j. differential geometry 64 (2003) 359-423

FROM CONSTANT MEAN CURVATURE HYPERSURFACES TO THE GRADIENT THEORY

OF PHASE TRANSITIONS

FRANK PACARD & MANUEL RITORÉ

Abstract

Given a nondegenerate minimal hypersurface Σ in a Riemannian manifold, we prove that, for all ε small enough there exists u_{ε}, a critical point of the Allen-Cahn energy E_{ε}(u) = ε^{2} ∫ |∇u|^{2} + ∫ (1 − u^{2})^{2}, whose nodal set converges to Σ as ε tends to 0. Moreover, if Σ is a volume nondegenerate constant mean curvature hypersurface, then the same conclusion holds with the function u_{ε} being a critical point of E_{ε} under some volume constraint.

1. Introduction

Let Ω ⊂ ℝ^{n+1}, n ⩾ 1, be an open bounded set with smooth boundary ∂Ω. For any ε > 0 and any function u : Ω → ℝ such that u ∈ H^{1}(Ω), we consider the energy

E_{ε}(u) := ε^{2} ∫_{Ω} |∇u|^{2} dx + ∫_{Ω}(1 − u^{2})^{2} dx,

being understood that E_{ε}(u) = ∞ if u ∉ L^{4}(Ω). We also consider the constraint

V (u) := ∫_{Ω} u dx.

Given c_{0} ∈ (−1, 1), we are interested in the critical points of E_{ε} subject to the constraint V (u) = c_{0} |Ω|, where |Ω| denotes the volume

Received 11/28/2002.

of Ω. Any critical point of this variational problem is a solution to

\begin{cases}ε^{2} Δu + 2 (u − u^{3}) = ε λ, & in Ω \\ ∂_{ν}u = 0, & on ∂Ω, \end{cases} (1.1)

where ν denotes a unit vector field normal to ∂Ω and where ε λ ∈ ℝ corresponds to the Lagrange multiplier associated to the constraint V (u) = c_{0} |Ω|. One can also ignore the volume constraint, in which case a critical point would satisfy Equation (1.1) with λ = 0.

Since classical methods of the calculus of variation apply, there is no difficulty in finding minimizers of E_{ε}. The real issue is the study of the asymptotic behavior of the minimizers (or more generally of the critical points) of E_{ε} as the parameter ε tends to 0. There has been a number of important work on this question over the last two decades and the basic result can be described as follows: Assume that (ε_{k})_{k⩾0} tends to 0 and let (u_{k})_{k⩾0} be a sequence of minimizers of E_{ε_{k}} under the constraint V (u) = c_{0} |Ω|. Then, up to a subsequence, one can assume that (|u_{k}|)_{k⩾0} converges a.e. to the constant function 1. In the definition of the energy E_{ε}, the role of the term

∫_{Ω}(1 − u^{2})^{2} dx,

is precisely to force the sequence of functions (|u_{k}|)_{k⩾0} to converge to 1 when the parameter ε_{k} tends to 0. Extracting subsequences if this is necessary, we can define Ω^{+} (resp. Ω^{−}) to be the set of points where u_{k} converges to +1 (resp. −1). The subsets Ω^{±} are not arbitrary since the constraint V (u_{k}) = c_{0} |Ω| forces Ω^{±} ⊂ Ω to satisfy

|Ω^{+}| − |Ω^{−}| = c_{0} |Ω|.

Now, the role of the Dirichlet integral

ε^{2} ∫_{Ω} |∇u|^{2} dx,

in the definition of E_{ε} forces the interface between the subsets Ω^{+} and Ω^{−} to be “as small as possible”, since this is where the gradient of the function u_{k} will concentrate when ε_{k} tends to 0. More precisely

N := ∂Ω^{+} ∩ Ω = ∂Ω^{−} ∩ Ω,

can be shown to be a minimizer of the isoperimetric problem: Minimize amongst all domains D ⊂ Ω the n-dimensional Hausdorff measure H^{n}(∂D) of the boundary ∂D subject to the volume constraint

|D| = \frac{1 + c_{0}}{2} |Ω|.

We refer to [16], [15], [12], [3], [4], [21] for more precise statements. From a purely analytic point of view, N can be understood as the limit of the nodal sets of the functions u_{k}, as k tends to +∞.

2. Statement of the problem

It is interesting to generalize the above problem first by considering instead of Ω ⊂ ℝ^{n+1}, any compact Riemannian manifold with or without smooth boundary and also by replacing the nonlinearity (1 − u^{2})^{2} by a more general one.

Hence, in this paper, we consider (M, g) to be a (n + 1)-dimensional compact Riemannian manifold with or without smooth boundary. In the case where ∂M, the boundary of M, is not empty, we can assume without loss of generality that M is a subdomain of a larger Riemannian manifold (M̃, g̃), with g̃_{|M} = g. In particular, ∂M is a smooth hypersurface of M̃.

Let W : ℝ → ℝ be a smooth function which is positive away from u = ±1. We assume that

W (±1) = 0, (2.1)

so that the infimum of W is achieved at the points u = ±1. Further assume that these points are nondegenerate critical points of W. In other words

W′′(±1) > 0. (2.2)

For any ε > 0 and any function u : M ⟶ ℝ, such that u ∈ H^{1}(M), we define the energy

E_{ε}(u) := ε^{2} ∫_{M} |∇u|_{g}^{2} dv_{g} + ∫_{M} W (u) dv_{g}, (2.3)

where ∇ denotes the gradient and dv_{g} the volume form on M associated to the Riemannian metric g. As usual, we agree that E_{ε}(u) = ∞ when W (u) ∉ L^{1}(M). We also define the volume constraint

V (u) := ∫_{M} u dv_{g}. (2.4)

Granted the above definitions, there are two closely related variational problems we can consider:

1. We can consider the critical points of the energy u ⟶ E_{ε}(u), which are solutions of

−ε^{2} Δ_{g}u + \frac{1}{2} W′(u) = 0, (2.5)

in M, where Δ_{g} is the Laplace-Beltrami operator in M. Moreover, if ∂M ≠ ∅ then the additional condition

∂_{ν_{∂M}} u = 0, (2.6)

must hold on ∂M, where ν_{∂M} denotes the unit vector field normal to ∂M. This problem is related to the Allen-Cahn equation [2] and it is well-known that, as ε tends to 0, the interfaces (i.e., the nodal sets of the solutions of (2.5)) converge to minimal hypersurfaces. Concerning this variational problem, the question we would like to address in this paper is the following:

(P-1) Assume that N ⊂ M is a minimal hypersurface. Does N appear as the limit, as the parameter ε tends to 0, of the nodal sets of a sequence of critical points of E_{ε}?

2. Given c_{0} ∈ (−1, 1), we can consider the critical points of the energy u ⟶ E_{ε}(u) under the constraint V (u) = c_{0} |M|, where |M| denotes the volume of M. This time, such a critical point u is solution of

−ε^{2} Δ_{g}u + \frac{1}{2} W′(u) = ε λ, (2.7)

in M, where ε λ ∈ ℝ corresponds to the Lagrange multiplier associated to the constraint V (u) = c_{0} |M|. Moreover, u satisfies (2.6) on ∂M if ∂M ≠ ∅. According to [19], [8], the energy E_{ε} corresponds to the total energy of a fluid within the Wan der Waals-Cahn-Hilliard theory of phase transitions. The Lagrange multiplier ε λ, which appears in (2.7), is known in the physics literature as the chemical potential of the density configuration u. Now, the question we would like to address becomes:

(P-2) Assume that N ⊂ M is a constant mean curvature hypersurface. Does N appear as the limit, as the parameter ε tends to 0, of the nodal sets of a se- quence of critical points of E_{ε} subject to the constraint V (u) = c_{0} |M|?

Before we proceed, let us observe that, in both problems, we are not only looking for minimizers of E_{ε} but more generally for critical points.

Remark 2.1. If the infimum of the function W is achieved at exactly two points u_{±}, there is no loss of generality in considering that u_{±} = ±1 since we can always reduce to this case by considering u ↦ W (au + b) where a and b are chosen appropriately.

3. Definitions and Preliminaries

3.1 Admissible hypersurfaces in M

Obviously if N is the nodal set of some function u which is defined in M and if 0 is a regular value of u then M − N is the union of

M^{+}(N) := u^{−1}((0, +∞))  and  M^{−}(N) := u^{−1}((−∞, 0)). (3.1)

We shall associate to N the unit normal vector field which points into M^{+}(N). In the case where M has a boundary, it may happen that N also has a boundary ∂N ⊂ ∂M. In this case, if N is the nodal set of the function u and if in addition the function u has 0 Neumann boundary condition on ∂M, then for all p ∈ ∂N ⊂ ∂M, the normal vector to N at p and the normal vector to ∂M at p are orthogonal. This later condition is standard in the study of minimal and constant mean curvature hypersurfaces. Indeed, it is well-known that smooth hypersurfaces N which are stationary points of the area functional (possibly with a volume constraint) and have a boundary ∂N ⊂ ∂M, satisfy the later orthogonality condition. This motivates the following:

Definition 3.1. A smooth embedded hypersurface N ⊂ M (not necessarily connected) is admissible if N is the nodal set of a smooth function u for which 0 is a regular value of u and which, in the case where M has a boundary, has 0 Neumann boundary condition.

A hypersurface N ⊂ M which separates M into two regions M^{±}(N), and which meets ∂M orthogonally in the case where N has a nonempty boundary, is easily shown to be admissible by using partitions of unity. Observe that N may have many connected components.

3.2 The Jacobi operator

Before we introduce our next definition, we recall a few basic facts about the study of constant mean curvature hypersurfaces N in a Riemannian manifold (M, g). To begin with, let us recall that the Jacobi operator, that is the linearized mean curvature operator about N, is given by

L_{N} := Δ_{N} + |A_{N}|^{2} + Ric_{g}(ν_{N}, ν_{N}), (3.2)

where Δ_{N} is the Laplace-Beltrami operator on N, |A_{N}|^{2} denotes the norm of the second fundamental form of N, Ric_{g} is the Ricci tensor of M and ν_{N} is a unit normal to N.

Given any (smooth) small function w on N, we can consider the hypersurface N(w) which is the normal graph on N of the function w (the image of N by the map p ∈ N ↦ exp_{p}(w(p)ν_{N}(p))). If H(w) denotes the mean curvature of N(w), defined as the arithmetic mean of the principal curvatures, then the linear operator L_{N} is the differential of w ↦ nH(w) at w ≡ 0.

When ∂N is empty, solutions of the homogeneous problem

L_{N} w = 0, (3.3)

on N are called Jacobi fields. When ∂N is not empty, we further assume that N meets ∂M orthogonally, then Jacobi fields are the solutions of L_{N} w = 0 in N which satisfy the boundary condition

B_{N} w := ∂_{ν_{∂M}} w + A_{∂M}(ν_{N}, ν_{N}) w = 0, (3.4)

on ∂N, where A_{∂M} is the second fundamental form of ∂M in M̃. Equation (3.4) has its origin in the requirement that all the hypersurfaces we are looking at meet ∂M orthogonally and this should be true for the hypersurfaces generated by the flow associated to a vector field X satisfying X = w ν_{N} on N.

Minimal hypersurfaces are critical points of the area functional while constant mean curvature hypersurfaces are critical points of the area functional with respect to deformations that keep constant the volume enclosed by the hypersurface. Consider a deformation of the hypersurface N by the flow generated by a vector field X. The second variation formula for the area functional is then given by

X ⟶ − ∫_{N} w L_{N}w da_{g} + ∫_{∂N} w B_{N} w ds_{g},

where the function w := g(ν_{N}, X). If one considers a deformation which is volume preserving up to first order, then the function w also has to satisfy

∫_{N} w da_{g} = 0.

We refer to [5] or [18] for a derivation of the second variation of the area functional in a Riemannian manifold. Here da_{g} and ds_{g} are the volume forms on N and ∂N which are induced by the metric g.

3.3 Nondegeneracy

The previous definitions being understood, we can now give the notions of nondegeneracy which are associated to the two problems we are interested in. To begin with let us define the notion of nondegenerate minimal hypersurface:

Definition 3.2. An admissible minimal hypersurface N is said to be nondegenerate if there are no nontrivial solutions w ∈ C^{2,α}(N) of

L_{N} w = 0,

in N, with B_{N} w = 0 on ∂N if N has a boundary.

The notion of nondegeneracy for minimal hypersurfaces is standard. Consider the Jacobi operator

L_{N} : [C^{2,α}(N)]_{0} ⟶ C^{0,α}(N), (3.5)

where the subscript 0 is meant to point out that functions in [C^{2,α}(N)]_{0} satisfy B_{N}w = 0 on ∂N when this latter is not empty. Nondegeneracy is equivalent to the fact that the operator L_{N} is injective. This operator being self-adjoint and elliptic, nondegeneracy is also equivalent to the invertibility of the operator L_{N} defined in (3.5). On a more geometric point of view, if N is a nondegenerate minimal hypersurface, the implicit function theorem ensures that it is possible to find a hypersurface Ñ which is close to N and whose mean curvature H̃ is prescribed, close to the mean curvature H of N.

We will also need the notion of volume-nondegenerate constant mean curvature hypersurface:

Definition 3.3. An admissible constant mean curvature hypersurface N is said to be volume-nondegenerate if there are no nontrivial solutions (w, c) ∈ C^{2,α}(N) × ℝ of

L_{N} w + c = 0,  and  ∫_{N} w dv_{g} = 0,

in N, with B_{N} w = 0 on ∂N if N has a boundary.

The notion of volume-nondegeneracy is less standard and perhaps requires some explanation. This time, we consider the extended-operator

\begin{aligned}L_{N} & : & [C^{2,α}(N)]_{0} × ℝ & ⟶ & C^{0,α}(N) × ℝ \\ & & (w, c) & ⟼ & (L_{N}w + c, ∫_{N} w da_{g}) . \end{aligned} (3.6)

Thus, volume-nondegeneracy is equivalent to the fact that the operator L_{N} is injective. Observe that L_{N} is self-adjoint with respect to the scalar product

⟨(v, c), (w, d)⟩ := ∫_{N} v w da_{g} + c d,

in L^{2}(N) × ℝ. The operator L_{N} being clearly elliptic, volume-nondege-neracy is also equivalent to the invertibility of the operator L_{N} defined in (3.6). From a geometric point of view, if N is a constant mean curvature volume-nondegenerate hypersurface, the implicit function theorem ensures that it is possible to find a hypersurface Ñ which is close to N, whose mean curvature H̃ is, up to a constant function, prescribed close to H the mean curvature of N and such that the volume enclosed by this hypersurface M^{+}(Ñ) is prescribed close to M^{+}(N), the volume enclosed by N. Hence, it is possible to prescribe the volume enclosed by Ñ and, up to a constant function, the mean curvature of Ñ.

4. Statement of the result

The previous definitions being understood, we can now state the results we have obtained concerning both (P-1) and (P-2).

We have the:

Theorem 4.1. Assume that N ⊂ M is an admissible nondegenerate minimal hypersurface. Then, there exists ε_{0} > 0 and for all ε ∈ (0, ε_{0}) there exists u_{ε}, critical point of u ⟶ E_{ε}(u), such that u_{ε} converges uniformly to 1 on compact subsets of M^{+}(N) (resp. to −1 on compact subsets of M^{−}(N)).

Let us mention the work of M. Kowalczyk [11] where a similar result is obtained when M is a two dimensional domain of ℝ^{2} and N is a line segment.

We will also prove the:

Theorem 4.2. Assume that N ⊂ M is an admissible volumenondegenerate constant mean curvature hypersurface. Then, there exists ε_{0} > 0 and for all ε ∈ (0, ε_{0}) there exists u_{ε}, critical point of u ⟶ E_{ε}(u) under the constraint V (u) = |M^{+}(N)| − |M^{−}(N)|, such that u_{ε} converges uniformly to 1 on compact subsets of M^{+}(N) (resp. to −1 on compact subsets of M^{−}(N)).

It is in general extremely hard to check whether a given minimal hypersurface (resp. constant mean curvature hypersurface) is nondegenerate (resp. volume-nondegenerate). Hopefully, first observe that both nondegeneracy and volume-nondegeneracy are “open conditions”, namely are stable under small perturbation of the metric. Moreover, in [23], B. White has proved that minimal hypersurfaces are nondegenerate for a generic choice of the metric. It follows from similar arguments that volume-nondegeneracy also holds for a generic choice of the metric.

The solutions constructed in Theorem 4.2 are solutions of

−ε^{2}Δu_{ε} + \frac{1}{2} W′(u_{ε}) = ε λ_{ε}.

As a byproduct of our construction, we obtain a precise expansion of u_{ε} in terms of ε. We also get the expansion of the Lagrange multiplier λ_{ε}

λ_{ε} = \frac{1}{2} c_{⋆} n H_{N} + O(ε),

where H_{N} is the mean curvature of the limit interface N and where the constant c_{⋆} is given by

c_{⋆} := ∫_{−1}^{+1} \sqrt{W (s)} ds.

Finally, in both problems, the expansion of the energy E_{ε}(u_{ε}) of the solutions we construct, is given by

E_{ε}(u_{ε}) = 2 ε c_{⋆} |N| + O(ε^{2}).

where |N| is the volume of the interface N. These expansions agree with the expansions which have already been obtained in [13] in the case where u_{ε} are minimizers of E_{ε} subject to the constraint V = |M^{+}(N)|− |M^{−}(N)|.

Unfortunately, in many interesting cases and despite the genericity of these notions, minimal hypersurfaces are degenerate and constant mean curvature hypersurfaces are volume-degenerate. This is for example the case when there is a nontrivial group of isometries acting on M. It is well-known that any (ϕ_{t})_{t∈(−1,1)} smooth one-parameter group of isometries of M (ϕ_{s}∘ϕ_{t} = ϕ_{s+t}) gives rise to a Jacobi field on N (when M has a nonempty boundary, we ask that these isometries preserve globally ∂M). Actually, the Jacobi field w is explicitly given by w := g(ν_{N}, X), where ν_{N} is the normal vector field to N and where X := ∂_{t}ϕ_{t}_{|t=0} is the Killing field corresponding to the one-parameter group of isometries {ϕ_{t}}_{t∈(−1,1)}. Observe that the isometries ϕ_{t} preserve the volume of the regions M^{±}(N). Therefore, it follows from the first variation of volume that the Jacobi field w has mean zero on N. In particular, w is a nontrivial solution of L_{N}w = 0 (resp. (w, 0) is a nontrivial solution of L_{N}(w, 0) = 0) and the hypersurface N is degenerate (resp. volumedegenerate).

