Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 31, 2006, 405–436

CALORIC MEASURE AND REIFENBERG FLATNESS

Kaj Nyström

Umeå University, Department of Mathematics SE-90187 Umeå, Sweden; kaj.nystrom@math.umu.se

Abstract. In this paper we study a (two-phase) free boundary regularity problem for caloric measure in parabolic δ_{0} -Reifenberg flat domains Ω ⊂ R^{n+1} . In particular for such a domain we define Ω^{1} = Ω ⊂ R^{n+1} , Ω^{2} = R^{n+1} \ \overline{Ω} and we let, for i ∈ \{1, 2\}, ω^{i}(X̂^{i}, t̂^{i}, ·) be the caloric measure at (X̂^{i}, t̂^{i}) ∈ Ω^{i} defined with respect to Ω^{i}. If t̂^{2} < t̂^{1} we assume that ω^{2}(X̂^{2}, t̂^{2}, ·) is absolutely continuous with respect to ω^{1}(X̂^{1}, t̂^{1}, ·) on ∂Ω and we denote by k(X̂^{1}, t̂^{1}, X̂^{2}, t̂^{2}, ·) = dω^{2}(X̂^{2}, t̂^{2}, ·)/dω^{1}(X̂^{1}, t̂^{1}, ·) the Radon–Nikodym derivative. Our main result states that there exists δ_{n} > 0 such that if δ_{0} < δ_{n} and if

log k(X̂^{1}, t̂^{1}, X̂^{2}, t̂^{2}, ·) ∈ VMO(dω^{1}(X̂^{1}, t̂^{1}, ·))

then C_{r}(X, t) ∩ ∂Ω is Reifenberg flat with vanishing constant whenever (X, t) ∈ ∂Ω and t̂^{2} > t + 4r^{2} .

1. Introduction

In this paper we study a free boundary regularity problem for caloric measure below the continous threshold. We consider unbounded domains Ω ⊂ R^{n+1} assuming that ∂Ω is δ_{0} -Reifenberg flat in the parabolic sense (this notion is defined below). As is described below the bounded continuous Dirichlet problem for the heat equation always has a unique solution in this type of domains. Let (X, t), X = (x_{0}, …, x_{n−1}), t ∈ R denote a point in R^{n+1} and for given r > 0 set C_{r}(X, t) = {(Y, s) : |Y − X| < r, |t − s| < r^{2}}. For fixed (X̂, t̂) ∈ Ω we let ω(X̂, t̂, ·) denote the parabolic measure (in this paper this measure is referred to as the caloric measure) for the heat equation obtained from the maximum principle and the Riesz representation theorem. Let Δ(X, t, r) = C_{r}(X, t) ∩ ∂Ω whenever (X, t) ∈ ∂Ω and r > 0. Given (X̂, t̂) ∈ Ω let (X, t) ∈ ∂Ω and suppose |X − X̂|^{2} ≤ A(t̂ − t) for some A ≥ 2. In [HLN2] it is proven that ω(X̂, t̂, ·) is, in the setting of Reifenberg flat domains as well as in the more general setting of parabolic NTA domains, a doubling measure in the sense that there exists a_{1} = a_{1}(n, A) such that if t̂− t ≥ 8r^{2} then

ω(X̂, t̂, Δ(X, t, 2r)) ≤ a_{1} ω(X̂, t̂, Δ(X, t, r)).

2000 Mathematics Subject Classification: Primary 35K05.

We let Ω^{1} = Ω ⊂ R^{n+1} , Ω^{2} = R^{n+1} \ \overline{Ω}. We also let (X̂^{i}, t̂^{i}) ∈ Ω^{i}, for i ∈ {1, 2}, t̂^{2} < t̂^{1} and define ω^{1}(X̂^{1}, t̂^{1}, ·) and ω^{2}(X̂^{2}, t̂^{2}, ·) to be the caloric measures defined with respect to Ω^{1} and Ω^{2} , respectively. In the following we will assume that ω^{2}(X̂^{2}, t̂^{2}, ·) is absolutely continuous with respect to ω^{1}(X̂^{1}, t̂^{1}, ·) on ∂Ω and that the Radon–Nikodym derivative

k(X̂^{1}, t̂^{1}, X̂^{2}, t̂^{2}, ·) = dω^{2}(X̂^{2}, t̂^{2}, ·)/dω^{1}(X̂^{1}, t̂^{1}, ·)

is such that log k(X̂^{1}, t̂^{1}, X̂^{2}, t̂^{2}, ·) ∈ VMO(dω^{1}).

VMO(dω^{1}) = VMO(dω^{1}(X̂^{1}, t̂^{1}, ·))

is the space of functions of vanishing mean oscillation defined with respect to the measure ω^{1}(X̂^{1}, t̂^{1}, ·). This space is defined in the bulk of the paper.

To formulate our main theorem we need to properly introduce the notion of δ_{0} -Reifenberg flat domains.

Definition 1. If Ω is a connected open set in R^{n+1} then we say that ∂Ω separates R^{n+1} and is δ_{0} -Reifenberg flat, 0 < δ_{0} ≤ 1/10, if given any (X, t) ∈ ∂Ω, R > 0, there exists an n-dimensional plane P̂ = P̂(X, t, R), containing (X, t) and a line parallel to the t-axis, having unit normal n̂ = n̂(X, t, R) such that

\begin{matrix}\{(Y, s) + rn̂ ∈ C_{R}(X, t) : (Y, s) ∈ P̂, r > δ_{0}R\} ⊂ Ω, \\ \{(Y, s) − rn̂ ∈ C_{R}(X, t) : (Y, s) ∈ P̂, r > δ_{0}R\} ⊂ R^{n+1} \ Ω. \end{matrix}

For short we say that ∂Ω separates R^{n+1} when the last two conditions hold for some δ_{0} .

Note that if ∂Ω separates R^{n+1} in the sense of Definition 1, then a line segment drawn parallel to n̂ and with endpoints in each of the sets stated in the definition also intersects ∂Ω. We will often refer to Ω as being a δ_{0} -Reifenberg flat domain if ∂Ω is δ_{0} -Reifenberg flat. We pose one more definition.

Definition 2. Let Ω be a connected open set in R^{n+1} , (X, t) ∈ ∂Ω, and r > 0. We say that C_{r}(X, t) ∩ ∂Ω is Reifenberg flat with vanishing constant in the parabolic sense, if for each ε > 0, there exists ϱ_{0} = ϱ_{0}(ε) > 0 with the following property. If (X̃ , t̃) ∈ C_{r}(X, t) ∩ ∂Ω and 0 < ϱ ≤ ϱ_{0} , then there exists a plane P′(X̃ , t̃, ϱ) containing a line parallel to the t axis such that the statement in Definition 1 holds with R, δ_{0} , P̂ replaced by ϱ, ε and P′.

We can now formulate our main result.

Theorem 1. Let Ω ⊂ R^{n+1} be a δ_{0} -Reifenberg flat domain and define Ω^{1} = Ω ⊂ R^{n+1} , Ω^{2} = R^{n+1} \ \overline{Ω}. Let (X̂^{i}, t̂^{i}) ∈ Ω^{i}, for i ∈ \{1, 2\}, t̂^{2} < t̂^{1} and assume that ω^{2}(X̂^{2}, t̂^{2}, ·) is absolutely continuous with respect to ω^{1}(X̂^{1}, t̂^{1}, ·) on ∂Ω and that the Radon–Nikodym derivative

k(X̂^{1}, t̂^{1}, X̂^{2}, t̂^{2}, ·) = dω^{2}(X̂^{2}, t̂^{2}, ·)/dω^{1}(X̂^{1}, t̂^{1}, ·)

is such that

log k(X̂^{1}, t̂^{1}, X̂^{2}, t̂^{2}, ·) ∈ VMO(dω^{1}(X̂^{1}, t̂^{1}, ·)).

Then there exists δ_{n} > 0 such that if δ_{0} < δ_{n} then C_{r}(X, t) ∩ ∂Ω is Reifenberg flat with vanishing constant whenever (X, t) ∈ ∂Ω and t̂^{2} > t + 4r^{2} .

In [KT3], Kenig and Toro consider the elliptic version of the two-phase problem stated in Theorem 1. In particular, they prove ([KT3, Corollary 4.1]) that if Ω ⊂ R^{n} is δ_{0} -Reifenberg flat (in the elliptic sense) for small δ_{0} > 0 and if k = dω^{2}/dω^{1} , log k ∈ VMO(dω^{1}), then ∂Ω is Reifenberg flat with vanishing constant. In the elliptic setting questions of this type have previous been addressed, from a slightly different point of view, in the case n = 2, i.e., in the plane, through the works of Bishop, Carleson, Garnett and Jones. See [B], [BCGJ] and [BJ]. We are not aware of any results of this type in the parabolic setting.

The rest of the paper is organized as follows. In Section 2 we put the problem considered in this paper in perspective and state that our main result, i.e. Theorem 1, can be seen as part of a program focusing on the understanding of certain parabolic one-phase and two-phase problems in parabolic Reifenberg flat domains. In Section 3 we in Section 3.1 list some basic estimates for solutions to the heat or adjoint heat equation in parabolic NTA domains. These estimates are then complemented, in Section 3.2, by a set of what we refer to as refined estimates, the latter being based on δ_{0} -Reifenberg flatness and an exploration of the condition log k(X̂^{1}, t̂^{1}, X̂^{2}, t̂^{2}, ·) ∈ VMO(dω^{1}(X̂^{1}, t̂^{1}, ·)). In Section 3.3 we clarify the notion of Green function with pole at infinity and the associated caloric measure. In Section 4, which is at the heart of the matter, our regularity assumption on the kernel k(X̂^{1}, t̂^{1}, X̂^{2}, t̂^{2}, ·) is explored in a blow-up argument. In the limit we encounter the problem of classification of what we refer to as global solutions to a specific two-phase free boundary problem. The section ends with a theorem giving us the appropriate classification and finally it is shown that Theorem 1 is a consequence of that classification theorem.

2. One and two-phase free boundary problems below the continous threshold

In this section we put Theorem 1 into perspective and briefly discuss how this result can be seen as part of a program focusing on the understanding of certain parabolic one-phase and two-phase problems in parabolic Reifenberg flat domains.

Given a Borel set F ⊂ R^{n+1} we let \overline{F}, ∂F denote the closure and the boundary of F respectively and define σ(F) = ∫_{F} dσ_{t} dt where dσ_{t} is n − 1dimensional Hausdorff measure on the time slice F ∩ (R^{n} × {t}). Let Ω be a connected open set in R^{n+1} .

Definition 3. We say that ∂Ω satisfies a (M, R) Ahlfors condition, M ≥ 4, if for all (X, t) ∈ ∂Ω and 0 < r ≤ R,

σ(∂Ω ∩ C_{r}(X, t)) ≤ Mr^{n+1}.

Combining the notion of Reifenberg flatness and the Ahlfors condition, the fact that Hausdorff measure does not increase under a projection we deduce that for 0 < r ≤ R, (X, t) ∈ ∂Ω,

(r/2)^{n+1} ≤ σ(∂Ω ∩ C_{r}(X, t)) ≤ Mr^{n+1},

whenever ∂Ω separates R^{n+1} and satisfies a (M, R) Ahlfors condition.

Let Ω ⊂ R^{n+1} be a δ_{0} -Reifenberg flat domain and define Ω^{1} = Ω ⊂ R^{n+1} , Ω^{2} = R^{n+1} \ \overline{Ω}. As above we let (X̂^{i}, t̂^{i}) ∈ Ω^{i}, for i ∈ \{1, 2\}, t̂^{2} < t̂^{1} and define ω^{1}(X̂^{1}, t̂^{1}, ·) and ω^{2}(X̂^{2}, t̂^{2}, ·) to be the caloric measures defined with respect to Ω^{1} and Ω^{2} , respectively. Based on this notation there are several problems of free boundary type that one can pose. As discussed in this paper we can assume that ω^{2}(X̂^{2}, t̂^{2}, ·) is absolutely continuous with respect to ω^{1}(X̂^{1}, t̂^{1}, ·) on ∂Ω and that the Radon–Nikodym derivative k(X̂^{1}, t̂^{1}, X̂^{2}, t̂^{2}, ·) = dω^{2}(X̂^{2}, t̂^{2}, ·)/dω^{1}(X̂^{1}, t̂^{1}, ·) is such that log k(X̂^{1}, t̂^{1}, X̂^{2}, t̂^{2}, ·) ∈ VMO(dω^{1}). The question is then what these conditions imply on the geometry and regularity of ∂Ω. Assuming that ∂Ω is δ_{0} -Reifenberg flat and satisfies a (M, R) Ahlfors condition it is also relevant to study the implication of similar conditions phrased in terms of the Poisson kernel. I.e., we could assume at least one of the caloric measures ω^{i}(X̂^{i}, t̂^{i}, ·) to be absolutely continuous with respect to σ on ∂Ω and hence define a Poisson kernel as k̃^{i}(X̂^{i}, t̂^{i}, ·) = dω^{i}(X̂^{i}, t̂^{i}, ·)/dσ. Using this notation the following natural problems, in the spirit of the one considered in this paper, can be formulated. VMO(dσ) is the space of functions of vanishing mean oscillation, defined with respect to dσ, defined in the bulk of the paper.

1. (One-phase problem). Assume that log k̃^{1}(X̂^{1}, t̂^{1}, ·) ∈ VMO(dσ). What implications does this condition have on ∂Ω?

2. (Two-phase problem). Assume that log k̃^{1}(X̂^{1}, t̂^{1}, ·) ∈ VMO(dσ) and that log k̃^{2}(X̂^{2}, t̂^{2}, ·) ∈ VMO(dσ). What implications do these conditions have on ∂Ω? In the following we briefly describe recent developments on these kind of

problems in the geometric settings of this paper. We first discuss the development in the elliptic situation before we focus on the parabolic situation.

2.1. Elliptic theory. In [D] B. Dahlberg showed that in a Lipschitz domain Ω the harmonic measure with respect to a fixed point, dω, and surface measure, dσ, are mutually absolutely continuous. In fact if k̃ = dω/dσ, then Dahlberg showed that k̃ is in a certain L^{2} reverse Hölder class from which it follows that log k̃ ∈ BMO(dσ), the functions of bounded mean oscillation with respect to surface area on ∂Ω. Jerison and Kenig [JK] showed for a C^{1} domain that log k̃ ∈ VMO(dσ), the functions in BMO(dσ) of vanishing mean oscillation. In [KT] this result was generalized to ‘chord arc domains with vanishing constant’. Concerning reverse conclusions, i.e., elliptic free boundary problems a classical result of Alt– Caffarelli states (for the definition of all the concepts we refer to [AC] and [KT2]) that if Ω ⊂ R^{n} is δ_{0} -Reifenberg flat with an Ahlfors regular boundary and if log k̃ ∈ C^{0,β}(∂Ω) for some β ∈ (0, 1), then Ω is a C^{1,α} -domain for some α ∈ (0, 1) which depends on β and n. In [J] Jerison proved that α = β. The conclusion is that the oscillation of the logarithm of the Poisson kernel controls the geometry and in particular the ‘flatness’ or the oscillation of the unit normal. Furthermore in [J] Jerison treated a case beyond the C^{0,β} situation for β > 0 under the assumption that the domain is locally given as the graph of a Lipschitz function and assuming that the normal derivative is continuous instead of having just vanishing mean oscillation. In the setting of domains not locally given by graphs, in [KT2], Kenig and Toro were able to prove the following theorem which is the analogue of the result of [AC] assuming vanishing oscillation of the logarithm of the Poisson kernel in an integral sense (VMO(dσ)) instead of in the classical pointwise sense.

Theorem 2. Assume that Ω ⊂ R^{n} is δ_{0} -Reifenberg flat for some small enough δ_{0} > 0 and assume that ∂Ω is Ahlfors regular. If log k̃ ∈ VMO(dσ) then Ω is a chord arc domain with vanishing constant, i.e., the measure theoretical normal n⃗ is in VMO(dσ).

This theorem can be seen as an answer to the elliptic one-phase type problem stated as Problem 1 above. In [KT3], Kenig and Toro consider the elliptic version of the two-phase problem we consider in Theorem 1. In particular, they prove ([KT3, Corollary 4.1]) that if Ω ⊂ R^{n} is a δ_{0} -Reifenberg flat (in the elliptic sense) for some small enough δ_{0} > 0 and if k = dω^{1}/dω^{2} , log k ∈ VMO(dω^{1}), then ∂Ω is Reifenberg flat with vanishing constant. In the elliptic setting questions of this type have previous been addressed in the case n = 2, i.e., in the plane, through the works of Bishop, Carleson, Garnett and Jones; see [B], [BCGJ] and [BJ].

Assuming that Ω ⊂ R^{n} is a two-sided chord arc domain (meaning that Ω^{1} and Ω^{2} are NTA-domains and that ∂Ω is Ahlfors) they also prove ([KT3, Corollary 5.2]) that if log k̃^{1} ∈ VMO(dσ) and log k̃^{2} ∈ VMO(dσ) then firstly ∂Ω is Reifenberg flat with vanishing constant and secondly Ω is a chord arc domain with vanishing constant, i.e., the measure theoretical normal n⃗ is in VMO(dσ). This result can be seen as an answer to the elliptic two-phase type problem stated as Problem 2 above. One interesting aspect of the last result is that by imposing the two-phase condition log k̃^{1} ∈ VMO(dσ) and log k̃^{2} ∈ VMO(dσ) the conclusion of Theorem 2 remains true without an assumption on Reifenberg flatness. I.e., the two-phase condition serves as a replacement for flatness.

2.2. Parabolic theory. Through the works in [LM], [HL] it has become clear that from the perspective of parabolic singular integrals and caloric measure the parabolic analogue of the notion of Lipschitz domains, explored in elliptic partial differential equations, is graph domains Ω = {(X, t) ∈ R^{n+1} : x_{0} > ψ(x, t)} where ψ = ψ(x, t): R^{n} → R has compact support and satisfies

|ψ(x, t) − ψ(y, t)| ≤ b_{1}|x − y|, x, y ∈ R^{n−1}, t ∈ R,  (1)
D_{1/2}^{t}ψ ∈ BMO(R^{n}), ∥D_{1/2}^{t}ψ∥_{∗} ≤ b_{2} < ∞.  (2)

Here D_{1/2}^{t}ψ(x, t) denotes the 1/2 derivative in t of ψ(x, ·), x fixed. This half derivative in time can be defined by way of the Fourier transform or by

D_{1/2}^{t}ψ(x, t) ≡ ĉ∫_{R} \frac{ψ(x, s) − ψ(x, t)}{|s − t|^{3/2}} ds

for properly chosen ĉ. ∥ · ∥_{∗} denotes the norm in parabolic BMO(R^{n}) (for a definition of this space see [HLN2]). One can prove that the conditions in (1) and (2) imply that ψ(x, t) is parabolically Lipschitz in the following sense,

|ψ(x, t) − ψ(y, s)| ≤ β(|x − y| + |t − s|^{1/2}) x, y ∈ R^{n} t, s ∈ R.

Under the smoothness assumptions on ψ stated in (1) and (2) it was proven in [LM] that the parabolic Poisson kernel is in a certain L^{p} reverse Hölder class for some p > 1. In particular ω(X̂, t̂, ·) is an A^{∞} weight (with respect to σ). The result of [LM] was later shown to be sharp in [HL] where examples are given of graph domains, as in [LM], with p arbitrarily close to 1. In [HL] the relevant L^{2} -result was established. Finally we note that examples of [KW] and [LS] show that caloric and adjoint caloric measure need not be absolutely continous with respect to the surface measure σ in graph Lip(1, 1/2) domain.

In [HLN2] the parabolic Poisson kernel was analyzed in domains not locally given by graphs. In this situation the geometry was controlled by a certain geometric square function, the boundedness of which implied that on every scale the boundary contained ‘big pieces of graph’, graph with the regularity stated in (1) and (2) (see [HLN1]). A fundamental assumption in [HLN2] is that ∂Ω is δ_{0} - Reifenberg flat and satisfies a (M, R) Ahlfors condition (as defined above) but to properly formulate the result in [HLN2] we need to introduce some more notation and concepts.

Let

d(F_{1}, F_{2}) = inf{|X − Y | + |s − t|^{1/2} : (X, t) ∈ F_{1}, (Y, s) ∈ F_{2}}

denote the parabolic distance between the sets F_{1} , F_{2} and for Ω (such that ∂Ω separates R^{n+1} and satisfies a (M, R) Ahlfors condition) we set

γ(Z, τ, r) = inf_{P}[r^{−n−3} ∫_{∂Ω∩C_{r}(Z,τ)} d({(Y, s)}, P)^{2} dσ(Y, s)].

Here the infimum is taken over all n-dimensional planes P containing a line parallel to the t axis. Let

dν(Z, τ, r) = γ(Z, τ, r) dσ(Z, τ) r^{−1}dr.

We say that ν is a Carleson measure on [∂Ω ∩ C_{R}(Y, s)] × (0, R) if there exists M_{1} < ∞ such that whenever (X, t) ∈ ∂Ω and C_{ϱ}(X, t) ⊂ C_{R}(Y, s), we have

(3)  ν([C_{ϱ}(X, t) ∩ ∂Ω] × (0, ϱ)) ≤ M_{1} ϱ^{n+1}.

The smallest such M_{1} is called the Carleson norm of ν on [∂Ω∩C_{R}(Y, s)]×(0, R) and we write ∥ν∥_{+} for the Carleson norm of ν if the inequality in (3) holds for all ϱ > 0. The following two definitions can be found in [HLN1] and [HLN2].

Definition 4. ∂Ω is said to be uniformly rectifiable (in the parabolic sense) if ∥ν∥_{+} < ∞ and (3) holds for all R > 0. If furthermore ∂Ω separates R^{n+1} and is uniformly rectifiable, then Ω is called a parabolic regular domain.

Definition 5. Ω is called a chord arc domain with vanishing constant if Ω is a parabolic regular domain and

(4)  sup_{{(X,t)∈∂Ω \atop 0<ϱ≤r}}[ϱ^{−(n+1)} ν([C_{ϱ}(X, t) ∩ ∂Ω] × (0, ϱ))] → 0 as r → 0.

In [HLN2] it is proven that if Ω is a parabolic regular domain with Reifenberg constant δ_{0} = δ_{0}(M, ∥ν∥_{+}), sufficiently small, then ω is an A^{∞} weight. Furthermore if Ω is a chord arc domain with vanishing constant and k̃(X̂, t̂, ·) = dω(X̂, t̂, ·)/dσ, then log k̃(X̂, t̂, ·) ∈ VMO(dσ).

To formulate the result in [HLN2] which is more relevant to the discussions in this paper let a = a(Δ(X, t, ϱ), f) denote the average of f = log k̃(X̂, t̂, ·) on Δ(X, t, ϱ) with respect to σ. Then we say that f ∈ VMO(dσ) provided for each compact K ⊂ ∂Ω ∩ {(Y, s) : s < t̂},

lim_{r→0} sup_{{(X,t)∈K \atop 0<ϱ≤r}} σ(Δ(X, t, ϱ))^{−1} ∫_{Δ(X,t,ϱ)} |f(Y, s) − a| dσ = 0.

Let (X, t) ∈ ∂Ω, and r, ϱ > 0 and put

Δ(X, t, r, ϱ) = {(Y, s) ∈ ∂Ω : |Y − X| < r, |s − t| < ϱ^{2}}.

In this notation Δ(X, t, r) = Δ(X, t, r, r). We say that ω(X̂, t̂, ·) is asymptotically optimal doubling if, whenever K ⊂ ∂Ω ∩ {(Y, s) : s < t̂} is compact and 0 < τ_{1}, τ_{2} < 1, we have

lim_{r→0} sup_{(X,t)∈K} \frac{ω(Δ(X, t, τ_{1}r, τ_{2}r))}{ω(Δ(X, t, r))} = lim_{r→0} inf_{(X,t)∈K} \frac{ω(Δ(X, t, τ_{1}r, τ_{2}r))}{ω(Δ(X, t, r))} = τ_{1}^{n−1}τ_{2}^{2}.

In [HLN2] the following theorem is proven.

Theorem 3. Let Ω be a parabolic regular domain and put k̃(X̂, t̂, ·) = dω(X̂, t̂, ·)/dσ. If ω(X̂, t̂, ·) is asymptotically optimal doubling, log k̃(X̂, t̂, ·) ∈ VMO(dσ) and ∥ν∥_{+} is small enough then (4) holds with ∂Ω replaced by any compact subset, F ⊂ ∂Ω ∩ {(Y, s) : s < t̂}.

Note that this result is weaker than the result proved in [KT2] as Theorem 3 uses the assumption that ω(X̂, t̂, ·) is asymptotically optimal doubling. In fact the proof in [KT2] uses the important result of [AC] for elliptic pde, whose potential generalization to the heat equation is currently unknown. In fact these ‘free boundary’ type problems appear harder in the caloric case. Similar problems have been considered in [ACS], [ACS1] under stronger assumptions.

Summarizing we can conclude that to a large extent the answers to Problem 1 and 2 in the parabolic setting remain unclear but that the main theorem of this paper, Theorem 1, gives a perfect parabolic analogue to the corresponding elliptic result ([KT3, Corollary 4.1]).

3. Estimates of caloric functions in parabolic NTA-domains

Recall from [LM, Chapter 3, Section 6] that Ω ⊂ R^{n+1} is an unbounded parabolic nontangentially accessible domain if ∂Ω separates R^{n+1} and if the following conditions are satisfied for some λ, γ ≥ 100. Given (X, t) ∈ ∂Ω and r > 0 there exist

\begin{aligned}\underline{A}_{r}^{1}(X, t) & = (U_{1}(X, t), t_{1}(X, t)) = (U_{1}, t_{1}) ∈ Ω ∩ C_{r}(X, t), \\ Ā_{r}^{1}(X, t) & = (U_{2}(X, t), t_{2}(X, t)) = (U_{2}, t_{2}) ∈ Ω ∩ C_{r}(X, t), \\ \underline{A}_{r}^{2}(X, t) & = (N_{1}(X, t), τ_{1}(X, t)) = (N_{1}, τ_{1}) ∈ (R^{n+1} \ \overline{Ω}) ∩ C_{r}(X, t), \\ Ā_{r}^{2}(X, t) & = (N_{2}(X, t), τ_{2}(X, t)) = (N_{2}, τ_{2}) ∈ (R^{n+1} \ \overline{Ω}) ∩ C_{r}(X, t), \end{aligned}

such that

λ^{−1}r^{2} ≤ min(t_{2} − t, t − t_{1}) ≤ λr^{2},
λ^{−1}r^{2} ≤ min(τ_{2} − t, t − τ_{1}) ≤ λr^{2},
r/λ ≤ min[d({(N_{i}, τ_{i})}, ∂Ω), d({(U_{i}, t_{i})}, ∂Ω)].

Here d( · , ·) denotes the parabolic distance defined in the previous section. As in [JK1] we refer to these conditions as the corkscrew condition. Next suppose (U_{i}, s_{i}) ∈ Ω, i = 1, 2, with (s_{2} − s_{1})^{1/2} > γ^{−1}d({(U_{1}, s_{1})}, {(U_{2}, s_{2})}). We say as in [JK1] that {C_{r_{i}}(X_{i}, t_{i})}_{1}^{l} is a Harnack chain from (U_{1}, s_{1}) to (U_{2}, s_{2}) with constant γ provided there exists c(γ) ≥ 1 such that

• (U_{1}, s_{1}) ∈ C_{r_{1}}(X_{1}, t_{1}), (U_{2}, s_{2}) ∈ C_{r_{l}}(X_{l}, t_{l}), and for i = 1, 2, …, l − 1, C_{r_{i+1}}(X_{i+1}, t_{i+1}) ∩ C_{r_{i}}(X_{i}, t_{i}) ≠ ∅,

• c(γ)^{−1}d({(X_{i}, t_{i})}, ∂Ω) ≤ r_{i} ≤ c(γ)d({(X_{i}, t_{i})}, ∂Ω), when i = 1, 2, …, l,

• t_{i+1} − t_{i} ≥ c(γ)^{−1}r_{i}^{2} , for i = 1, 2, …, l,

• l ≤ c(γ) log(2 + \frac{d(\{(U_{1}, s_{1})\}, \{(U_{2}, s_{2})\})}{min[d(\{(U_{1}, s_{1})\}, ∂Ω), d(\{(U_{2}, s_{2})\}, ∂Ω)]}).

l is referred to as the length of the Harnack chain. Our first lemma states that δ -Reifenberg flat domains are examples of parabolic NTA-domains. This lemma is also proven in [HLN2].

Lemma 4. Let Ω be δ_{0} -Reifenberg flat. If δ_{0} > 0 is sufficiently small, then Ω is a parabolic NTA domain for λ = 100 and any γ ≥ 100.

Proof. Let (X, t) ∈ ∂Ω and r > 0. The definition of Reifenberg flatness (Definition 1) implies that for δ_{0} > 0 sufficiently small and with λ = 100, the corkscrew conditions are fulfilled with

\begin{aligned}\underline{A}_{r}^{1}(X, t) & = (U_{1}, t_{1}) = (X + rn̂, t − r^{2}), Ā_{r}^{1}(X, t) = (U_{2}, t_{2}) = (X + rn̂, t + r^{2}), \\ \underline{A}_{r}^{2}(X, t) & = (N_{1}, τ_{1}) = (X − rn̂, t − r^{2}), Ā_{r}^{2}(X, t) = (N_{2}, τ_{2}) = (X − rn̂, t + r^{2}). \end{aligned}

Here n̂ = n̂(X, t, r). To prove, for any γ ≥ 100, the existence of Harnack chains we follow [KT1] and for (U_{1}, s_{1}), (U_{2}, s_{2}) ∈ Ω, as above, we choose points (P_{1}, t_{1}), (P_{2}, t_{2}) ∈ ∂Ω with

ϱ_{i} = d({(P_{i}, t_{i})}, {(U_{i}, t_{i})}) = d({(U_{i}, t_{i})}, ∂Ω), i = 1, 2.

If ϱ = d({(U_{1}, t_{1})}, {(U_{2}, t_{2})}), then using that Ω is δ_{0} -Reifenberg flat for δ_{0} > 0 small, we can choose a Harnack chain of length

l ≤ c(γ) log(2 + \frac{ϱ}{min\{ϱ_{1}, ϱ_{2}\}})

joining (U_{1}, s_{1}) to (U_{2}, s_{2}). If for example ϱ > 1000 max(ϱ_{1}, ϱ_{2}) and l_{0} is the smallest positive integer greater than log(ϱ/ϱ_{1}), then from the Reifenberg flatness it follows that we can choose {(X_{i}, t_{i})} with (X_{i}, t_{i}) ∈ C_{e^{i}ϱ_{1}}(P_{1}, t_{1}) for 2 ≤ i ≤ l_{0} and then (X_{i}, t_{i}) ∈ C_{e^{l_{0}−i}ϱ}(P_{2}, t_{2}) for l_{0}+1 ≤ i ≤ l_{0}+l_{1} , where l_{1} ≤ log(2ϱ/ϱ_{2}).

In this section we will assume, in order to ensure that Lemma 4 is valid, that Ω is a δ_{0} -Reifenberg flat with small constant. For (X, t) ∈ ∂Ω and r > 0 we define the following points located in Ω,

\underline{A}_{r}(X, t) = (X +rn̂, t−r^{2}), A_{r}(X, t) = (X +rn̂, t), Ā_{r}(X, t) = (X +rn̂, t+r^{2}).

Again here n̂ = n̂(X, t, r). The existence of these points is a consequence of the δ_{0} -Reifenberg flatness and we will make use of these points throughout the section.

3.1. Basic estimates. In this section we state some basic estimates for certain solutions to the heat and adjoint heat equation in parabolic NTA domains. An outline of the proofs of these lemmas valid in the current situation can be found in [LM, Chapter 3, Section 6] and [HLN2]. Apart from these references many of the relevant ideas used in the proofs can also be found in [FS], [FSY] and [N]. In particular, in [N] all relevant estimates are stated and proved, in Lip(1, 1/2)- domains, in the general setting of second order parabolic equations in divergence form.

Note that the characteristics of a parabolic NTA-domain is described by the parameters λ and γ and hence basically all constants appearing below will depend on these two parameters. I.e., below c = c(λ, γ) but the constants often also depend on other parameters and we will not always indicate the dependence on λ and γ.

We start by a lemma on Hölder decay at the boundary of non-negative solutions vanishing on the boundary. The lemma is proved using standard comparison arguments and the fact that the complement of Ω is uniformly ‘fat’.

Lemma 5. Let Ω ⊂ R^{n+1} be a parabolic NTA-domain with parameters λ and γ. Let (X, t) ∈ ∂Ω and suppose that u is a non-negative solution to either the heat or the adjoint heat equation in Ω∩C_{2r}(X, t) which vanishes continuously on ∂Ω∩C_{2r}(X, t). Then there exists α = α(λ, γ), 0 < α < \frac{1}{2} , and c = c(λ, γ) ≥ 1 such that whenever (Y, s) ∈ Ω ∩ C_{r}(X, t)

u(Y, s) ≤ c[\frac{d(\{(Y, s)\}, \{(X, t)\})}{r}]^{α} sup_{(Z,τ)∈Ω∩C_{r}(X,t)} u(Z, τ).

The next lemma is a standard Carleson type lemma.

Lemma 6. Let u, Ω and (X, t) be as in the previous lemma. If (Y, s) ∈ Ω ∩ C_{r/2}(X, t), then

u(Y, s) ≤ cu(Ā_{r}(X, t))

when u is a solution to the heat equation while

u(Y, s) ≤ cu(\underline{A}_{r}(X, t))

when u is a solution to the adjoint heat equation in C_{2r}(X, t) ∩ Ω.

Given (Y, s) ∈ Ω, let G( · , Y, s) denote Green’s function for the heat equation in Ω with pole at (Y, s). That is

\frac{∂}{∂t}G(X, t, Y, s) − ΔG(X, t, Y, s) = δ((X, t) − (Y, s)) in Ω and G ≡ 0 on ∂Ω.

Here δ denotes the Dirac delta function and Δ is the Laplacian in X. We note that G(Y, s, ·) is Green’s function for the adjoint heat equation with pole at (Y, s) ∈ Ω (i.e. −(∂/∂t)G(Y, s, ·)−ΔG(Y, s, ·) = δ( ·−(Y, s)). Let ω, ω̂ be the corresponding caloric and adjoint caloric measures for the heat/adjoint heat equation in Ω. We note that ω(Y, s, ·), ω̂(Y, s, ·) are the Riesz measures associated with G(·, Y, s), G(Y, s, ·) by way of the Riesz representation theorem for sub caloric/adjoint caloric functions in R^{n+1} \ {(Y, s)} (see [Do]). From this theorem we have that

∫_{∂Ω} ϕ dω(Y, s, ·) = ∫_{Ω} G(Y, s, ·)(Δϕ − \frac{∂ϕ}{∂s}) dZ dτ

for all ϕ ∈ C_{0}^{∞}(R^{n+1} \ {(Y, s)}). A similar formula holds for ω̂. Estimates for caloric/adjoint caloric measure in terms of Green’s function and vice versa are given by the following lemma. The proof follows by standard arguments.

Lemma 7. Let Ω and (X, t) be as in the previous lemma. Let A ≥ 100 and assume that (Y, s) ∈ Ω with |Y − X|^{2} ≤ A|s − t| and |s − t| ≥ 4r^{2} . There exists c = c(A) ≥ 1 such that if s > t, then

c^{−1}r^{n}G(Y, s, Ā_{r}(X, t)) ≤ ω(Y, s, Δ(X, t, r/2)) ≤ cr^{n}G(Y, s, \underline{A}_{r}(X, t))

while if s < t,

c^{−1}r^{n}G(\underline{A}_{r}(X, t), Y, s) ≤ ω̂(Y, s, Δ(X, t, r/2)) ≤ cr^{n}G(Ā_{r}(X, t), Y, s).

Next we have the following backward Harnack inequality.

Lemma 8. Let Ω and (X, t) be as in the previous lemma. Let A ≥ 100 and assume that |Y − X|^{2} ≤ A|s − t| and |s − t| ≥ 5r^{2} . There exists c = c(A) ≥ 1 such that

G(Y, s, \underline{A}_{r}(X, t)) ≤ cG(Y, s, Ā_{r}(X, t))

when s > t while if s < t, then

G(\underline{A}_{r}(X, t), Y, s) ≤ cG(Ā_{r}(X, t), Y, s).

Combining the previous two lemmas the doubling property of caloric/adjoint caloric measure can be proven.

Lemma 9. Let Ω, (X, t), (Y, s) and A be as in the previous lemma. Then

ω^{∗}(Y, s, Δ(X, t, r)) ≤ c(A)ω^{∗}(Y, s, Δ(X, t, r/2))

where ω^{∗} = ω when s > t while ω^{∗} = ω̂ when s < t.

Let (X, t) ∈ ∂Ω, ϱ > 0 and R > 0. u > 0 is said to satisfy a strong Harnack inequality in C_{R}(X, t) ∩ Ω provided that u is a solution to either the heat or adjoint heat equation in C_{R}(X, t) ∩ Ω and

u(X̂, t̂) ≤ λ̃u(X̃ , t̃) whenever (X̂, t̂), (X̃ , t̃) ∈ C_{ϱ}(Z, τ)
and C_{2ϱ}(Z, τ) ⊂ C_{R}(X, t) ∩ Ω.

Here λ̃, 1 ≤ λ̃ < ∞, is independent of C_{2ϱ}(Z, τ) ⊂ C_{R}(X, t) ∩ Ω. For (X, t), ϱ as above and A > 0 we define

\begin{aligned}ΓA^{+}(X, t, ϱ) & = Ω ∩ \{(Y, s) : |Y − X|^{2} ≤ A|s − t|, |s − t| ≥ 5ϱ^{2}, s > t\}, \\ ΓA^{−}(X, t, ϱ) & = Ω ∩ \{(Y, s) : |Y − X|^{2} ