Pennsylvania State University Department of Mathematics University Park, PA 16802 USA and Inst. Math. Romanian Acad. PO BOX 1-764, 014700 Bucharest Romania nistor@math.psu.edu Ludmil T. Zikatanov Pennsylvania State University Department of Mathematics University Park, PA 16802 USA ltz@math.psu.edu 746

Documenta Mathematica 15 (2010)