In some cases, it is possible to reduce to a nondegenerate (or volumenondegenerate) problem by working in the space of functions and hypersurfaces which are equivariant with respect to the action of some finite group of symmetries. If this can be done, then the above theorems apply mutatis mutandis. We give here a short list of examples.

1. Consider M = S^{n+1} the unit (n + 1)-dimensional sphere with the standard metric and N = S^{n}(r) the meridian at height \sqrt{1 − r^{2}}. The hypersurface N has constant mean curvature and is volumedegenerate since there are nontrivial Jacobi fields w_{i}(x) = x · e_{i}, for i = 1, …, n coming from the action of the orthogonal group. Here e_{1}, …, e_{n+1} is an orthonormal basis of ℝ^{n+1}. However, one may work with hypersurfaces and functions which are invariant under the action of the n hyperplanar symmetries

I_{i} : (x_{1}, …, x_{i}, …, x_{n+1}) ⟶ (x_{1}, …, −x_{i}, …, x_{n+1})

for i = 1, …, n. Namely, hypersurfaces Ñ ⊂ S^{n+1} such that I_{i}(Ñ) = Ñ and functions u : S^{n+1} ⟶ ℝ such that u ∘ I_{i} = u. Since none of the Jacobi fields is invariant under all the symmetries I_{i}, our construction applies and the conclusion of Theorem 4.2 is still valid. Moreover, when r = 1, the equator is a minimal hypersurface and Theorem 4.1 is also valid.

2. Consider M = B^{n+1} the unit ball of ℝ^{n+1} endowed with the induced metric and N is a spherical cap. This example can be dealt like the previous one and the result of Theorem 4.2 holds. Moreover, when N is the horizontal hyperplane, Theorem 4.1 also holds.

3. Consider a flat torus T^{n+1} and let N ⊂ T^{n+1} be a pair of parallel hyperplanes. For the sake of simplicity, assume that T^{n+1} = ℝ^{n+1}/ℤ^{n+1} and is identified with [−\frac{1}{2}, \frac{1}{2}]^{n+1}. Finally, assume that N is given by x_{n+1} = ±α for some fixed α ∈ (0, 1/2). The action of vertical translations induce a nontrivial Jacobi field. Again, one may work with hypersurfaces and functions which are invariant under the action of the n + 1 hyperplanar symmetry

I_{n+1} : (x_{1}, …, x_{i}, …, x_{n+1}) ⟶ (x_{1}, …, x_{n}, −x_{n+1}),

and reduce to a volume-nondegenerate problem to show that the conclusion of Theorem 4.1 and Theorem 4.2 are valid.

5. Comments

We state here a number of comments, open problems and directions for further investigations:

1. In [14], A. Malchiodi and M. Montenegro have constructed solutions of

ε^{2} Δu − u + u^{p} = 0, (5.1)

which are defined on a 2 dimensional domain and which have 0 Neumann boundary condition. These solutions have the property that they concentrate along the boundary of the domain and they can be obtained for ε belonging to some sequence of intervals which converge to 0. Behind this result, lies a very interesting bifurcation phenomena which somehow prevents the construction to work for all ε close enough to 0. The analysis of A. Malchiodi and M. Montenegro relies on the precise estimate of the least eigenvalue of the linearized operator about an approximate solution, and this forces them to work in 2 dimensional domains. It would be very interesting to construct solutions of (5.1) which concentrate along a geodesic of a two dimensional manifold and also to extend their result to higher dimensional domains.

2. As already mentionned, M. Kowalczyk [11] has obtained a similar result when M is a two dimensional domain of ℝ^{2} and N is a line segment. Our result can be also be compared to the result of C.H. Taubes [22] where solutions of the Seiberg-Witten equation concentrating along holomorphic curves are constructed, though our analysis is completely different. Very closely related to our result, is the one of S. Brendle [7] on the construction of solutions of the Ginzburg-Landau equation which concentrate along codimension 2 minimal submanifolds. In all these results, solutions to nonlinear partial differential equation which concentrate along smooth submanifolds are constructed. The concentration set is always a minimal submanifold, which has to be assumed to be nondegenerate. Hence the construction holds for a generic choice of the background metric.

3. Let us also mention the work of S. Brendle concerning Yang-Mills connections in higher dimensions [6]. However, in this result, the construction does not hold for a generic choice of the metric but rather for a fairly restricted set of metrics.

4. We have not studied the case where the hypersurfaces are singular. For example, it is known that stable minimal cones do exist in dimension n + 1 ⩾ 8 and it would be very interesting to develop the corresponding analysis in this case.

5. We have not studied the case where there is a nontrivial group of isometries acting on M and where the problem cannot be reduced to a nondegenerate problem by working in the space of functions and hypersurfaces which are equivariant with respect to some finite group of symmetries. For example, one may consider the case where M is the (n + 1)-sphere with the standard metric and N is the equator. Now, let us perturb slightly the metric on S^{n+1} in some neighborhood of the north pole. Unless the perturbed metric is invariant under the action of I_{i}, for i = 1, …, n which are defined in Example (1), our method does not apply in this setting.

6. The canonical profile

In this section, we consider the case where M = ℝ and λ = 0. In this case, Equation (2.7) reduces to the following second order ordinary differential equation

∂_{t}^{2}u − \frac{1}{2} W′(u) = 0. (6.1)

Observe that

H(u, ∂_{t}u) := (∂_{t}u)^{2} − W (u),

is constant along solutions of (6.1). Using this property it is easy to check that there exists a solution of (6.1), which will be denoted by u_{⋆} in the remaining of the paper, and which satisfies

lim_{t→±∞} u_{⋆}(t) = ±1,  and  u_{⋆}(0) = 0.

This solution corresponds to H(u, ∂_{t}u) ≡ 0 and is implicitly defined by

t = ∫_{0}^{u_{⋆}(t)} \frac{dx}{\sqrt{W (x)}}.

We define the indicial roots γ_{±} > 0 by

γ_{±}^{2} := \frac{1}{2} W′′(±1), (6.2)

(observe that ±1 are minimizers of W and are assumed to be nondegenerate, hence W′′(±1) > 0 and this implies that γ_{±} are well-defined). The asymptotics of the function u_{⋆} as t tends to ±∞ are easy to derive by linearizing (6.1) about u ≡ ±1. We find that, for all k ∈ ℕ, there exists c_{k} > 0 such that

|∂_{t}^{k}(u_{⋆}(t) + 1)| ⩽ c_{k} e^{γ_{−}t}  for all  t ⩽ 0, (6.3)

and

|∂_{t}^{k}(u_{⋆}(t) − 1)| ⩽ c_{k} e^{−γ_{+}t}  for all  t ⩾ 0. (6.4)

7. Injectivity results

We prove some injectivity results for ordinary differential operators and partial differential operators whose potential is defined using the function u_{⋆}.

7.1 Preliminary results

For all ζ ∈ ℝ, we set

L_{ζ} := − ∂_{t}^{2} + ζ + \frac{1}{2} W′′(u_{⋆}).

All the injectivity results rely on the fact that, when ζ = 0, the function

w_{⋆} := ∂_{t}u_{⋆},

is a bounded positive solution of the homogeneous problem

L_{0} w_{⋆} = 0.

Furthermore w_{⋆} decays exponentially at both +∞ and −∞. We introduce, for ζ ⩾ 0, the indicial roots of the operator L_{ζ} at ±∞ by

γ_{±}(ζ) := \sqrt{ζ + γ_{±}^{2}}, (7.1)

where γ_{±} have been defined in (6.2). These indicial roots are related to the asymptotic expansion near ±∞ of the solutions of the homogeneous problem L_{ζ} w = 0. For example, it follows from Cauchy’s existence result for solutions of ordinary differential equations that there exist \underline{w} and \overline{w} solutions of L_{ζ}w = 0, which satisfy

lim_{t→+∞} e^{−γ_{+}(ζ)t} \overline{w}(t) = 1,  and  lim_{t→+∞} e^{γ_{+}(ζ)t} \underline{w}(t) = 1.

Our first injectivity result reads:

Lemma 7.1. Assume that ζ ⩾ 0 and let w be a solution of L_{ζ} w = 0 which is defined on (t_{1}, t_{2}). Further assume that w(t_{i}) = 0, for i = 1, 2. Then w ≡ 0.

Proof. We argue by contradiction. Given η ∈ ℝ, observe that

L_{ζ}(w_{⋆} + η w) = ζ w_{⋆}. (7.2)

Now, since we have assumed that w is not identically equal to 0, one can choose η ∈ ℝ such that the infimum of the function W := w_{⋆} + η w, over [t_{1}, t_{2}] is equal to 0. Observe that, since w(t_{1}) = w(t_{2}) = 0, the infimum of W is achieved at some point t_{0} ∈ (t_{1}, t_{2}) and L_{ζ}W ⩽ 0 at this point. When ζ > 0, this last inequality clearly contradicts (7.2). When ζ = 0, observe that W is a solution of some second order linear ordinary differential equation and that W (t_{0}) = ∂_{t}W (t_{0}) = 0, hence W ≡ 0. This contradicts the fact that W (t_{i}) ≠ 0 for i = 1, 2. q.e.d.

Our second injectivity result classifies the set of ζ for which there exists a bounded solution of L_{ζ}w = 0, which is defined on ℝ or on a half line. Given δ_{±} ∈ ℝ, we define δ := (δ_{−}, δ_{+}) and the function

φ_{δ}(t) := (1 + e^{t})^{δ_{+}} (1 + e^{−t})^{δ_{−}}. (7.3)

In particular, φ_{δ}(t) ∼ e^{δ_{+}t} at +∞ and φ_{δ}(t) ∼ e^{−δ_{−}t} at −∞. This definition being understood, we now prove the:

Lemma 7.2. Assume that ζ > 0. Let w be a solution of L_{ζ} w = 0 which is defined on ℝ (or on (−∞, t_{0}) or (t_{0}, +∞), in which case we ask that w(t_{0}) = 0). Further assume that |w| is bounded by a constant times the function φ_{δ} for δ_{±} < γ_{±}(ζ). Then w ≡ 0.

Proof. The proof of this result is almost identical to the proof of the previous one. The key observation is that, under the above assumptions, any solution of L_{ζ}w = 0 defined on a half line decays faster than w_{⋆} at infinity. This follows at once from the fact that the indicial roots γ_{±}(ζ) of L_{ζ} are larger than the indicial roots γ_{±} of L_{0}.

For example, assume that w is defined on (t_{0}, +∞). Any solution of the homogeneous problem L_{ζ}w = 0 is a linear combination of \overline{w}, a solution which blows up exponentially at +∞ like e^{γ_{+}(ζ)t}, and \underline{w}, the solution which decays exponentially at +∞ like e^{−γ_{+}(ζ)t}. Since |w| is bounded by e^{δ_{+}t} for some δ_{+} < γ_{+}(ζ), we conclude that w is collinear to \underline{w}. Now, γ_{+}(ζ) > γ_{+} hence w decays faster than w_{⋆} at +∞. Once this is known, the proof of the result reduces to the proof of Lemma 7.1.

q.e.d.

The set of solutions of L_{ζ}w = 0 is two dimensional and there exists a unique w_{ζ}^{−} solution of L_{ζ}w_{ζ}^{−} = 0 which is defined on all ℝ and which satisfies

lim_{t→−∞} e^{−γ_{−}(ζ) t} w_{ζ}^{−}(t) = 1.

As already mentioned, this essentially follows from Cauchy’s existence result for solutions of ordinary differential equations. When ζ > 0, the previous Lemma implies that the function w_{ζ}^{−} does not vanish and furthermore blows up exponentially at +∞ like t → e^{γ_{+}(ζ) t}. Similarly, there exists a unique w_{ζ}^{+} solution of L_{ζ}w_{ζ}^{+} = 0 on ℝ which satisfies

lim_{t→+∞} e^{γ_{+}(ζ) t} w_{ζ}^{+}(t) = 1.

Again, when ζ > 0, this function does not vanish and blows up exponentially at −∞ like t → e^{−γ_{−}(ζ) t}.

7.2 Injectivity results

Assume that (N, h) is a compact n-dimensional Riemannian manifold with or without boundary. Further assume that (N, h) is at least C^{1,α}. This means that one can choose local coordinate charts on N in which the coefficients of the metric h are C^{1,α} functions. We define on the product space ℝ × N the partial differential operator

𝔏_{h} := −∂_{t}^{2} − Δ_{h} + \frac{1}{2} W′′(u_{⋆}), (7.4)

where Δ_{h} is the Laplace-Beltrami operator on (N, h). Using the result of Lemma 7.1, we get the:

Corollary 7.3. Assume that w is a solution of 𝔏_{h} w = 0 which is defined on (t_{1}, t_{2}) × N. Further assume that w = 0 on {t_{i}} × N, for i = 1, 2 and that w has 0 Neumann boundary data on (t_{1}, t_{2}) × ∂N if ∂N is not empty. Then w ≡ 0.

Proof. We denote by (ϕ_{j}, λ_{j})_{j⩾0} the eigendata of Δ_{h} (with Neumann boundary conditions when the boundary of N is not empty). Namely

Δ_{h}ϕ_{j} = −λ_{j} ϕ_{j},

with λ_{j} ⩽ λ_{j+1}. We also assume that the eigenfunctions are normalized so that their L^{2} norm on N is 1. We decompose the function w defined on (t_{1}, t_{2}) × N as

w(t, y) = ∑_{j∈ℕ}w_{j}(t) ϕ_{j}(y).

Then w_{j} is a solution of L_{λ_{j}}w_{j} = 0 and the result of Lemma 7.1 implies that w_{j} ≡ 0. This completes the proof. q.e.d.

Using similar arguments together with Lemma 7.2, we also get the:

Corollary 7.4. Assume that w is a solution of 𝔏_{h} w = 0 which is defined on ℝ × N. Further assume that w is bounded by a constant times φ_{δ} for some δ_{±} < γ_{±}. Then w only depends on t and there exists a constant c ∈ ℝ such that w = c w_{⋆}.

Before we proceed, let us observe that the above result holds under a slightly more general assumption, namely that δ_{±} < γ_{±}(λ_{1}), where λ_{1} is the first nonzero eigenvalue of the Laplace-Beltrami operator Δ_{h}. Also the function φ_{δ} is the natural extension to ℝ × N of the function defined in (7.3).

Proof. Again, we decompose the function w defined on ℝ × N as

w(t, y) = ∑_{j∈ℕ}w_{j}(t) ϕ_{j}(y)

Then w_{j} is a solution of L_{λ_{j}}w_{j} = 0 and the result of Lemma 7.2 implies that w_{j} ≡ 0, for all j ≠ 0. When j = 0, all bounded solutions of L_{0}w = 0 have to be collinear to w_{⋆}. This completes the proof. q.e.d.

Let Δ denote the Laplacian in ℝ^{n} endowed with the Euclidean metric. We define the elliptic operator

𝔏 := −∂_{t}^{2} − Δ + \frac{1}{2} W′′(u_{⋆}).

The result of Lemma 7.2 also implies the:

Corollary 7.5. Assume that w ∈ L^{∞}(ℝ × ℝ^{n}) is a solution of 𝔏 w = 0. Then w only depends on t and there exists a constant c ∈ ℝ such that w = c w_{⋆}.

This result seems to be standard and for example appeared (without a proof) in [17]. We have not been able to find a precise reference for it and since this is a key result for the forthcoming argument, we give here a detailed proof.

Proof. We denote by S(ℝ^{k}), the space of smooth rapidly decaying functions which are defined on ℝ^{k}. This space is endowed with the family of semi-norms

[ ϕ]_{k,l} := ∥(1 + |z|^{k}) ∇^{l}ϕ∥_{L^{∞}},

for all k, l ∈ ℕ, where z denotes the variable in ℝ^{k}. The dual space S′(ℝ^{k}) is the space of tempered distributions [20].

Let F denote the Fourier transform in ℝ^{n} and \overline{F} the inverse Fourier transform. We define, for all ϕ ∈ S(ℝ × ℝ^{n}),

T : ϕ ⟶ ∫_{ℝ} ⟨w(t, ·), \overline{F}(ϕ(t, ·))⟩_{S′,S} dt.

This clearly defines a tempered distribution T ∈ S′(ℝ × ℝ^{n}). Let us denote by ξ ∈ ℝ^{n} the dual variable of z ∈ ℝ^{n}. Using the fact that w is a solution of 𝔏 w = 0, we get

⟨L_{|ξ|^{2}} T, Ψ⟩_{S′,S} := ⟨T, L_{|ξ|^{2}}Ψ⟩_{S′,S} = 0, (7.5)

for any smooth function Ψ ∈ S(ℝ × ℝ^{n}).

We claim that the support of T is included in ℝ × {0}. Indeed, choose any smooth function ψ : ℝ × ℝ^{n} ⟶ ℝ with compact support in ℝ × (ℝ^{n} − {0}). We define

Ψ(t, ξ) := \frac{1}{α(ξ)}(w_{|ξ|^{2}}^{−}(t) ∫_{t}^{+∞} w_{|ξ|^{2}}^{+}(s)ψ(s, ξ)ds
+ w_{|ξ|^{2}}^{+}(t) ∫_{−∞}^{t} w_{|ξ|^{2}}^{−}(s)ψ(s, ξ)ds),

where

α(ξ) := w_{|ξ|^{2}}^{−}(t) ∂_{t}w_{|ξ|^{2}}^{+}(t) − ∂_{t}w_{|ξ|^{2}}^{−}(t) w_{|ξ|^{2}}^{+}(t),

is the Wronskian of the two independent solutions w_{|ξ|^{2}}^{±} of the homogeneous problem L_{|ξ|^{2}}w = 0 which have been defined at the end of §7.1 (hence α does not vanish and does not depend on t !). We claim that L_{|ξ|^{2}}Ψ = ψ and also that Ψ ∈ S(ℝ × ℝ^{n}). The first claim follows at once from the fact that w_{|ξ|^{2}}^{±} are solutions of L_{|ξ|^{2}}w = 0. For the second claim, observe that the function ψ has compact support in ℝ×(ℝ^{n}−{0}), hence Ψ(t, ξ) ≡ 0 for all |ξ| large enough (say |ξ| ⩾ c) and |ξ| small enough (say |ξ| ⩽ 1/c). To show that Ψ is rapidly decaying in t when 1/c ⩽ |ξ| ⩽ c, we use the fact that, for ξ ≠ 0, the function w_{|ξ|^{2}}^{+} is exponentially decaying at +∞ and the function w_{|ξ|^{2}}^{−} is exponentially decaying at −∞. This implies at once that Ψ ∈ S(ℝ × ℝ^{n}). Therefore, we conclude that

⟨T, ψ⟩_{S′,S} = 0,

for all ψ : ℝ × ℝ^{n} ⟶ ℝ with compact support in ℝ × (ℝ^{n} − {0}). This proves the claim.

By a classical result in the theory of distributions [20], we know that T is the linear combination of derivatives (of bounded order), with respect to t and ξ_{j}, j = 1, …, n, of measures with support on ℝ × {0}. Performing the Fourier transform backward in the ξ variable, we see that the function w(t, ·) depends polynomially on the coordinates z_{j} of z ∈ ℝ^{n}. This, together with the fact that w is bounded in the z variable, implies that the function w only depends on t and hence w = c w_{⋆}. This completes the proof of the result. q.e.d.

Given γ ∈ ℝ, we define the elliptic operator

𝔏_{γ} := −∂_{t}^{2} − Δ + γ.

Following the proof of Lemma 7.5, we have:

Lemma 7.6. Assume that γ > 0 and that u ∈ L^{∞}(ℝ × ℝ^{n}) is a solution of 𝔏_{γ} w = 0. Then w ≡ 0.

8. Mapping properties of a model operator

In this section we study the mapping properties of the operator 𝔏_{h}, given in (7.4), when this operator is defined between weighted Hölder spaces.

8.1 Function spaces

Assume that (N, h) is a compact n-dimensional compact Riemannian manifold with or without boundary. To begin with, let us define the weighted spaces we will work with:

Definition 8.1. Given ℓ ∈ ℕ, α ∈ (0, 1) and δ := (δ_{−}, δ_{+}) ∈ ℝ^{2}, we define the weighted space C_{δ}^{ℓ,α}(ℝ×N) to be the space of functions which are ℓ times differentiable, whose ℓ-th partial derivatives are Hölder of exponent α and for which the weighted norm

∥u∥_{C_{δ}^{ℓ,α}(ℝ×N)} := ∥φ_{−δ} u∥_{C^{ℓ,α}(ℝ×N)},

is finite. Here by definition

∥u∥_{C^{ℓ,α}(ℝ×N)} := ∑_{j=0}^{ℓ} ∥∇^{j}u∥_{L^{∞}(ℝ×N)} + sup_{p≠q ∈ ℝ×N} \frac{|∇^{ℓ}u(p) − ∇^{ℓ}u(q)|}{d(p, q)^{α}},

is the standard (unweighted) Hölder norm and d is the geodesic distance in ℝ × N, for the product metric.

Roughly speaking, this is the space of functions which, together with their partial derivatives, are bounded by e^{δ_{+}t} on (0, +∞) × N and are bounded by e^{−δ_{−}t} on (−∞, 0) × N.

We finally define a 1-codimensional subspace of C_{δ}^{ℓ,α}(ℝ × N).

Definition 8.2. Given ℓ ∈ ℕ, α ∈ (0, 1) and δ := (δ_{−}, δ_{+}) ∈ ℝ^{2} such that δ_{±} < γ_{±}. The space D_{δ}^{ℓ,α}(ℝ × N) is defined to be the closed subspace of functions u ∈ C_{δ}^{ℓ,α}(ℝ × N) which are L^{2} orthogonal to w_{⋆}, i.e.,

∫_{ℝ×N} u(t, y) w_{⋆}(t) dt da_{h} = 0. (8.1)

Naturally, this space is endowed with the induced norm.

The restriction δ_{±} < γ_{±} is needed to ensure the convergence of the integral in (8.1), i.e., that u w_{⋆} ∈ L^{1}(ℝ × N).

In the case where N has a boundary, we define

[D_{δ}^{ℓ,α}(ℝ × N)]_{0} ,

to be the subspace of functions of D_{δ}^{ℓ,α}(ℝ × N) which have 0 Neumann boundary condition on ℝ × ∂N. In the subsequent sections, it will be convenient to adopt the notation [D_{δ}^{ℓ,α}(ℝ × N)]_{0} for D_{δ}^{ℓ,α}(ℝ × N) even when N has no boundary, being understood that the condition on the boundary data is void in this latter case.

8.2 Mapping properties

Recall that we have defined on the product space ℝ × N the partial differential operator

𝔏_{h} := −∂_{t}^{2} − Δ_{h} + \frac{1}{2} W′′(u_{⋆}), (8.2)

where Δ_{h} is the Laplace-Beltrami operator on (N, h). We now assume that (N, h) is at least C^{1,α}(N). This means that, for all y ∈ N, there exists local coordinate (z_{1}, …, z_{n}) ∈ ℝ^{n} (or in ℝ_{+}^{n} := {(z_{1}, …, z_{n}) ∈ ℝ^{n} : z_{n} > 0} in the case where y ∈ ∂N) in some neighborhood of y for which the coefficients h_{ij} of the metric

h := ∑_{i,j} h_{ij} dz_{i} ⊗ dz_{j},

are C^{1,α} functions. Clearly, the operator

𝔏_{h} : [D_{δ}^{2,α}(ℝ × N)]_{0} ⟶ D_{δ}^{0,α}(ℝ × N), (8.3)

is well-defined and bounded, for any δ ∈ ℝ^{2}. It is well known that the mapping properties of the above defined operator depends crucially on the choice of the weight parameter δ. Indeed, the set of indicial roots of 𝔏_{h} at +∞ (resp. −∞) is defined by I_{+} (resp. I_{−}) where

I_{±} := {γ_{±}(λ_{j}) : j ⩾ 0},

and λ_{j} are the eigenvalues of Δ_{h} on N. Now, if δ_{±} ∉ I_{±}, the operator (8.3) can be shown to have closed range and to be Fredholm. While, if δ_{−} or δ_{+} is an indicial root, then the operator (8.3) does not have close range, and hence is not Fredholm.

The result of Corollary 7.4 yields the injectivity of the operator 𝔏_{h} when δ_{±} < γ_{±}. This, together with a ”duality argument” implies that the operator (8.3) is surjective provided δ_{±} > −γ_{±} and δ_{±} ∉ I_{±} (the duality argument does hold stricto sensu when the operator is defined between weighted Lebesgue spaces and the corresponding result in weighted Hölder spaces is then obtained through elliptic regularity theory). In particular, the operator (8.3) is an isomorphism if the weight δ_{±} ∈ (−γ_{±}, γ_{±}). Most of these mapping properties of 𝔏_{h} will not be needed. Indeed, we will only need the latter claim on the range of weights for which the operator is an isomorphism. Therefore, in the next Proposition, we concentrate on the proof of this fact.

In addition, we will also show that the inverse of 𝔏_{h} is bounded independently of the choice of the metric h on N, provided an uniform ellipticity condition is fulfilled. Hence, we now assume that there exists Λ > 0 such that:

(i) For all y ∈ N, there exists a local chart

Y_{y} : B^{n}(Λ) ⟶ N (8.4)

where B^{n}(Λ) denotes the ball or radius Λ centered at the origin in ℝ^{n} (resp. Y_{y} is defined on B_{+}^{n}(Λ) the half ball of radius Λ centered at the origin in ℝ_{+}^{n}, if y ∈ ∂N).

(ii) For all z ∈ B^{n}(Λ) (or all z ∈ B_{+}^{n}(Λ)), the metric h at the point of coordinates z is given by

h := ∑_{i,j} h_{ij}(z) dz_{i} ⊗ dz_{j}, (8.5)

where the coefficients h_{ij} are of class C^{1,α} and satisfy h_{ij}(0) = δ_{i}^{j}.

(iii) For all z ∈ B^{n}(Λ) (or all z ∈ B_{+}^{n}(Λ)), the metric h satisfies

∑_{i,j} h^{ij}(z)ξ_{i} ξ_{j} ⩾ Λ |ξ|^{2}, (8.6)

for all ξ ∈ ℝ^{n}.

(iv) We have

∥h∥_{C^{1,α}(N)} ⩽ \frac{1}{Λ}. (8.7)

where the C^{1,α} norm is the usual one, once a section of the symmetric 2 tensors, is fixed as in (8.5).

The second condition will ensure that the Laplace-Beltrami operator on N is uniformly elliptic while the third condition will ensure uniform Hölder estimates, independent of the choice of the metric.

We prove the following:

Proposition 8.3. Assume that h satisfies (8.6) and (8.7). Further assume that δ_{±} ∈ (−γ_{±}, γ_{±}). Then, the operator 𝔏_{h} defined in (8.3) is an isomorphism and there exists a constant c > 0, only depending on Λ, such that, for all w ∈ [D_{δ}^{2,α}(ℝ × N)]_{0} we have

∥w∥_{C_{δ}^{2,α}(ℝ×N)} ⩽ c ∥𝔏_{h} w∥_{C_{δ}^{0,α}(ℝ×N)}.

Proof. As already mentioned, the injectivity of the operator 𝔏_{h} follows at once from Corollary 7.4. Indeed, given the range in which we have chosen δ_{±}, any solution of 𝔏_{h}w = 0 which is bounded by φ_{δ} has to be collinear to w_{⋆}. Since functions in [D_{δ}^{2,α}(ℝ × N)]_{0} are orthogonal to w_{⋆} in L^{2}(ℝ × N), we conclude that w ≡ 0.

We now prove that the operator 𝔏_{h} defined in (8.3) is surjective. To this aim, we decompose any function f ∈ D_{δ}^{0,α}(ℝ × N) as

f(t, y) = f_{0}(t) + f̃(t, y),

where, for each t, the function f̃(t, ·) is orthogonal to the constant function 1 in L^{2}(N). The proof is now decomposed into 5 steps.

Step 1. We define

w_{0}(t) := w_{⋆}(t) (α_{0} + ∫_{0}^{t} w_{⋆}^{−2}(s) ∫_{−∞}^{s} w_{⋆}(r) f_{0}(r) dr ds) , where the constant α_{0} ∈ ℝ is chosen so that

∫_{ℝ} w_{0} w_{⋆} dt = 0.

Using the fact that w_{⋆} ∼ e^{γ_{−}t} at −∞, together with −γ_{−} < δ_{−} < γ_{−}, we conclude that

sup_{(−∞,0]} e^{δ_{−}t} |w_{0}| ⩽ c_{0} sup_{ℝ}e^{δ_{−}t} |f_{0}|.

Sincef ∈ D_{δ}^{ℓ−2,α}(ℝ × N), we have

∫_{ℝ} f_{0} w_{⋆} dt = 0.

Therefore, we have the alternative definition of w_{0} as

w_{0}(t) := w_{⋆}(t) (α_{0} − ∫_{0}^{t} w_{⋆}^{−2}(s) ∫_{s}^{+∞} w_{⋆}(r) f_{0}(r) dr ds) ,

and, using this time the fact that w_{⋆} ∼ e^{−γ_{+}t} at +∞ together with −γ_{+} < δ_{+} < γ_{+}, we conclude that

sup_{ℝ}e^{−δ_{+}t} |w_{0}| ⩽ c_{0} sup_{ℝ}e^{−δ_{+}t} |f_{0}|,

for some constant c_{0} > 0 which does not depend on f_{0}.

Step 2. Making use of the result of Corollary 7.3, we can solve, for each T > 0

𝔏_{h}w̃_{T} = f̃,  in  (−T, T) × N,

with w̃_{T} = 0 on {±T }×N and 0 Neumann boundary data on (−T, T)× ∂N if N has a boundary. The superscript ∼ is meant to recall that we are working with functions (t, y) → g̃(t, y) for which the function g̃(t, ·) is for each t orthogonal to the constant function 1 in L^{2}(N).

Step 3. Choose t_{0} > 0 large enough so that

𝔭 := min ( inf_{(−∞,−t_{0})} (\frac{1}{2} W′′(u_{⋆}) − δ_{−}^{2}) , inf_{(t_{0},+∞)} (\frac{1}{2} W′′(u_{⋆})) − δ_{+}^{2}) > 0,

then, the potential in the operator 𝔏_{h} is bounded from below by 𝔭 in (ℝ − [−t_{0}, t_{0}]) × N and hence 𝔏_{h} satisfies the maximum principle in this set. Moreover, we have

𝔏_{h} e^{−δ_{−}t} = (\frac{1}{2} W′′(u_{⋆}) − δ_{−}^{2}) e^{−δ_{−}t} ⩾ 𝔭 e^{−δ_{−}t},

in (−∞, −t_{0}) × N and

𝔏_{h} e^{δ_{+}t} = (\frac{1}{2} W′′(u_{⋆}) − δ_{+}^{2}) e^{δ_{+}t} ⩾ 𝔭 e^{δ_{+}t},

in (t_{0}, +∞) × N. Hence, the function t → e^{δ_{+}t} (resp. t → e^{−δ_{−}t}) can be used as barrier in (t_{0}, +∞) × N (resp. in (−∞, t_{0}) × N) to prove, for all T > t_{0}, the inequality

∥φ_{−δ}w̃∥_{L^{∞}((−T,T)×N)} (8.8)

⩽ c (∥φ_{−δ}w̃∥_{L^{∞}([−t_{0},t_{0}]×N)} + ∥φ_{−δ} f̃∥_{L^{∞}((−T,T)×N)}) ,

for some constant c > 0 which only depends on δ and on 𝔭.

Step 4. We claim that

∥φ_{−δ} w̃_{T} ∥_{L^{∞}((−T,T)×N)} ⩽ c ∥φ_{−δ} f̃∥_{L^{∞}(ℝ×N)},

for some constant c > 0 which depends on Λ but neither depends on T ⩾ 1 nor on f̃. We argue by contradiction and assume that there exists a sequence of metrics h_{k} satisfying (8.6) and (8.7), a sequence T_{k} ⩾ 1, a sequence of functions f̃_{k}, and a sequence of solutions of 𝔏_{h}w̃_{k} = f̃_{k} in (−T_{k}, T_{k}) × N, with w̃_{k} = 0 on {±T_{k}} × N and 0 Neumann boundary data on (−T_{k}, T_{k}) × ∂N if N has a boundary, such that

∥φ_{−δ} w̃_{k}∥_{L^{∞}((−T_{k},T_{k})×N)} = 1, and lim_{k→∞} ∥φ_{−δ} f̃_{k}∥_{L^{∞}(ℝ×N)} = 0.

Furthermore, f̃(t, ·) and w̃(t, ·) are, for all t, orthogonal to 1 in L^{2}(N).

Observe that the claim is certainly true when T_{k} remains bounded, hence we may well assume that the sequence T_{k} ⩾ t_{0} tends to +∞. Since both w̃_{k} and f̃_{k} are uniformly bounded on compact subsets, we deduce from elliptic estimates that the sequence of functions w̃_{k} is uniformly bounded in C^{1,α} topology on any compact of ℝ × N. Now, Ascoli’s Theorem together with a standard diagonal argument implies that, up to a subsequence and for some α′ < α, the sequence of metrics h_{k} converges (in C^{1,α′}(N) topology) to h_{∞}, the sequence of functions w̃_{k} converges (in C^{1,α′}(N) topology) to w̃_{∞}. Passing to the limit in the equation satisfied by w̃_{k}, we conclude that w̃_{∞} is a weak solution of 𝔏_{h_{∞}} w̃_{∞} = 0 in ℝ × N, which is bounded by φ_{δ}. But Corollary 7.4 and the choice of the parameter δ_{±} in (−γ_{±}, γ_{±}) imply that w̃_{∞} = c w_{⋆}, for some constant c ∈ ℝ. Since the function w̃_{∞}(t, ·) is by construction orthogonal to 1 in L^{2}(N), we conclude that c = 0, hence w̃_{∞} ≡ 0.

Finally, (8.8) implies that, for each k ∈ ℕ sufficiently large (so that T_{k} ⩾ t_{0}) we have

∥φ_{−δ}w̃_{k}∥_{L^{∞}([−T_{k},T_{k}]×N)}
= 1 ⩽ c (∥φ_{−δ}w̃_{k}∥_{L^{∞}([−t_{0},t_{0}]×N)} + ∥φ_{−δ}f̃_{k}∥_{L^{∞}(ℝ×N)}) .

Passing to the limit as k tends to ∞, we conclude that

1 ⩽ c ∥φ_{−δ} w̃_{∞}∥_{L^{∞}([−t_{0},t_{0}]×N)},

which implies that w̃_{∞} is not identically equal to 0. A contradiction. This ends the proof of the claim.

Step 5. Using the results of Steps 2 and 4, standard elliptic estimates and Ascoli’s Theorem, we can pass to the limit as T tends to +∞ in the sequence w̃_{T} and obtain a function w̃ solution of 𝔏_{h} w̃ = f̃ in ℝ × N. Furthermore, the result of Step 4 implies that

∥φ_{−δ}w̃∥_{L^{∞}(ℝ×N)} ⩽ c ∥φ_{−δ}f̃∥_{L^{∞}(ℝ×N)},

for some constant c > 0 which only depends on Λ. Collecting this result and the result of Step 1, we conclude that w = w_{0} + w̃ is a solution of 𝔏_{h} w = f which satisfies (8.1). Furthermore, using Schauder’s estimates, we also conclude that

∥w∥_{C_{δ}^{2,α}(ℝ×N)} ⩽ c ∥f∥_{C_{δ}^{0,α}(ℝ×N)},

for some constant c > 0 which only depends on Λ. In particular w ∈ [D_{δ}^{2,α}(ℝ × N)]_{0}. This completes the proof of the result. q.e.d.

Once Proposition 8.3 is proven, it is easy to see that:

Proposition 8.4. Assume that δ_{±} < γ_{±}. If w ∈ [D_{δ}^{2,α}(ℝ × N)]_{0} and f ∈ D_{δ}^{0,α}(ℝ × N) satisfy 𝔏_{h} w = f and if in addition

∫_{ℝ} f(t, y) w_{⋆}(t) dt = 0,

for all y ∈ N, then

∫_{ℝ} w(t, y) w_{⋆}(t) dt = 0,

for all y ∈ N.

Proof. Recall that (ϕ_{j}, λ_{j})_{j⩾0} are the eigendata of Δ_{h} (with 0 Neumann boundary condition if ∂N is not empty). For all j ⩾ 1, we multiply the equation 𝔏_{h} w = f by w_{⋆} ϕ_{j} and integrate by parts over ℝ × N. We obtain

∫_{ℝ×N} w 𝔏_{h}(ϕ_{j} w_{⋆}) dt da_{h} = 0.

Since 𝔏_{h}(w_{⋆} ϕ_{j}) = λ_{j} w_{⋆} ϕ_{j}, we conclude that

λ_{j} ∫_{N} ϕ_{j} (∫_{ℝ} w w_{⋆} dt) da_{h} = 0.

When j ≠ 0, λ_{j} ≠ 0, hence this implies that the function

Φ(y) := ∫_{ℝ} w(t, y) w_{⋆}(t) dt,

is orthogonal to ϕ_{j} in L^{2}(N). By construction this function is also orthogonal to ϕ_{0}, which is the constant function, in L^{2}(N). Since the (ϕ_{j})_{j⩾0} form a Hilbert basis of L^{2}(N), we conclude that Φ ≡ 0. q.e.d.

Before we proceed further, let us comment on our choice of function spaces. Observe that, in (8.3), we could have replaced the spaces D_{δ}^{ℓ,α}(ℝ × N) by the spaces C_{δ}^{ℓ,α}(ℝ × N) and we could have defined

𝔏̃_{h} : [C_{δ}^{2,α}(ℝ × N)]_{0} ⟶ C_{δ}^{0,α}(ℝ × N).

Then all the above results about the set of weights for which the operator is Fredholm remain true. However, this time, the injectivity of 𝔏̃_{h} only holds provided δ_{±} < −γ_{±} and, using a “duality argument”, this implies that the operator 𝔏̃_{h} is now surjective when δ_{±} > γ_{±} are not indicial roots. Hence, this choice of function spaces would force us to work in a space of functions which blow up exponentially at ±∞ and this would not be suitable for the forthcoming nonlinear analysis.

9. Mapping properties of a singularly perturbed linear operator

9.1 Function spaces

Assume that (N, h) is a n-dimensional compact Riemannian manifold, with or without boundary and further assume that the metric h on N satisfies (8.6) and (8.7), for some fixed constant Λ > 0. We now turn to the study of the operator

𝔏_{ε} := −ε^{2} (∂_{t}^{2} + Δ_{h}) + \frac{1}{2} W′′(u_{⋆}(·/ε)), (9.1)

where ε ∈ (0, 1) is a fixed parameter. This operator still depends on h but, since we now focus our attention on its dependence with respect to the parameter ε, we omit the subscript h. Taking the parameter ε into account, we now define:

Definition 9.1. Given ℓ ∈ ℕ, α ∈ (0, 1), δ := (δ_{−}, δ_{+}) ∈ ℝ^{2} and ε ∈ (0, 1), we define the weighted space C_{δ,ε}^{ℓ,α}(ℝ × N) to be the space of functions which are ℓ times differentiable, whose ℓ-th partial derivaties are Hölder of exponent α and for which the weighted norm

∥u∥_{C_{δ,ε}^{ℓ,α}(ℝ×N)} := ∥φ_{−δ}(·/ε) u∥_{C_{ε}^{ℓ,α}(ℝ×N)},

is finite. Here, by definition

∥u∥_{C_{ε}^{ℓ,α}(ℝ×N)} := ∑_{j=0}^{ℓ} ε^{j} ∥∇^{j}u∥_{L^{∞}(ℝ×N)} + ε^{ℓ+α} ∥∇^{ℓ}u∥_{C^{0,α}(ℝ×N)}.

When δ_{±} < γ_{±}, we also define the spaces D_{δ,ε}^{ℓ,α}(ℝ × N) as the 1codimensional subspace of functions u ∈ C_{δ,ε}^{ℓ,α}(ℝ × N) which are L^{2} orthogonal to the function w_{⋆}(·/ε). For fixed ε the results of §8 hold and this shows that

𝔏_{ε} : [D_{δ,ε}^{2,α}(ℝ × N)]_{0} ⟶ D_{δ,ε}^{0,α}(ℝ × N),

is an isomorphism, provided δ_{±} ∈ (−γ_{±}, γ_{±}). Our aim is now to understand the mapping properties of 𝔏_{ε} as the scaling parameter ε tends to 0. Unfortunately, if we work with the function spaces D_{δ,ε}^{2,α}(ℝ × N), the norm of the inverse of the above defined operator blows up as ε tends to 0.

To get a better understanding of the underlying phenomena, we digress slightly and study the spectrum of the self-adjoint operator 𝔏_{ε} acting on functions which are defined on [−1, 1] × N, have 0 Dirichlet boundary conditions on {±1}×N and 0 Neumann boundary conditions on [−1, 1]×∂N if ∂N is not empty. Recall that there is an eigenfunction decomposition of the functions w defined on [−1, 1] × N as

w(t, y) = ∑_{j=0}^{∞} w_{j}(t) ϕ_{j}(y),

where (ϕ_{j}, λ_{j}) are the eigendata of Δ_{h}. This splitting induces a splitting of the operator 𝔏_{ε} into a sequence of second order ordinary differential operators

L_{ε,j} := −ε^{2} ∂_{t}^{2} + ε^{2} λ_{j} + \frac{1}{2} W′(u_{⋆}(·/ε)),

acting on functions defined on [−1, 1]. The spectrum of 𝔏_{ε} is then the union of the spectra of the operators L_{ε,j}. Namely

Spec (𝔏_{ε}) = ⋃_{j∈ℕ} Spec (L_{ε,j}).

Furthermore, the spectrum of L_{ε,j} is equal to the spectrum of L_{ε,0} shifted by ε^{2} λ_{j}. Now, the spectrum of L_{ε,0} is given by

Spec L_{ε,0} := {μ_{0,ε} < μ_{1,ε} < μ_{2,ε} … }.

All the eigenvalues are simple since the operator is a second order ordinary differential operator. Moreover, arguing as in the proof of Lemma 7.1, one can show that μ_{0,ε} > 0. The existence of the function w_{⋆} also implies easily that μ_{0,ε} decays exponentially fast to 0 as ε tends to 0. Furthermore, if w_{0,ε} denote the eigenfunctions associated to μ_{0,ε}, which are normalized so that w_{0,ε}(0) = 1, the sequence of rescaled functions (w_{0,ε}(ε ·))_{ε>0} converges on compacts to w_{⋆}, as ε tends to 0. Concerning the second eigenvalue μ_{1,ε}, one can show that

lim_{ε→0} μ_{1,ε} > 0.

To prove this fact, one can argue by contradiction as in the proof of Proposition 8.3. Since these results are not needed in our analysis, we leave the proofs to the reader.

On a heuristic level, the above study shows that, as ε tends to 0, the number of small eigenvalues of 𝔏_{ε} which are close to 0 tends to +∞. Moreover the corresponding eigenfunctions are of the form w_{0,ε} ϕ_{j}, where w_{0,ε}(ε·) tends to w_{⋆} as ε tends to 0. Since we want to work with an operator whose inverse is uniformly bounded as ε tends to 0, we need to replace the spaces we work with by much smaller function spaces. It is standard to work orthogonally to all the eigenfunctions which correspond to small eigenvalues to recover an operator whose inverse is uniformly bounded. Since the eigenfunctions corresponding to small eigenvalues tend to w_{⋆}(·/ε) ϕ_{j}, as ε tends to 0, it should be enough to work orthogonally to all the functions of the form w_{⋆}(·/ε) ϕ_{j}, in order to recover an operator whose inverse is uniformly bounded.

The above should be enough to justify why we replace the spaces D_{δ,ε}^{ℓ,α}(ℝ×N) by the much smaller function spaces which we now describe.

Definition 9.2. Given, ℓ ∈ ℕ, α ∈ (0, 1), δ := (δ_{−}, δ_{+}) ∈ ℝ^{2} and ε ∈ (0, 1) such that δ_{±} < γ_{±}, we define the weighted space E_{δ,ε}^{ℓ,α}(ℝ × N) to be the space of functions u ∈ C_{δ,ε}^{ℓ,α}(ℝ × N), which satisfy

∀ y ∈ N, ∫_{ℝ} u(t, y) w_{⋆}(t/ε) dt = 0. (9.2)

This space is endowed with the induced norm.

Observe that we now impose infinitely many constraints on the functions u, so E_{δ,ε}^{ℓ,α}(ℝ × N) has infinite codimension in C_{δ,ε}^{ℓ,α}(ℝ × N). The restriction δ_{±} < γ_{±} is needed to ensure the convergence of the integrals in (9.2).

In the case where N has a boundary, we define [E_{δ,ε}^{ℓ,α}(ℝ × N)]_{0}, to

be the subspace of functions of E_{δ,ε}^{ℓ,α}(ℝ × N) which have 0 Neumann boundary condition on ℝ × ∂N. Again, we keep the same notation for this space whether N has a boundary or not.

9.2 Mapping properties

It will be interesting to take into account the parameter ε in order to relax slightly the assumptions on the metric h. Indeed, we will still consider that h satisfies (8.6) but instead of (8.7) we now assume that ℓ ⩾ 2 is fixed and that h satisfies the weaker condition

∥h∥_{C_{ε}^{ℓ−1,α}(N)} ⩽ \frac{1}{Λ}, (9.3)

where by definition

(9.4)  ∥h∥_{C_{ε}^{ℓ,α}(N)} := ∥h_{ij}∥_{L^{∞}(N)} + ∑_{j=1}^{ℓ} ε^{j−1} ∥∇^{j}h_{ij}∥_{L^{∞}(N)}
+ ε^{ℓ−1+α} ∥∇^{ℓ}h_{ij}∥_{C^{0,α}(N)}.

To define these, we use the local charts defined in (8.4) and the sections of the symmetric two tensors defined in (8.5).

Remark 9.3. At first glance, this condition does not seem very natural since, paralleling the definition of the spaces C_{ε}^{ℓ,α}(ℝ × N), it would have been more natural to require that

|h|_{C_{ε}^{ℓ,α}(N)} ⩽ \frac{1}{Λ}, (9.5)

where

|h|_{C_{ε}^{ℓ,α}(N)} := ∑_{j=0}^{ℓ} ε^{j} ∥∇^{j}h_{ij}∥_{L^{∞}(N)} + ε^{ℓ+α} ∥∇^{ℓ}h_{ij}∥_{C^{0,α}(N)}.

However, as we will see in the proof of the next Proposition, we will need a control on the C^{1} norm of h which is slightly better then the one which is provided by (9.5).

Indeed, if Λ is assumed to be fixed, ε ∇h_{ij} converges uniformly to 0 as ε tends to 0 when h is assumed to satisfy (9.3), while this fact is not guaranteed by (9.5). Now the fact that ε ∇h_{ij} converges uniformly to 0 as ε tends to 0 will be used in the proof of the next Proposition and this will allow us to use the result of Corollary 7.5. It is possible to use (9.5) instead of (9.3) and this simplifies some of the forthcoming statements. However, we would have to pay a price, namely we would have to prove a result similar to Corollary 7.5 when Δ is replaced by a more general elliptic operator on ℝ^{n}.

Observe that, granted the above definitions, the operator

𝔏_{ε} : [E_{δ,ε}^{2,α}(ℝ × N)]_{0} ⟶ E_{δ,ε}^{0,α}(ℝ × N), (9.6) is well-defined. Indeed, given w ∈ E_{δ,ε}^{2,α}(ℝ × N) we have

∫_{ℝ} 𝔏_{ε}w w_{⋆}(·/ε) dt (9.7)

= ∫_{ℝ} (−ε^{2}(∂_{t}^{2} + Δ_{h}) + \frac{1}{2}W′′(w_{⋆}(·/ε))) w w_{⋆}(·/ε) dt

= ∫_{ℝ} (−ε^{2}∂_{t}^{2} + \frac{1}{2}W′′(w_{⋆}(·/ε))) w w_{⋆}(·/ε) dt

− ε^{2} Δ_{h} (∫_{ℝ} w w_{⋆}(·/ε) dt)

= ∫_{ℝ} (−ε^{2}∂_{t}^{2} + \frac{1}{2}W′′(w_{⋆}(·/ε))) w_{⋆}(·/ε) w dt

= 0.

Hence, 𝔏_{ε} w ∈ E_{ε}^{0,α}(ℝ×N). Recall that we have assumed that the metric h satisfies (8.6) and (8.7). This implies that 𝔏_{ε} is uniformly bounded as ε varies in (0, 1). Applying the results of §8, we see that 𝔏_{ε}, defined in (9.6) is an isomorphism, provided δ_{±} ∈ (−γ_{±}, γ_{±}). In the next result, we show that the inverse of 𝔏_{ε} is also uniformly bounded as ε varies in (0, 1).

Proposition 9.4. Assume that h satisfies (8.6), (9.3) and further assume that δ_{±} ∈ (−γ_{±}, 0]. There exists a constant c > 0 (only depending on Λ), such that, for all ε ∈ (0, 1) and for all w ∈ [E_{δ,ε}^{2,α}(ℝ × N)]_{0}, we have

∥w∥_{C_{δ,ε}^{2,α}(ℝ×N)} ⩽ c ∥𝔏_{ε} w∥_{C_{δ,ε}^{0,α}(ℝ×N)}.

The restriction δ_{±} ⩽ 0 will be needed to apply the result of Corollary 7.5.

Proof. We claim that, there exists a constant c > 0, such that if w and 𝔏_{ε} w are functions bounded by φ_{δ}(·/ε) (and w has 0 Neumann boundary condition on ℝ × ∂N if N has a boundary) and if w satisfies (9.2), then

∥φ_{−δ}(·/ε) w∥_{L^{∞}(ℝ×N)} ⩽ c ∥φ_{−δ}(·/ε) 𝔏_{ε} w∥_{L^{∞}(ℝ×N)}.

As in the proof of Proposition 8.3, the proof of the claim relies on the observation that there exists t_{0} > 0 and c > 0 such that, if w and 𝔏_{ε} w are functions bounded by φ_{δ}(·/ε) (and w has 0 Neumann boundary condition on ℝ × ∂N if N has a boundary), then

∥φ_{−δ}(·/ε) w∥_{L^{∞}(ℝ×N)} (9.8)

⩽ c (∥w∥_{L^{∞}([−t_{0}ε,t_{0}ε]×N)} + ∥φ_{−δ}(·/ε) 𝔏_{ε} w∥_{L^{∞}(ℝ×N)}) .

see (8.8). In order to prove the claim, we argue by contradiction. If the claim were not true, there would exist a sequence ε_{k} tending to 0, a sequence of metrics h_{k} satisfying (8.6) and (9.3), a sequence of functions w_{k} satisfying (9.2) such that

∥φ_{−δ}(·/ε_{k})w_{k}∥_{L^{∞}(ℝ×N)} = 1, and

lim_{k→∞} ∥φ_{−δ}(·/ε_{k}) 𝔏_{ε_{k}} w_{k}∥_{L^{∞}(ℝ×N)} = 0.

Moreover, the functions w_{k} have 0 Neumann boundary condition on ℝ × ∂N if N has a boundary.

Observe that (9.8) implies that, for each k ∈ ℕ

1 ⩽ c (∥φ_{−δ}(·/ε_{k}) w_{k}∥_{L^{∞}([−t_{0}ε_{k},t_{0}ε_{k}]×N)} + ∥φ_{−δ}(·/ε_{k}) 𝔏_{ε_{k}} w_{k}∥_{L^{∞}(ℝ×N)}) .

Furthermore, since the functions y ⟶ sup_{[−t_{0}ε_{k},t_{0}ε_{k}]} φ_{−δ}(t/ε_{k})|w_{k}(t, y)| are continuous, for each k ⩾ 0, one can choose a point y_{k} ∈ N such that

(9.9)  1 ⩽ c(∥φ_{−δ}(·/ε_{k}) w_{k}(·, y_{k})∥_{L^{∞}([−t_{0}ε_{k},t_{0}ε_{k}])}
+ ∥φ_{−δ}(·/ε_{k}) 𝔏_{ε_{k}} w_{k}∥_{L^{∞}(ℝ×N)}).

The hypersurface N being compact, we can assume without loss of generality that the sequence y_{k} converges in N.

We define, for all s ∈ ℝ and for all z ∈ ℝ^{n} (or ℝ_{+}^{n}) close enough to the origin

w̃_{k}(s, z̃) := w_{k}(ε_{k} s, Y_{y_{k}}(ε_{k} z̃)).

In the case where N has a boundary, we define

ρ_{k} := d(y_{k}, ∂N)/ε_{k},

where d denotes the geodesic distance on N for the metric h_{k}. We now distinguish a few cases according to the behavior of the sequence (ρ_{k})_{k}, which can be assumed to converge to ρ_{∞} ∈ [0, ∞].

Case 1. Assume that either N has no boundary or N has a boundary and ρ_{∞} = +∞. Since both w_{k} and 𝔏_{ε_{k}} w_{k} are uniformly bounded on compact sets, we deduce from elliptic estimates that the sequence of functions w̃_{k} is bounded in C^{1,α} topology on any compact of ℝ × ℝ^{n}. Now, Ascoli’s Theorem together with a standard diagonal argument implies that, up to a subsequence and for some α′ < α the sequence of functions w̃_{k} converge (in C^{1,α′} topology) to w̃_{∞}. Furthermore, w_{∞} is bounded by φ_{δ}. Passing to the limit in the equation satisfied by w̃_{k}, we claim that the function w̃_{∞} is a weak solution of

(∂_{t}^{2} + Δ − \frac{1}{2} W′′(u_{⋆})) w̃_{∞} = 0,

in ℝ×ℝ^{n}. This is precisely at this point that we use the full strength of (9.3) which guarantees that the C^{1} norm of the coefficients of the metrics h_{k} are uniformly bounded, hence the sequence of dilated metrics converges to the flat metric. While the weaker more natural condition given in Remark 9.3 would only have ensured that the sequence of dilated metrics converges to some metric on ℝ^{n}. Indeed, observe that, in order to define the functions w̃_{k}, we have chosen normal coordinates ε_{k} z to parameterize N in some neighborhood of the point y_{k}. In these coordinates the metric h_{k} = ε_{k}^{2} h̃_{k} where

h̃_{k} := ∑_{ij} h̃_{k,ij}(z̃) dz̃_{i} ⊗ dz̃_{j},

satisfies h̃_{k,ij}(0) = δ_{i}^{j} and

∥∇^{m}h̃_{k,ij}∥_{L^{∞}} ⩽ c ε_{k},

for m = 1, …, 2, thanks to (9.3). Here all norms are evaluated on a fixed compact, c > 0 only depends on Λ and the partial derivatives are taken with respect to the z̃_{i}. This in turn implies that the Laplace-Beltrami operator ε_{k}^{2} Δ_{h_{k}} can be expanded as

ε_{k}^{2} Δ_{h_{k}} = Δ + O(ε_{k} |z̃| ∂z̃_{i} ∂z̃_{j}) + O(ε_{k} ∂z̃_{i}).

This explains why the equation satisfied by w̃_{∞} involves the operator Δ.

Case 2. Assume that N has a boundary and that ρ_{∞} < +∞. Applying the above argument, we can assume that, up to a subsequence, the sequence of functions w̃_{k} converges to w̃_{∞}. Furthermore, w̃_{∞} is a weak solution of

(∂_{t}^{2} + Δ − \frac{1}{2} W′′(u_{⋆})) w̃_{∞} = 0,

in ℝ × ℝ_{+}^{n} which is bounded by φ_{δ}. In addition, w has 0 Neumann boundary data and we can then extend w̃_{∞} to ℝ × ℝ^{n} by reflection across the hyperplane z_{n} = 0 to reduce to Case 1.

In either case, Lebesgue’s convergence Theorem implies that w̃_{∞} satisfies

∫_{ℝ×ℝ^{n}} w̃_{∞} w_{⋆} dt dz = 0. (9.10)

We give the proof of this fact when ρ_{∞} = +∞, obvious modifications are needed to handle the case where ρ_{∞} < +∞. Given a function η : ℝ^{n} ⟶ ℝ, with compact support we use the fact that, for all z in ℝ^{n} and all k large enough

∫_{ℝ} w̃_{k}(s, z) w_{⋆}(s) ds = 0.

Provided k is chosen large enough so that the domain of definition of z → w̃_{k}(s, z) includes the support of η, we multiply this identity by η and integrate over z to get

∫_{ℝ×ℝ^{n}} η(z) w̃_{k}(s, z) w_{⋆}(s) ds da_{h_{k}} = 0.

Passing to the limit as k tends to ∞, we conclude that

∫_{ℝ×ℝ^{n}} η(z) w̃_{∞}(s, z) w_{⋆}(s) ds dz = 0.

This identity being valid for all η, we finally obtain, for all z ∈ ℝ^{n}

∫_{ℝ} w̃_{∞}(s, z) w_{⋆}(s) ds = 0.

By construction, the function w̃_{∞} is bounded by φ_{δ} and since we have assumed that δ_{±} ⩽ 0, this implies that w̃_{∞} is bounded. We now apply the result of Corollary 7.5 which shows that w̃_{∞} only depends on t. Since all bounded solutions of L_{0} w = 0 are collinear to w_{⋆}, we can write w̃_{∞}(s, z) = c w_{⋆}(s) for some constant c ∈ ℝ, and (9.10) implies that c = 0. Therefore, we have shown that w̃_{∞} ≡ 0.

Finally, (9.9) implies that, for each k ∈ ℕ

1 ⩽ c (∥φ_{−δ} w̃_{k}(·, 0)∥_{L^{∞}([−t_{0},t_{0}]×N)} + ∥φ_{−δ}(·/ε_{k}) 𝔏_{ε_{k}} w_{k}∥_{L^{∞}(ℝ×N)}) ,

passing to the limit as k tends to ∞ in this inequality, we conclude that

1 ⩽ c ∥φ_{−δ} w̃_{∞}(·, 0)∥_{L^{∞}([−t_{0},t_{0}]×N)},

which implies that w̃_{∞} is not identically equal to 0. Since we have reached a contradiction, the proof of the claim is complete.

The claim being proved the result of the Proposition follows immediately from Schauder’s estimates which are applied in geodesic balls of radius ε in ℝ × N. q.e.d.

Obviously, this result could have been obtained directly without any reference to the result of Proposition 8.3. However, we feel that the decomposition of the proof into two different steps sheds light on the choice of the function spaces and in particular explains where the conditions (8.1) and (9.2) enter into play.

10. Fermi coordinates

Assume that N ⊂ M is an admissible hypersurface. To begin with, let us assume that N has no boundary. We will explain in the last paragraph of this section the modifications which are needed to handle the case where the boundary of N is not empty.

For any p ∈ M we define d_{N}(p) to be the signed geodesic distance from p to N. This means that d_{N}(p) is the geodesic distance from p to N if p ∈ M^{+}(N) and that d_{N}(p) is equal to minus the geodesic distance from p to N if p ∈ M^{−}(N). We define

V_{τ} (N) := {p ∈ M : d_{N}(p) ∈ (−τ, τ)}. (10.1)

It is well-known that for all τ small enough, the set V_{τ} (N) is a tubular neighborhood of N in M, provided N is at least C^{2}. In order to measure the regularity of a function u which is defined in V_{τ} (N), we define for ℓ ⩾ 0, α ∈ (0, 1) and ε ∈ (0, 1)

∥u∥_{C_{ε}^{ℓ,α}(V_{τ} (N))} := ∑_{j=0}^{ℓ} ε^{j} ∥∇^{j}u∥_{L^{∞}(V_{τ} (N))}

+ ε^{ℓ+α} sup_{p≠q ∈ V_{τ} (N)} \frac{|∇^{ℓ}u(p) − ∇^{ℓ}u(q)|}{d(p, q)^{α}},

where d is the geodesic distance in M.

In the previous sections, we have asked that the metrics h we consider on the manifold N satisfy (8.6) and (8.7) or (9.3). This was needed to guarantee that the results we have obtained are independent of the choice of h, but might depend on Λ. We now restrict our attention to admissible hypersurfaces N which are embedded in M and which are close to some reference smooth admissible hypersurface N_{0}. To make things precise, we consider a smooth admissible hypersurfaces N_{0} and assume that any admissible hypersurface N we consider can be written as a geodesic normal graph over N_{0} for some function ψ_{N} ∈ C^{ℓ,α}(N_{0}).

Definition 10.1. The C_{ε}^{ℓ,α} norm of the hypersurface N, with respect to the hypersurface N_{0}, is defined to be

∥N∥_{C_{ε}^{ℓ,α}(N_{0})} := ∥ψ_{N}∥_{C_{ε}^{ℓ,α}(N_{0})},

where

\begin{aligned}∥ψ∥_{C_{ε}^{ℓ,α}(N_{0})} & := & ∥ψ∥_{L^{∞}(N_{0})} + ∥∇ψ∥_{L^{∞}(N_{0})} \\ & + & ∑_{j=2}^{ℓ} ε^{j−2}∥∇^{j}ψ∥_{L^{∞}(N_{0})} + ε^{ℓ−2+α}∥∇^{ℓ}ψ∥_{C^{0,α}(N_{0})}, \end{aligned} (10.2)

and where ψ_{N} is the function whose normal geodesic graph over N_{0} is N.

Since the hypersurface N is assumed to be embedded in M, the metric h we consider on N is just the metric induced by the metric g of M. Now, given Λ > 0 and ℓ ⩾ 2, one can find Λ′ > 0 such that : If the C_{ε}^{ℓ,α}(N_{0}) norm of ψ_{N} is bounded by Λ′, then conditions (8.6) and (9.3) are fulfilled and N is embedded.

Remark 10.2. As in (9.4) the definition of the weighted norm of a hypersurface is not the natural one, namely

|N|_{C_{ε}^{ℓ,α}(N_{0})} := |ψ_{N}|_{C_{ε}^{ℓ,α}(N_{0})},

where

|ψ|_{C_{ε}^{ℓ,α}(N_{0})} := ∑_{j=0}^{ℓ} ε^{j} ∥∇^{j}ψ∥_{L^{∞}(N_{0})} + ε^{ℓ+α} ∥∇^{ℓ}ψ∥_{C^{0,α}(N_{0})}. (10.3)

In order to justify this choice, observe that, as already mentioned, Definition 10.1 is consistent with the definition of the norm of the induced metric which is used in §9.2, while the above definition is consistent with the definition used in Remark 9.3.

In order to define Fermi coordinates in some neighborhood of N we set for all t ∈ [−τ, τ] and all y ∈ N

Z_{N}(t, y) := exp_{y}(t ν_{y}),

where ν_{y} is the normal to N at the point y, which is assumed to point towards M^{+}(N). The parameter τ is chosen small enough so that Z_{N} is a diffeomorphism onto its image.

Definition 10.3. The coordinates (t, y), defined in V_{τ} (N), are called Fermi coordinates relative to the hypersurface N.

We will need the following classical result which states that the distance function to N is well-defined and as smooth as N in some tubular neighborhood of N whose width is bounded from below. The proof of this result can be found in [10] or in [9].

Lemma 10.4. Assume that ℓ ⩾ 2 and that Λ′ > 0 is fixed small enough. Then, there exists τ_{0} > 0 only depending on Λ′ such that if N is a hypersurface whose C_{ε}^{ℓ,α}(N_{0}) norm is bounded by Λ′, then Z_{N} is a C^{ℓ−1,α} diffeomorphism from (−τ_{0}, τ_{0}) × N onto its image. Moreover the norm of ∇d_{N} in C_{ε}^{ℓ−1,α}(V_{τ_{0}}(N)) is bounded by a constant which only depends on Λ′.

Proof. The existence of the constant τ_{0} follows from the fact that the hypersurfaces we consider are uniformly bounded in C^{2} topology. This implies that the principal curvatures of N are bounded from above by some constant only depending on Λ′ and, in turn, this shows that the size of the tubular neighborhood over which Z_{N} is a diffeomorphism is bounded from below.

The fact that d_{N} is as regular as N follows from standard arguments. Indeed, the gradient of d_{N} at the point of coordinates (t, y) is given by the tangent vector to the geodesic starting from y with vector speed ν_{y}. Hence the gradient of d_{N} is as regular as the Gauss map of N, which in turn is as regular as ∇ψ_{N}. Hence ∇d_{N} is bounded in C_{ε}^{ℓ−1,α}(V_{τ_{0}}) by Λ′ (we even know that ∇^{2}d_{N} is bounded in C_{ε}^{ℓ−2,α}(V_{τ_{0}}) by some constant only depending on Λ′, which is a slightly stronger statement). q.e.d.

Let us recall that

|∇ d_{N}|_{g} = 1, (10.4)

a.e. in M.

From now on, we assume that τ_{0} is chosen as in Lemma 10.4. For all t ∈ [−τ_{0}, τ_{0}], we define the hypersurface N_{t} to be the hypersurface which is parallel to N, at distance t. This means that N_{t} is the normal geodesic graph over N for the constant function ψ(y) ≡ t. This definition being understood, we recall the expression of the Laplace-Beltrami operator in Fermi coordinates.

Lemma 10.5. Denote by (t, y) the Fermi coordinates relative to N, which are defined in V_{τ_{0}}(N). Then, at any point of Z_{N}(t, y) ∈ N_{t} we have

Δ_{g} = ∂_{t}^{2} + Δ_{h_{t}} − n H_{N_{t}} ∂_{t}.

where h_{t} is the metric induced by g on N_{t} and where H_{N_{t}}(t, y) denotes the mean curvature of the hypersurface N_{t} at the point Z_{N}(t, y).

Proof. We provide the proof of the formula when t = 0. The general case can be treated similarly, up to notation changes. Let e_{1}, …, e_{n} be an orthonormal frame field on N and ν be the normal vector field.

The Laplace-Beltrami operator on M is defined by

Δ_{g} = ∑_{i=1}^{n} (e_{i} e_{i} − D_{e_{i}}e_{i}) + ν ν − D_{ν}ν,

where D is the Levi-Civita connection on M. Let D^{N} denote the Levi-Civita connection on N, by construction, we have

D_{e_{i}}e_{i} = D_{e_{i}}^{N}e_{i} + g(D_{e_{i}}e_{i}, ν) ν.

Therefore

Δ_{g} = ∑_{i=1}^{n} (e_{i} e_{i} − D_{e_{i}}^{N}e_{i}) + ∑_{i=1}^{n} g(e_{i}, D_{e_{i}}ν) ν + ν ν − D_{ν}ν.

By definition ν ν = ∂_{t}^{2} and ν = ∂_{t}. Furthermore D_{ν}ν = 0 and

∑_{i=1}^{n} g(e_{i}, D_{e_{i}}ν) = −n H_{N},

where H_{N} is the mean curvature of N. Hence we conclude that

Δ_{g} = ∂_{t}^{2} + Δ_{h} − n H_{N} ∂_{t}.

Which is the desired expression. q.e.d.

Applying the expansion given in the previous Lemma to the function d_{N}(t, y) = t, we obtain the important formula

Δ_{g} d_{N}(t, y) = −n H_{N_{t}}(t, y). (10.5)

When t = 0 we recover the well-known fact that the Laplacian of the distance function to the hypersurface N, computed at a point of N, is equal to (minus) the sum of the principal curvatures of the hypersurface at this point [9].

We will also need the expansion of the Laplace-Beltrami operator Δ_{g} in Fermi coordinates. To do so, we need slightly better control on the regularity of the hypersurfaces N which should be at least bounded in C_{ε}^{3,α}(N_{0}).

Proposition 10.6. Assume that Λ′ > 0 is fixed small enough. If N is a hypersurface whose C_{ε}^{3,α}(N_{0}) norm is bounded by Λ′, then there exists a second order operator in ∂_{z_{j}}

L_{2} := t O_{C_{ε}^{1,α}}(Λ′) ∇^{2},

without any first or zero-th order term, and a first order operator in ∂_{t} and ∂_{z_{j}}

L_{1} := O_{C_{ε}^{0,α}}(Λ′) ∇ + t ε^{−1} O_{C_{ε}^{0,α}}(Λ′) ∇,

without zero-th order term, such that

Δ_{g} = ∂_{t}^{2} + Δ_{h} + L_{1} + L_{2},

where h is the metric on N, induced by the metric g on M. The notation O_{C_{ε}^{k,α}}(Λ′) refers to the fact that the coefficients of the operators are bounded in C_{ε}^{k,α}(V_{τ_{0}}(N)) by a constant which only depends on Λ′.

Proof. This result follows from the expansion of the metric in Fermi coordinates (t, y) relative to N. Indeed, if we write the metric g as

g = g_{tt} dt ⊗ dt + ∑_{j=1}^{n} g_{tj} dt ⊗ dz_{j} + ∑_{j,j′=1}^{n} g_{jj′} dz_{j} ⊗ dz_{j′},

the coefficients of the metric can be expanded as

g_{tt} = 1,  g_{tj} = 0  and  g_{ij} = h_{ij} + t O_{C_{ε}^{ℓ−1,α}}(Λ′),

where h_{ij} are the coefficients of the induced metric on N. This is precisely at this point that we use the definition (10.2). Indeed, the regularity of the coefficients of the metric g in Fermi coordinates is the same as the regularity of the gradient of the Gauss map of N, which in turn can be estimated by the second derivatives of the function whose graph is N. Hence, the C_{ε}^{1,α}(V_{τ_{0}}(N)) norm of the coefficients of the metric is bounded by the C_{ε}^{ℓ+1,α}(N_{0}) norm of N.

Similar expansions hold for g^{αβ}, the coefficients of g^{−1}. This implies that

\sqrt{det g} = \sqrt{det h} + t O_{C_{ε}^{1,α}}(Λ′).

With these expansions, we get

\frac{1}{\sqrt{det g}} ∂_{t} (\sqrt{det g} g^{tt} ∂_{t}) = ∂_{t}^{2} + (O_{C_{ε}^{1,α}}(Λ′) + t ε^{−1} O_{C_{ε}^{0,α}}(Λ′)) ∂_{t},

and

\frac{1}{\sqrt{det g}} ∂_{z_{i}} (\sqrt{det g} g^{ij} ∂_{z_{j}}) = \frac{1}{\sqrt{det h}} ∂_{z_{i}} (\sqrt{deth} h^{ij} ∂_{z_{j}})
+ t ε^{−1} O_{C_{ε}^{0,α}}(Λ′) ∂_{z_{j}} + t O_{C_{ε}^{1,α}}(Λ′) ∂_{z_{i}} ∂_{z_{j}}.

The result now follows at once. q.e.d.

As promised, we now explain the modifications needed when ∂N ≠ ∅. Observe that, when ∂N = ∅, the geodesics starting from a point y ∈ N with initial velocity ν_{N}(y) induce a fibration of a tubular neighborhood of N. In the case where N has a boundary this property is not true anymore. To overcome this problem, we define what we call ”twisted” Fermi coordinates. Recall that we have assumed in the introduction that M is a smooth domain of a Riemannian manifold M̃. An admissible hypersurface N in M can be extended to a hypersurface Ñ whose C_{ε}^{ℓ,α} norm is bounded by a constant (independent of ε) times the C_{ε}^{2,α} norm of N.

Observe that, provided τ_{0} is chosen small enough, we can define Fermi coordinates in

V_{τ_{0}}(Ñ) := {ZÑ(t, y) ∈ M̃ : y ∈ Ñ, t ∈ (−τ_{0}, τ_{0})}.

Let X_{1} be the vector field ∂_{t}, which is defined in V_{τ_{0}}(Ñ). Reducing τ_{0} if this is necessary, we can assume that ν_{∂M}, the inner normal to ∂M, does not coincide with X_{1} in V_{τ_{0}}(Ñ) (recall that N is assumed to meet ∂M in an orthogonal way).

We set X_{2} to be the extension of ν_{∂M} which is given by the unit normal vector fields to the family of parallel hypersurfaces to ∂M. This vector field is well-defined in a fixed tubular neighborhood of radius ε_{0} > 0 of ∂M. Finally, we consider a smooth cut-off function η which is identically equal to 1 in (−∞, ε_{0}/2) and equal to 0 in (ε_{0}, ∞), and let

χ̃ := η(d_{∂M}),

where d_{∂M} is the distance function to ∂M. We define the vector field

X := \frac{X_{1} − ⟨X_{1}, χ̃ X_{2}⟩_{g} χ̃X_{2}}{1 − ⟨X_{1}, χ̃ X_{2}⟩_{g}^{2}} .

It is straightforward to check the following properties of X:

(i) X is tangent to ∂M.

(ii) X = X_{1} at a point p whenever d_{∂M}(p) ⩾ ε_{0}.

(iii) ⟨X, X_{1}⟩_{g} ≡ 1.

We now define {F_{t}}_{t} to be the flow associated to X. It follows from (i) that F_{t}(N) is contained in M. Also, (i) and (iii) imply that (t, y) → F_{t}(y) is in fact a diffeomorphism from (−τ_{0}, τ_{0}) × N onto V_{τ_{0}}(Ñ) ∩ M.

The “twisted” Fermi coordinates (t, y) relative to N are defined by:

Y_{N}(t, y) := F_{t}(y).

Observe that (ii) implies that Y_{N}(t, y) = Z_{N}(t, y) away from a tubular neighborhood of radius ε_{0} around ∂M. Moreover, (iii) together with the fact that, N being an admissible hypersurface, N meets ∂M orthogonally, imply that in a neighborhood of radius ε_{0} of ∂M, if (t, y) are the “twisted” Fermi coordinates of a point p (relative to N) and if (t̃, ỹ) are the Fermi coordinates of the same point p (relative to Ñ), we have

t = t̃,  and  d(ỹ, y) = O(t^{2}).

Using these properties, one checks that all the results we have obtained in this section do hold when Fermi coordinates are replaced by “twisted” Fermi coordinates.

One last comment about the modifications which are needed when ∂N is not empty. In the case where ∂N is empty, one can parameterize any hypersurface Ñ close enough to N as a normal geodesic graph, that is, using the flow Z_{N}. In turn, when ∂N is not empty, we agree that we will parameterize any hypersurface Ñ close enough to N using the flow Y_{N}. With slight abuse of terminology we will say that Ñ is a “normal graph” over N.

11. The linear problem

Assume that (M, g) is a smooth (n + 1)-dimensional Riemannian manifold, with or without boundary and that N_{0} ⊂ M is a fixed admissible, nondegenerate minimal hypersurface or a volume-nondegenerate constant mean curvature hypersurface.

11.1 Preliminary assumptions and definitions

From now on, we assume that ℓ ⩾ 2 and that Λ′ > 0 is fixed small enough. We also agree that any admissible hypersurface N we consider satisfies:

(H(ℓ, Λ′)) The C_{ε}^{ℓ,α}(N_{0}) norm of the function ψ_{N}, whose normal graph is the hypersurface N, is bounded by Λ′.

We assume that Λ′ is chosen so that whenever N satisfies the above conditions, then N is embedded and (8.6) and (9.3) are fulfilled. Furthermore, reducing Λ′ if this is necessary, we can assume that N ⊂ V_{τ_{0}/4}(N_{0}) where τ_{0} is chosen as in Lemma 10.4.

We choose a cutoff function χ which is identically equal to 1 in (−τ_{0}/2, τ_{0}/2) and identically equal to 0 outside [−τ_{0}, τ_{0}]. We define

u_{ε,N} := χ(d_{N}) \frac{d_{N}}{|d_{N}|} + (1 − χ(d_{N})) u_{⋆}(d_{N}/ε). (11.1)

11.2 Function spaces

Paralleling what we have done in §9, we define weighted Hölder spaces adapted to our problem. We first give the definition of the Hölder spaces which take into account the scaling parameter ε.

Definition 11.1. Given ℓ ∈ ℕ, α ∈ (0, 1) and ε ∈ (0, 1), we define the space C_{ε}^{ℓ,α}(M) to be the space of functions which are ℓ times differentiable and whose ℓ-th partial derivatives are Hölder of exponent α. This space is endowed with the norm

∥u∥_{C_{ε}^{ℓ,α}(M)} := ∑_{j=0}^{ℓ′} ε^{j} ∥∇^{j}u∥_{L^{∞}(M)} + ε^{ℓ+α} sup_{p≠q ∈ M} \frac{|∇^{ℓ}u(p) − ∇^{ℓ}u(q)|}{d(p, q)^{α}},

where d is the geodesic distance in M.

This definition being understood, we define a projection operator:

Definition 11.2. Denote by (t, y) the (twisted) Fermi coordinates relative to N. For all u ∈ C^{ℓ,α}(M), we define the function Π_{ε,N}(u) by

Π_{ε,N}(u)(t, y) := u(t, y) − \frac{∫_{ℝ} χ(s) u(s, y) w_{⋆}(s/ε) ds}{∫_{ℝ} χ^{2}(s) w_{⋆}^{2}(s/ε) ds} χ(t) w_{⋆}(t/ε), (11.2)

in V_{τ_{0}}(N) and Π_{ε,N}(u) = u in M − V_{τ_{0}}(N).

Observe that Π_{ε,N} is an involution since, by construction Π_{ε,N} ∘ Π_{ε,N} = Π_{ε,N}. It is straightforward to check the:

Lemma 11.3. Assume that (H(3, Λ′)) holds. Then, for all ℓ = 0, …, 2

Π_{ε,N} : C_{ε}^{ℓ,α}(M) ⟶ C_{ε}^{ℓ,α}(M),

is well-defined and uniformly bounded for ε ∈ (0, 1).

Proof. The fact that this operator is well-defined follows from the fact that Fermi coordinates induce a local C^{2,α} diffeomorphism D when N is a C^{3,α} hypersurface.

The fact that Π_{ε,N} is bounded uniformly between the weighted spaces follows at once from the definition of the C_{ε}^{3,α}(N_{0}) norm of N which ensures that the diffeomorphism D is uniformly bounded in L^{∞}(V_{τ_{0}}(N)) and has a differential which is uniformly bounded in C_{ε}^{1}(V_{τ_{0}}(N)). q.e.d.

We now define weighted function spaces on M which parallel the spaces E_{δ,ε}^{ℓ,α}(ℝ × N), when the weights δ_{±} are chosen to be equal to 0.

Definition 11.4. Assume that (H(ℓ + 1, Λ′)) holds for some ℓ ⩾ 2. Given 0 ⩽ ℓ′ ⩽ ℓ, α ∈ (0, 1) and ε ∈ (0, 1), we define the space E_{ε,N}^{ℓ′,α}(M) to be the space of functions u ∈ C_{ε}^{ℓ′,α}(M) which satisfy

Π_{ε,N}(u) = u.

This space is endowed with the induced norm.

As usual, we also define [C_{ε}^{ℓ′,α}(M)]_{0} (resp. [E_{ε,N}^{ℓ′,α}(M)]_{0}) to be the subspace of functions of C_{ε}^{ℓ′,α}(M) (resp. E_{ε,N}^{ℓ′,α}(M)) which have 0 Neumann boundary condition on ∂M, if this boundary is not empty.

11.3 The linear problem

Assume that N is an admissible hypersurface which satisfies (H(2, Λ′)). Let u_{ε,N} be defined as in (11.1), the linearized operator we are interested in reads

𝕃_{ε,N} := −ε^{2} Δ_{g} + \frac{1}{2} W′′(u_{ε,N}).

In this section, we would like to construct a right inverse for this operator. To do so, we first define and study the auxiliary operator

𝕃_{0} := −ε^{2} Δ_{g} + \frac{1}{2} Γ,

where the potential Γ is chosen to interpolate smoothly between 2 γ_{−}^{2} = W′′(−1) in M^{−}(N) and 2 γ_{+}^{2} = W′′(1) in M^{+}(N). More precisely, let ξ denote a smooth cutoff function equal to 1 on (1, +∞) and equal to 0 on (−∞, −1), with ξ ⩾ 0. If (t, y) are (twisted) Fermi coordinates relative to N, we define

Γ(t, y) := 2 ((1 − ξ(t/ε)) γ_{−}^{2} + ξ(t/ε) γ_{+}^{2}) ,

in V_{τ_{0}}(N) and

Γ := 2 γ_{±}^{2},

in M^{±}(N) − V_{τ_{0}}(N). We have:

Lemma 11.5. There exists ε_{0} > 0 only depending on Λ′ such that, for all ε ∈ (0, ε_{0}), the operator

𝕃_{0} : [C_{ε}^{0,α}(M)]_{0} ⟶ C_{ε}^{2,α}(M),

is an isomorphism the norm of whose inverse is bounded by some constant which depends on Λ′ but does not depend on ε nor on N satisfying (H(2, Λ′)).

Proof. The fact that 𝕃_{0} is an isomorphism, clearly follows from the fact that the potential Γ is bounded from below by a positive constant independent of ε. In particular, the constant function 1 can be used as a barrier to show that

∥w∥_{L^{∞}(M)} ⩽ c ∥𝕃_{0} w∥_{L^{∞}(M)},

for some constant c > 0 which does not depend on ε. Then, the estimates for the derivatives of w are consequences of Schauder’s estimates on geodesic balls of radius ε. q.e.d.

Collecting the results of the previous sections we construct for the operator 𝕃_{ε,N} a right inverse whose norm is uniformly bounded as ε tends to 0. The construction of the right inverse relies on all our former analysis. To be more precise, we glue together local parametrizes given by Proposition 9.4 and Lemma 11.5 and obtain the right inverse for 𝕃_{ε,N} by applying a perturbation argument together with Proposition 10.6. As will become clear in the proof and in the statement of the next result, we need to assume that the submanifold N has one degree of regularity higher than would be expected. Indeed, a natural guess would be that, in order to solve the equation 𝕃_{ε,N}w = f ∈ C^{k−2,α} in some Hölder space C^{k,α}, one would need to assume that the hypersurface N is itself C^{k,α}. However, our construction relies on the use of Proposition 10.6 together with the use of the projection operator Π_{ε,N} which both require N to be a C^{k+1,α} hypersurface. We now state the main technical result of our paper:

Proposition 11.6. Assume N satisfies (H(3, Λ′)). Then, there exists ε_{0} > 0 only depending on Λ′ such that, for all ε ∈ (0, ε_{0}), there exists an operator

𝔾_{ε,N} : E_{ε,N}^{0,α}(M) ⟶ [E_{ε,N}^{2,α}(M)]_{0},

satisfying

Π_{ε,N} ∘ 𝕃_{ε,N} ∘ 𝔾_{ε,N} = I.

Furthermore the norm of 𝔾_{ε,N} is bounded by some constant which depends on Λ′, but does not depend on ε ∈ (0, ε_{0}) nor on N.

Proof. The construction of 𝔾_{ε,N} is decomposed in 4 steps. As already mentioned we “glue” together different parametrizes which have been defined in the previous sections. In the first step, we use the result of Lemma 11.5 to reduce the solvability of Π_{ε,N} ∘ 𝕃_{ε,N}w = f to the case where f is supported in V_{τ_{0}}(N). This allows us to use, in Step 2 the result of Proposition 9.4 where a right inverse for the operator 𝔏_{ε} has been constructed. In Step 3, we use the expansion of the Laplace-Beltrami operator in Fermi coordinates which is provided by Proposition 10.6 to estimate the difference between the operators 𝕃_{ε,N} and the operator 𝔏_{ε}. At this point we have produced a bounded operator G which is almost a right inverse. In Step 4, we apply a standard perturbation argument to find a right inverse for Π_{ε,N} ∘ 𝕃_{ε,N}.

In the proof, the constant c > 0, which may vary from line to line, may depend on Λ′, but does not depend on ε (provided this parameter is chosen small enough), does not depend on f, nor on N chosen to satisfy (H(3, Λ′)).

Step 1. Thanks to the result of Lemma 11.5, we find w_{1} ∈ [C_{ε}^{2,α}(M)]_{0} solution of 𝕃_{0}w_{1} = f. Furthermore, we know that

∥w_{1}∥_{C_{ε}^{2,α}(M)} ⩽ c ∥f∥_{C_{ε}^{0,α}(M)}.

Now, the result of Lemma 11.3 implies that we also have

∥Π_{ε,N} w_{1}∥_{C_{ε}^{2,α}(M)} ⩽ c ∥f∥_{C_{ε}^{0,α}(M)}. (11.3)

Observe that, as mentioned, we already need to assume that the hypersurface N has bounded norm in C_{ε}^{3,α}(N_{0}).

We define the function

g := f − 𝕃_{ε,N} ∘ Π_{ε,N} w_{1}.

Since, away from V_{τ_{0}}(N), we have Π_{ε,N}u = u and 𝕃_{0} = 𝕃_{ε,N}, we conclude that, the function g is supported in V_{τ_{0}}(N). If we identify V_{τ_{0}}(N) with (−τ_{0}, τ_{0}) × N via (twisted) Fermi coordinates, we can extend the function g by 0 to all ℝ × N. We claim that

∥g∥_{C_{−γ,ε}^{0,α}(ℝ×N)} ⩽ c ∥f∥_{C_{ε}^{0,α}(M)}, (11.4) where γ := (γ_{−}, γ_{+}) are the indicial roots defined in (6.2). Indeed, using (6.3) and (6.4), we see that u_{⋆} converges exponentially to ±1 at a rate which is dictated by the indicial roots γ_{±}. This in turn shows that the difference between the potential of the operators 𝕃_{ε,N} and 𝕃_{0} converges exponentially to 0 at a rate which is dictated by the indicial roots 2 γ_{±}. Namely

∥Γ − W′′(u_{⋆}(·/ε))∥_{C_{−2γ,ε}^{0,α}(ℝ×N)} ⩽ c.

This inequality already implies that

∥𝕃_{ε,N} w_{1} − f∥_{C_{−2γ,ε}^{0,α}(ℝ×N)} ⩽ c ∥f∥_{C_{ε}^{0,α}(M)}.

Furthermore, using the definition of Π_{ε,N} together with the fact that

∥χ w_{⋆}(·/ε)∥_{C_{−γ,ε}^{2,α}(ℝ×N)} ⩽ c, (11.5)

we get

∥𝕃_{ε,N} (w_{1} − Π_{ε,N}w_{1})∥_{C_{−γ,ε}^{0,α}(ℝ×N)} ⩽ c ∥f∥_{C_{ε}^{0,α}(M)}.

This completes the proof of the claim.

Step 2. We define the projection operator

Π_{ε,N}^{0}(u)(t, y) := u(t, y) − \frac{∫_{ℝ} u(s, y) w_{⋆}(s/ε) ds}{∫_{ℝ} w_{⋆}^{2}(s/ε) ds} w_{⋆}(t/ε).

Clearly,

Π_{ε,N}^{0} : C_{−γ,ε}^{0,α}(ℝ × N) ⟶ E_{−γ,ε}^{0,α}(ℝ × N),

is bounded uniformly in ε. Hence, using (11.4), we conclude that

∥Π_{ε,N}^{0} g∥_{C_{−γ,ε}^{0,α}(ℝ×N)} ⩽ c ∥f∥_{C_{ε}^{0,α}(M)}. (11.6)

We use the result of Proposition 9.4, with some fixed weights δ := (δ_{−}, δ_{+}) satisfying −γ_{±} < δ_{±} < 0, to define w_{2} ∈ [E_{δ,ε}^{2,α}(ℝ×N)]_{0} solution of

𝔏_{ε} w_{2} = Π_{ε,N}^{0} g,

where we recall that

𝔏_{ε} := −ε^{2}(∂_{t}^{2} + Δ_{N}) + \frac{1}{2} W′′(u_{⋆}(·/ε)).

Since the inverse of 𝔏_{ε} has been shown to be uniformly bounded, we get the estimate

∥w_{2}∥_{C_{δ,ε}^{2,α}(ℝ×N)} ⩽ c ∥Π_{ε,N}^{0} g∥_{C_{δ,ε}^{0,α}(ℝ×N)} ⩽ c ∥Π_{ε,N}^{0} g∥_{C_{−γ,ε}^{0,α}(ℝ×N)},

which, together with (11.6) implies that

∥w_{2}∥_{C_{δ,ε}^{2,α}(ℝ×N)} ⩽ c ∥f∥_{C_{ε}^{0,α}(M)}. (11.7)

Step 3. We claim that

∥Π_{ε,N} (𝕃_{ε,N} ∘ Π_{ε,N} (χw_{2}) − g) ∥_{C_{ε}^{0,α}(M)} ⩽ c ε ∥f∥_{C_{ε}^{0,α}(M)}. (11.8)

Since w_{2} and g are exponentially decaying in terms of |t|/ε, we readily have

∥(1 − χ) Π_{ε,N} (𝕃_{ε,N} ∘ Π_{ε,N} (χw_{2}) − g) ∥_{C_{ε}^{0,α}(M)} ⩽ c e^{−κ_{1}/ε} ∥f∥_{C_{ε}^{0,α}(M)},

for some fixed κ_{1} > 0 only depending on γ, δ and τ_{0}. Hence, identifying V_{τ_{0}}(N) with [−τ_{0}, τ_{0}] × N as above, we only need to show that

∥χ Π_{ε,N} (𝕃_{ε,N} ∘ Π_{ε,N} (χw_{2}) − g) ∥_{C_{ε}^{0,α}(ℝ×N)} ⩽ c ε ∥f∥_{C_{ε}^{0,α}(M)}. (11.9)

The proof of this inequality essentially follows from the result of Proposition 10.6 which gives the expansion of Δ_{g} in Fermi coordinates relative to N. Observe that, again, we need to assume that the hypersurface N has bounded norm in C_{ε}^{3,α}(N_{0}) in order to apply the result of Proposition 10.6 with the correct regularity.

To begin with we replace Π_{ε,N} by Π_{ε,N}^{0} in the left-hand side of (11.9). This introduce a discrepancy which we now estimate. We set

D_{1} := Π_{ε,N} ∘ 𝕃_{ε,N} ∘ Π_{ε,N} (χw_{2}) − Π_{ε,N}^{0} ∘ 𝕃_{ε,N} ∘ Π_{ε,N}^{0} (χw_{2}),

and

D_{2} := Π_{ε,N}^{0} g − Π_{ε,N} g.

Using the fact that w_{⋆}(·/ε) is exponentially decaying in terms of |t|/ε, we see that the same is true for D_{1} and D_{2}, so

∥χ (D_{1} + D_{2}) ∥_{C_{ε}^{0,α}(ℝ×N)} ⩽ c e^{−κ_{2}/ε} ∥f∥_{C_{ε}^{0,α}(M)},

for some fixed positive κ_{2} which only depends on γ, δ and τ_{0}.

Having replaced Π_{ε,N} by Π_{ε,N}^{0}, we now replace χ w_{2} by w_{2} in (11.9). This introduces yet another discrepancy which we now estimate. We set

D_{3} := Π_{ε,N}^{0} ∘ 𝕃_{ε,N} ∘ Π_{ε,N}^{0}((1 − χ)w_{2}).

Since w_{2} is exponentially decaying in terms of |t|/ε, we also have

∥χ D_{3}∥_{C_{ε}^{0,α}(ℝ×N)} ⩽ c e^{−κ_{3}/ε} ∥f∥_{C_{ε}^{0,α}(M)},

for some fixed positive κ_{3} which only depends on γ, δ and τ_{0}.

Having done the above modifications, we are left with the estimate of

D_{4} := Π_{ε,N}^{0} ∘ 𝕃_{ε,N} ∘ Π_{ε,N}^{0} w_{2} − Π_{ε,N}^{0} g
= Π_{ε,N}^{0} ∘ 𝕃_{ε,N} w_{2} − 𝔏_{ε}w_{2}
= Π_{ε,N}^{0} (𝕃_{ε,N} w_{2} − 𝔏_{ε}w_{2}) .

We have used the fact that Π_{ε,N}^{0} w_{2} = w_{2} since w_{2} ∈ E_{δ,ε}^{2,α}(ℝ × N) and 𝔏_{ε} w_{2} = Π_{ε}^{0} g to obtain the first identity. In order to obtain the second identity, we have used the fact that 𝔏_{ε}w_{2} = Π_{ε,N}^{0} 𝔏_{ε}w_{2}, by construction of the inverse of 𝔏_{ε}. At this point, we refer to the result of Proposition 10.6 which gives the expansion of Δ_{g} in Fermi coordinates about N to get

𝕃_{ε,N} − L_{ε} = L_{1} + L_{2},

where the properties of the operators L_{1} and L_{2} are stated in Proposition 10.6. These properties, together with (11.7), imply that

∥χ D_{4}∥_{C_{ε}^{0,α}(ℝ×N)} ⩽ c ε ∥f∥_{C_{ε}^{0,α}(M)}.

This completes the proof of the claim.

Step 4. Collecting (11.8) together with the definition of g, we conclude that

∥Π_{ε,N} ∘ 𝕃_{ε,N} ∘ Π_{ε,N}(w_{1} + χw_{2}) − f∥_{C_{ε}^{0,α}(M)} ⩽ c ε ∥f∥_{C_{ε}^{0,α}(M)}.

Now, collecting (11.3) and (11.7) together with Lemma 11.3), we obtain

∥Π_{ε,N}(w_{1} + χw_{2})∥_{C_{ε}^{2,α}(M)} ⩽ c ∥f∥_{C_{ε}^{0,α}(M)}.

At this point, the existence of 𝔾_{ε,N} easily follows from a classical perturbation argument. q.e.d.

Our next proposition gives an estimate of the norm of the operator

(I − Π_{ε,N}) ∘ 𝕃_{ε,N} ∘ 𝔾_{ε,N},

which, in some sense, measures the distance between the operator 𝕃_{ε,N} ∘ 𝔾_{ε,N} and the identity in E_{ε}^{ℓ−2,α}(M). We define the operator S_{ε,N} by

S_{ε,N}(u)(y) := ∫_{ℝ} χ(s) u(t, y) w_{⋆}(t/ε) dt, (11.10)

where (t, y) are (twisted) Fermi coordinates relative to N and u is a function defined in M. Since N is assumed to be a normal graph over N_{0}, any function on N can be identified with a function on N_{0} and its norm can be evaluated using the norms defined in (10.2) or in (10.3), provided N is regular enough. This being understood, we have:

Proposition 11.7. Assume N satisfies (H(3, Λ′)). Then, there exists c > 0 only depending on Λ′ such that, for all w ∈ [E_{ε,N}^{2,α}(M)]_{0}

|S_{ε,N} ∘ 𝕃_{ε,N} w|_{C_{ε}^{0,α}(N_{0})} ⩽ c ε^{2} ∥w∥_{C_{ε}^{2,α}(M)}, (11.11)

where the norm | · |_{C_{ε}^{ℓ,α}(N_{0})} is the one defined in (10.3).

Proof. We have to estimate

y ⟶ ∫_{ℝ} χ(t) w_{⋆}(t/ε)L_{ε,N} w(t, y) dt.

We set

E_{1} := ∫_{ℝ} χ(t) w_{⋆}(t/ε)(L_{ε,N} − 𝔏_{ε}) w(t, y) dt,

where 𝔏_{ε} is defined as in Step 2 of the previous proof. Using once more the result of Proposition 10.6, we obtain as in Step 3 of the previous proof

|E_{1}|_{C_{ε}^{0,α}(N_{0})} ⩽ c ε^{2} ∥w∥_{C_{ε}^{2,α}(M)}.

We set

E_{2} := ∫_{ℝ} w_{⋆}(t/ε) [χ, 𝔏_{ε}] w(t, y) dt,

where [A, B] is the commutator of the operators A and B. Using the fact that w_{⋆} decays exponentially fast at ±∞, we conclude that E_{2} decays exponentially fast in terms of |t|/ε. In particular

|E_{2}|_{C_{ε}^{0,α}(N_{0})} ⩽ c e^{−κ_{4}/ε} ∥w∥_{C_{ε}^{2,α}(M)},

for some fixed κ_{4} > 0 which only depends on γ and τ_{0}.

Having estimated E_{1} and E_{2}, it remains to estimate

y ⟶ ∫_{ℝ} w_{⋆}(t/ε) 𝔏_{ε}(χw)(t, y) dt.

We set

E_{3} := ∫_{ℝ} w_{⋆}(t/ε) 𝔏_{ε} ∘ (Π_{ε,N} − Π_{ε,N}^{0})(χw)(t, y) dt,

and

E_{4} := ∫_{ℝ} w_{⋆}(t/ε) 𝔏_{ε} ∘ (I − Π_{ε,N})(χw)(t, y) dt.

Using the fact that w_{⋆}(·/ε) is exponentially decaying in terms of |t|/ε, we get

|E_{3}|_{C_{ε}^{0,α}(N_{0})} ⩽ c e^{−κ_{5}/ε} ∥u∥_{C_{ε}^{2,α}(M)},

and, using in addition the fact that (I − Π_{ε,N}) w = 0, we also have

|E_{4}|_{C_{ε}^{0,α}(N_{0})} ⩽ c e^{−κ_{5}/ε} ∥u∥_{C_{ε}^{2,α}(M)},

for some fixed κ_{5} > 0 which only depends on γ and τ_{0}.

Having estimated E_{1} through E_{4}, it finally remains to estimate

y ⟶ ∫_{ℝ} w_{⋆}(t/ε) 𝔏_{ε} ∘ Π_{ε,N}^{0}(χu)(t, y) dt.

But, as we have already done in (9.7), one checks that this quantity is identically equal to 0. This completes the proof of (11.11). q.e.d.

We would like to obtain similar results when N is less regular than what is required in the last two results. To do so we introduce smoothing operators which will alow us to find N_{⋆}, smooth enough so that we can apply the previous results, and close enough to N so that a perturbation argument can be applied.

We recall from [1], page 97, that there exists a one parameter family of smoothing operators (R_{θ})_{θ⩾1} and C > 0 such that

∥R_{θ} u∥_{C^{k,α}} ⩽ C ∥u∥_{C^{k′,α′}}  for k + α ⩽ k′ + α′
∥R_{θ} u∥_{C^{k,α}} ⩽ C θ^{k+α−k′−α′} ∥u∥_{C^{k′,α′}}  for k + α ⩾ k′ + α′
∥u − R_{θ} u∥_{C^{k,α}} ⩽ C θ^{k+α−k′−α′} ∥u∥_{C^{k′,α′}}  for k + α ⩽ k′ + α′.

(11.12)

These operators act on functions defined in ℝ^{n} but they can be localized and extended to functions defined on smooth manifolds using a partition of unity. We use R_{θ} to improve the regularity of a given hypersurface N which is assumed to be a normal graph over N_{0}. We further assume that (H(ℓ, Λ′′)) is satisfied with Λ′′ smaller than the usual constant Λ′. Hence, N is the normal graph over N_{0} for some function ψ which satisfies

∥ψ∥_{C_{ε}^{ℓ,α}(N_{0})} ⩽ Λ′′,

for some ℓ ⩾ 2. We define the function ψ_{⋆} := R_{1/ε} ψ, and we define the hypersurface N_{⋆} to be the graph of the function ψ_{⋆} over N_{0}. We will write

N_{⋆} = R_{1/ε} N.

We claim that,

|ψ_{⋆} − ψ|_{C_{ε}^{ℓ,α}(N_{0})} ⩽ c ε^{2} ∥ψ∥_{C_{ε}^{ℓ,α}(N_{0})}, (11.13)

and, for all ℓ′ ⩾ ℓ

∥ψ_{⋆}∥_{C_{ε}^{ℓ′,α}(N_{0})} ⩽ c ∥ψ∥_{C_{ε}^{ℓ,α}(N_{0})}, (11.14)

where the constant c depends on ℓ′ and ℓ but does not depend on Λ′.

Indeed, using the first property of R_{θ} in (11.12), we get ∥ψ_{⋆}∥_{C^{2}(N_{0})} ⩽ c ∥ψ∥_{C^{2}(N_{0})} and, for all 2 ⩽ k ⩽ ℓ, we have

∥∇^{k}ψ_{⋆}∥_{L^{∞}(N_{0})} ⩽ c ∥ψ∥_{C^{k,0}(N_{0})} ⩽ c ε^{2−k} ∥ψ∥_{C_{ε}^{ℓ,α}(N_{0})},

Now, for all ℓ ⩽ k ⩽ ℓ + α, we use the second property of R_{θ} in (11.12) to get

∥∇^{k}ψ_{⋆}∥_{L^{∞}(N_{0})} ⩽ c ε^{ℓ+α−k} ∥ψ∥_{C^{ℓ,α}(N_{0})} ⩽ c ε^{2−k} ∥ψ∥_{C_{ε}^{ℓ,α}(N_{0})},

which implies the second estimate (11.13). Finally, for all ℓ′ ⩽ ℓ, we use the third property of R_{θ} in (11.12) to show that

∥∇^{k}(ψ_{⋆} − ψ)∥_{L^{∞}(N_{0})} ⩽ c ε^{ℓ+α−k} ∥ψ∥_{C^{ℓ,α}(N_{0})} ⩽ c ε^{2−k} ∥ψ∥_{C_{ε}^{ℓ,α}(N_{0})},

which proves (11.14). The proof of the claim is therefore complete.

In the case where N has a nonempty boundary, this construction can be modified so that we can assume that N_{⋆} is an admissible hypersurface. Thanks to (11.13) and (11.14), we can apply all the above results to N_{⋆}, provided Λ′′ is chosen small enough. This yields the:

Proposition 11.8. Assume that N satisfies (H(2, Λ′′)) and let N_{⋆} = R_{1/ε} N. Then, there exists ε_{0} > 0 such that, for all ε ∈ (0, ε_{0}), there exists an operator

𝔾_{ε,N} : E_{ε,N_{⋆}}^{0,α}(M) ⟶ [E_{ε,N_{⋆}}^{2,α}(M)]_{0},

satisfying

Π_{ε,N_{⋆}} ∘ 𝕃_{ε,N} ∘ 𝔾_{ε,N} = I.

Furthermore,

∥𝔾_{ε,N} w∥_{C_{ε}^{2,α}(M)} ⩽ c ∥w∥_{C_{ε}^{0,α}(M)},

and

|S_{ε,N_{⋆}} ∘ 𝕃_{ε,N} ∘ 𝔾_{ε,N} w|_{C_{ε}^{0,α}(N_{0})} ⩽ c ε^{2} ∥w∥_{C_{ε}^{0,α}(M)}, (11.15)

for some constant c > 0 which depends on Λ′′ but does not depend on ε nor on N.

Proof. We claim that

∥(𝕃_{ε,N_{⋆}} − 𝕃_{ε,N}) w∥_{C_{ε}^{0,α}(M)} ⩽ c ε ∥w∥_{C_{ε}^{0,α}(M)}. (11.16)

This follows from the fact that the difference of these two operators is equal to the difference of their potentials which involve functions depending on the distance to N and N_{⋆}. The result is then a consequence of the fact that difference ψ_{⋆} − ψ satisfies the estimate (11.13).

Now, we apply the results of Proposition 11.6 to get for all f ∈ E_{ε,N_{⋆}}^{0,α}(M) a solution w ∈ [E_{ε,N_{⋆}}^{2,α}(M)]_{0} of

Π_{ε,N_{⋆}} ∘ 𝕃_{ε,N_{⋆}} w = f,

with

∥w∥_{C_{ε}^{2,α}(M)} ⩽ c ∥f∥_{C_{ε}^{0,α}(M)}.

The existence of 𝔾_{ε,N} then follows from (11.16) together with a simple perturbation argument. It remains to check (11.15). But this follows at once from (11.16) together with the result of Proposition 11.7. q.e.d.

Observe that the mapping 𝔾_{ε,N} depends continuously on N. This essentially follows from the fact that, each step of the construction of this operator depends continuously on N.

12. Moving the nodal set

Assume that we are given an admissible hypersurface N_{0}. We define for convenience

ℚ_{ε}(N) := −ε^{2} Δu_{ε,N} + \frac{1}{2} W′(u_{ε,N}), (12.1)

where N is an admissible hypersurface which can be written as a normal graph over N_{0}. If N_{⋆} is the regularized hypersurface defined at the end of the previous section, we define, as in (11.10), the operator S_{ε,N_{⋆}} by

S_{ε,N_{⋆}}(u)(y) := ∫_{ℝ} χ(t) u(t, y) w_{⋆}(t/ε) dt, (12.2)

where (t, y) are (twisted) Fermi coordinates relative to N_{⋆} and u is a function defined in V_{τ_{0}}(N). Since N_{⋆} is assumed to be a normal graph over N_{0}, any function on N_{⋆} can be identified with a function on N_{0} and its norm can be evaluated using the norms defined in (10.2) or (10.3).

In this section we exploit the notion of nondegeneracy which have been defined in Definition 3.2 and Definition 3.3. There are two results which correspond to the two different notions of nondegeneracy. To start with let us assume that we are dealing with an admissible nondegenerate minimal hypersurface N_{0}. We have:

Lemma 12.1. Assume that N_{0} is an admissible nondegenerate minimal hypersurface and that c > 0 is fixed. Then, there exists ε_{0} ∈ (0, 1) such that, for all f ∈ C^{0,α}(N_{0}) satisfying

|f|_{C_{ε}^{0,α}(N_{0})} ⩽ c ε^{2},

one can find an admissible hypersurface N satisfying

S_{ε,N_{⋆}} ℚ_{ε}(N) = ε^{2} f

where N_{⋆} = R_{1/ε} N. Moreover, if ψ_{N} is the function whose graph is N, we have

∥ψ_{N}∥_{C_{ε}^{2,α}(N_{0})} ⩽ c ε^{−α} (ε^{2} + |f|_{C_{ε}^{0,α}(N_{0})}),

for some constant c > 0 which does not depend on f nor on ε.

Proof. The proof of this result follows from the implicit function theorem. Granted the definition of S_{ε,N_{⋆}}, we have to find and admissible hypersurface N such that

∫_{ℝ} χ(t) (−ε^{2} Δu_{ε,N} + \frac{1}{2} W′(u_{ε,N})) w_{⋆}(t/ε) dt = ε^{2} f,

where (t, y) are Fermi coordinates relative to N_{⋆}. Using (10.5) we compute

−ε^{2} Δu_{ε,N} + \frac{1}{2} W′(u_{ε,N}) = ε n H_{N_{s(t,y)}} w_{⋆}(t/ε)

where s(t, y) denotes the distance to of the point (t, y) to N and N_{s} is the hypersurface parallel to N at distance s. We now write N as a graph over N_{0} for some function ψ and hence N_{⋆} is the normal graph over N_{0} for the function ψ_{⋆} = R_{1/ε} ψ. We define

A_{ε}(ψ) := n ∫_{ℝ} χ(ε t) H_{N_{s(εt,y)}} w_{⋆}^{2}(t) dt.

so that the equation we have to solve reads

A_{ε}(ψ) = f.

We claim that

A_{ε}(ψ) = c_{⋆} L_{N_{0}} ψ + Q(ε, ψ),

where L_{N_{0}} is the Jacobi operator about N_{0}, where

c_{⋆} = ∫_{ℝ} w_{⋆}^{2} dt,

and where Q satisfies

|Q(ε, 0)|_{C_{ε}^{0,α}(N_{0})} ⩽ c ε^{2}, (12.3)

and

|Q(ε, ψ′) − Q(ε, ψ)|_{C_{ε}^{0,α}(N_{0})} (12.4)

⩽ c (ε^{2} + ∥ψ∥_{C_{ε}^{2,α}(N_{0})} + ∥ψ′∥_{C_{ε}^{2,α}(N_{0})}) ∥ψ′ − ψ∥_{C_{ε}^{2,α}(N_{0})},

for some constant c > 0 which depends on c but does not depend on ε nor on ψ, ψ′ satisfying

∥ψ∥_{C_{ε}^{2,α}(N_{0})} + ∥ψ′∥_{C_{ε}^{2,α}(N_{0})} ⩽ 1.

To obtain this expansion, we first use the second estimate in (11.13) to reduce to the case where N_{⋆} is replaced by N, i.e., A_{ε}(ψ) is replaced by

Ã_{ε}(ψ) := n ∫_{ℝ} χ(ε s) H_{N_{εs}} w_{⋆}^{2}(s) ds.

If D_{ε}(ψ) := Ã_{ε}(ψ) − A_{ε}(ψ), we have D_{ε}(0) = 0 and it follows from (11.13) that

|D_{ε}(ψ′) − D_{ε}(ψ)|_{C_{ε}^{0,α}(N_{0})} ⩽ c ε^{2} ∥ψ′ − ψ∥_{C_{ε}^{2,α}(N_{0})},

We now use the expansion

H_{N_{t}} = H_{N} + t Q_{1}(ψ; y) + t^{2} Q_{2}(ψ; t, y),

where Q_{1}(ψ; y) does not depend on t. In particular

n ∫_{ℝ} χ(ε s) ε s Q_{1}(ψ; y) w_{⋆}^{2}(s) ds ≡ 0,

and, if we define

E_{ε}(ψ) := n ε^{2} ∫_{ℝ} χ(ε s) s^{2} Q_{2}(ψ; εs, y) w_{⋆}^{2}(s) ds,

we obtain |E_{ε}(0)|_{C_{ε}^{0,α}(N_{0})} ⩽ c and the estimate

|E_{ε}(ψ′) − E_{ε}(ψ)|_{C_{ε}^{0,α}(N_{0})} ⩽ c ε^{2} ∥ψ′ − ψ∥_{C_{ε}^{2,α}(N_{0})},

Finally, we use the fact that

n H_{N} = n H_{N_{0}} + L_{N_{0}}ψ + Q(ψ),

and we check that

F_{ε}(ψ) := n Q(ψ) ∫_{ℝ} χ(ε s) w_{⋆}^{2}(s) ds,

satisfies Q(0) = 0 and

|F_{ε}(ψ′) − F_{ε}(ψ)|_{C_{ε}^{0,α}(N_{0})}
⩽ c (∥ψ∥_{C_{ε}^{2,α}(N_{0})} + ∥ψ′∥_{C_{ε}^{2,α}(N_{0})}) ∥ψ′ − ψ∥_{C_{ε}^{2,α}(N_{0})},

We have assumed that the operator

L_{N_{0}} : [C^{2,α}(N_{0})]_{0} ⟶ C^{0,α}(N_{0}),

is an isomorphism. Here the subscript 0 means that we are considering functions which satisfy B_{N_{0}}(ψ) = 0 on ∂N_{0}, if this boundary is not empty. This also implies that

L_{N_{0}} : [C_{ε}^{2,α}(N_{0})]_{0} ⟶ C_{ε}^{0,α}(N_{0}),

is an isomorphism, the norm of whose inverse is bounded by a constant times ε^{−α} when the space C_{ε}^{2,α}(N_{0}) is endowed with the norm ∥·∥_{C_{ε}^{2,α}(N_{0})} defined in (10.2) and the space C_{ε}^{0,α}(N_{0}) is endowed with the norm | · |_{C_{ε}^{0,α}(N_{0})} defined in (10.3). Indeed, if L_{N_{0}} v = g and if |g|_{C_{ε}^{0,α}(N_{0})} ⩽ 1, then ∥g∥_{C^{0,α}(N_{0})} ⩽ ε^{−α} and hence ∥v∥_{C^{2,α}(N_{0})} is bounded by a constant times ε^{−α}. In particular, ∥v∥_{C_{ε}^{2,α}(N_{0})} is also bounded by a constant times ε^{−α}.

The proof of the result is now a corollary of the fixed point theorem for contraction mappings, provided ε is chosen small enough. q.e.d.

We now state the corresponding result for volume-nondegenerate constant mean curvature hypersurfaces.

Lemma 12.2. Assume that N_{0} is an admissible constant mean curvature hypersurface which is volume-nondegenerate. We define

λ_{⋆} := \frac{1}{2} c_{⋆} n H_{N_{0}}

where

c_{⋆} := ∫_{−1}^{1} \sqrt{W (s)} ds.

Then, there exists ε_{0} ∈ (0, 1) and c > 0 such that for all f ∈ C^{0,α}(N_{0}) and all μ ∈ ℝ, satisfying

|f|_{C_{ε}^{0,α}(N_{0})} + |μ| ⩽ c ε,

one can find an admissible hypersurface N and a constant λ such that

S_{ε,N_{⋆}} (ℚ_{ε}(N) − ε λ) = ε^{2} f,

and

∫_{M} u_{ε,N} dv_{g} = c_{0} |M| + μ.

Moreover, if ψ_{N} is the function whose graph is N, we have

∥ψ_{N}∥_{C_{ε}^{2,α}(N_{0})} ⩽ c (|f|_{C_{ε}^{0,α}(N_{0})} + |μ|),

for some constant c > 0 which does not depend on the data f, μ nor on ε.

Proof. We now define

A_{ε}(ψ) := n ∫_{ℝ} χ(ε t) H_{N_{s(εt,y)}} w_{⋆}^{2}(t) dt − λ ∫_{ℝ} χ(ε t) w_{⋆}(t) dt,

which can be expanded as

A_{ε}(ψ) = n H_{N_{0}} ∫_{ℝ} w_{⋆}^{2}(t) dt − ε λ ∫_{ℝ} w_{⋆}(t) dt + c_{⋆} L_{N_{0}} ψ + Q(ε, ψ),

where Q(ε, ψ) satisfies (12.3) and (12.4). Now, observe that

∫_{ℝ} w_{⋆} dt = 2,

and also that

∫_{ℝ} w_{⋆}^{2} dt = ∫_{ℝ} \sqrt{W′(u_{⋆})} ∂_{t}u_{⋆} dt = ∫_{−1}^{1} \sqrt{W′(s)} ds = c_{⋆}.

Granted the definition of λ_{⋆}, we can write

A_{ε}(ψ) = c_{⋆} L_{N_{0}} ψ − 2 (λ − λ_{⋆}) + Q(ε, ψ).

We now expand

∫_{M} u_{ε,N} dv_{g} = c_{0} |M| − 2 ∫_{N_{0}} ψ da_{g} + O(ε, ψ),

where the operator O satisfies

|O(ε, 0)| ⩽ c ε^{2}, and |O(ε, ψ′) − O(ε, ψ)| ⩽ c ε^{2} ∥ψ′ − ψ∥_{C_{ε}^{2,α}(N_{0})},

for some constant c > 0 which depends on c but does not depend on ε nor on ψ, ψ′ satisfying

∥ψ∥_{C_{ε}^{2,α}(N_{0})} + ∥ψ′∥_{C_{ε}^{2,α}(N_{0})} ⩽ 1.

The proof now follows exactly the proof of the previous Lemma when volume-nondegeneracy replaces nondegeneracy. We omit the details.

q.e.d.

13. The nonlinear problem

13.1 The proof of Theorem 4.1

Assume that we are given a volume-nondegenerate admissible minimal hypersurface N_{0} in M. We would like to solve the nonlinear problem

−ε^{2} Δ_{g}(u_{ε,N} + v) + \frac{1}{2} W′(u_{ε,N} + v) = 0, (13.1)

in M. In addition, we want u_{ε,N} + v to have 0 Neumann boundary data on ∂M if this later is not empty. This means that, for all ε small enough, we would like to find an admissible hypersurface N close to N_{0} and a function v close to 0 satisfying (13.1).

Recall that we have defined the nonlinear operator

ℚ_{ε}(N) := −ε^{2} Δ_{g}u_{ε,N} + \frac{1}{2} W′(u_{ε,N}), (13.2)

which corresponds to the error we produce when we consider u_{ε,N} as a solution of (13.1). We define

ℚ̃_{ε}(N, v) := \frac{1}{2} (W′(u_{ε,N} + v) − W′(u_{ε,N}) − W′′(u_{ε,N}) v) , (13.3)

which is nothing but the Taylor expansion of the nonlinearity W′ at u_{ε,N}.

These definitions being understood, the equation we have to solve reads

ℚ_{ε}(N) + 𝕃_{ε,N} v + ℚ̃_{ε}(N, v) = 0.

Using the projection Π_{ε,N_{⋆}} defined in (11.2) and the operator S_{ε,N_{⋆}} defined in (12.2), we conclude that (13.1) is equivalent to the system

\{ \begin{matrix}Π_{ε,N_{⋆}} ∘ 𝕃_{ε,N} v & = & −Π_{ε,N_{⋆}} (ℚ_{ε}(N) + ℚ̃_{ε}(N, v)) \\ S_{ε,N_{⋆}} ℚ_{ε}(N) & = & −S_{ε,N_{⋆}} (𝕃_{ε,N} v + ℚ̃_{ε}(N, v)) , \end{matrix} (13.4)

where N_{⋆} = R_{1/ε} N is the hypersurface obtained at the end of §11 from N.

This system will be solved using a fixed point argument. The key result which allows one to apply a fixed point theorem is the following estimate:

Lemma 13.1. There exists a constant c_{0} > 0 such that, for all ε ∈ (0, 1)

∥ℚ_{ε}(N_{0})∥_{C_{ε}^{0,α}(M)} ⩽ \frac{c_{0}}{2} ε^{2}, (13.5)

and, given c_{2} > 0, there exists ε_{0} > 0 such that, for all ε ∈ (0, ε_{0}) and all admissible hypersurface N satisfying

∥N∥_{C_{ε}^{2,α}(N_{0})} ⩽ c_{2} ε^{2−α},

we have

∥ℚ_{ε}(N)∥_{C_{ε}^{0,α}(M)} ⩽ c_{0} ε^{2}. (13.6)

Proof. It follows from (10.5) that

ℚ_{ε}(N) := −ε^{2} Δ_{g}u_{ε,N} + \frac{1}{2} W′(u_{ε,N}) = n ε H_{N_{t}} w_{⋆}(t/ε),

in V_{τ_{0}/2}(N), where N_{t} is the hypersurface parallel to N at distance t. Moreover ℚ_{ε}(N) ≡ 0 in M − V_{τ_{0}}(N). Since N_{0} is minimal and ∥N∥_{C_{ε}^{2,α}(N_{0})} ⩽ c_{2} ε^{2−α}, we estimate

|H_{N_{t}}| ⩽ c (|t| + c_{2} ε^{2−α}),

V_{τ_{0}}(N) − V_{τ_{0}/2}(N) and this already implies that

∥H_{N_{t}} w_{⋆}(t/ε)∥_{L^{∞}(M)} ⩽ c (ε + c_{2} ε^{2−α}).

(The estimate in V_{τ_{0}}(N) − V_{τ_{0}/2}(N) is easy to get since ℚ_{ε}(N) is exponentially small in this set). The estimates for the Hölder derivative follows similarly.

It now suffices to apply this estimate to N_{0} itself to obtain (13.5), while (13.6) follows at once by taking ε to be small enough. q.e.d.

Assume that we are given v ∈ C_{ε}^{0,α}(M) and N an admissible hypersurface close to N_{0}. To make things quantitatively precise, we assume that

∥v∥_{C_{ε}^{0,α}(M)} ⩽ c_{1} ε^{2},  and  ∥N∥_{C_{ε}^{2,α}(M)} ⩽ c_{2} ε^{2−α}. (13.7)

For some constant c_{1} and c_{2} which will be fixed shortly. We apply the result of Proposition 11.8 and Lemma 12.1 to find ṽ and Ñ solutions of

\{ \begin{matrix}Π_{ε,N_{⋆}} ∘ 𝕃_{ε,N} ṽ & = & −Π_{ε,N_{⋆}} (ℚ_{ε}(N) + ℚ̃_{ε}(N, v)) \\ S_{ε,Ñ_{⋆}} ℚ_{ε}(Ñ) & = & −S_{ε,N_{⋆}} (𝕃_{ε,N} v + ℚ̃_{ε}(N, v)) . \end{matrix} (13.8)

Using the fact that Q̃_{ε} is quadratic in v together with Lemma 13.1, we easily get the estimates

∥ṽ∥_{C_{ε}^{2,α}(M)} ⩽ c (c_{0} ε^{2} + C_{c_{1}} ε^{4}), and
∥Ñ∥_{C_{ε}^{2,α}(M)} ⩽ c ε^{−α} (c_{1} ε^{2} + C_{c_{1}} ε^{3}).

provided ε is chosen small enough. Here the constant c > 0 neither depends on c_{1} nor on c_{2} and the constant C_{c_{1}} depends on c_{1}.

It follows at once that, if

c_{1} = 2 c c_{0}  and  c_{2} = 2 c c_{1},

and if ε is chosen small enough, this produces a continuous mapping from the set of (v, N) ∈ C_{ε}^{2,α}(M) × C_{ε}^{2,α}(N_{0}) satisfying (13.7) into itself. If this mapping were compact, we would obtain a fixed point through Schauder’s fixed point theorem. However, in our case the mapping just fails to be compact since no regularity is gained through the iteration process. To overcome this problem, we use once more the smoothing operators R_{θ} which have been introduced in the proof of Proposition 11.8. We define instead ṽ and Ñ to be the solutions of

\{ \begin{matrix}Π_{ε,N_{⋆}} ∘ 𝕃_{ε,N} ṽ & = & −Π_{ε,N_{⋆}} ∘ R_{θ} (ℚ_{ε}(N) + ℚ̃_{ε}(N, v)) \\ S_{ε,Ñ_{⋆}} ℚ_{ε}(Ñ) & = & −R_{θ} ∘ S_{ε,N_{⋆}} (𝕃_{ε,N} v + ℚ̃_{ε}(N, v)) . \end{matrix} (13.9)

We fix α′ > α and choose θ so that C θ^{α′−α} = 2, where C is the constant which appears in the estimate of the smoothing operator R_{θ}. If c_{1} = 4 c c_{0} and c_{2} = 4 c c_{1}, this produces, for all ε small enough, a continuous mapping from the set of (v, N) ∈ C_{ε}^{2,α}(M) × C_{ε}^{2,α}(N_{0}) satisfying (13.7) into itself, but this time the mapping is compact. We conclude that there exists a fixed point (v_{θ}, N_{θ}). Finally, we pass to the limit as θ tends to +∞ (i.e., as α′ tends to α). The solutions (v_{θ}, N_{θ}) of (13.9) being uniformly bounded in C_{ε}^{2,α}(M)×C_{ε}^{2,α}(N_{0}), we can extract a subsequence which converges to (v, N) in C_{ε}^{2,β}(M) × C_{ε}^{2,β}(N_{0}), for some fixed β < α. The limit (v, N) is then a solution of our problem.

13.2 The proof of Theorem 4.2

Assume that we are given a volume-nondegenerate admissible constant mean curvature hypersurface N_{0} in M. We would like to solve the nonlinear problem

−ε^{2} Δ(u_{ε,N} + v) + \frac{1}{2} W′(u_{ε,N} + v) = ε λ, (13.10)

in M, with u_{ε,N} + v having 0 Neumann boundary data if ∂M is not empty. This equation has to be complimented with the constraint

∫_{M}(u_{ε,N} + v) dv_{g} = c_{0} |M|, (13.11) where the constant c_{0} ∈ (−1, 1) is fixed so that

c_{0} |M| = |M^{+}(N_{0})| − |M^{−}(N_{0})|.

Again, in order to solve (13.10), we write the equation as a fixed point problem

\{ \begin{matrix}Π_{ε,N_{⋆}} ∘ 𝕃_{ε,N} v & = & −Π_{ε,N_{⋆}} (ℚ_{ε}(N) − ε λ + ℚ_{ε}(N, v)) \\ S_{ε,N_{⋆}} (ℚ_{ε}(N) − ε λ) & = & −S_{ε,N_{⋆}} (𝕃_{ε,N} v + ℚ̃_{ε}(N, v)) \\ ∫_{M} u_{ε,N} dv_{g} & = & c_{0} |M| − ∫_{M} v dv_{g}. \end{matrix} (13.12)

The proof of the existence of a fixed point is identical to the proof of the previous result, with Lemma 12.1 replaced by Lemma 12.2. The only difference being that the hypersurface N_{0} does not have 0 mean curvature anymore and this implies that Lemma 13.1 has to be replaced by

Lemma 13.2. There exists a constant c_{0} > 0 such that, for all ε ∈ (0, 1)

∥ℚ_{ε}(N_{0})∥_{C_{ε}^{0,α}(M)} ⩽ \frac{c_{0}}{2} ε,

and, given c_{2} > 0, there exists ε_{0} > 0 such that, for all ε ∈ (0, ε_{0}) and all admissible hypersurface N satisfying

∥N∥_{C_{ε}^{2,α}(N_{0})} ⩽ c_{2} ε^{1−α},

we have

∥ℚ_{ε}(N)∥_{C_{ε}^{0,α}(M)} ⩽ c_{0} ε.

The fact that we do not get an estimate as good as the one obtained in Lemma 13.1 is a consequence of the fact that the mean curvature of N_{0} is not necessarily equal to 0.

This implies that the condition (13.7) has to be replaced by

∥v∥_{C_{ε}^{0,α}(M)} ⩽ c_{1} ε,  and  ∥N∥_{C_{ε}^{2,α}(M)} ⩽ c_{2} ε^{1−α}.

Details are left to the reader.

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Centre de Mathématiques

Faculté de Sciences et Technologie Université Paris XII - Val de Marne

94 010 Creteil Cedex France

Departamento de Geometría y Topología

Facultad de Ciencias Universidad de Granada

E–18071 Granada España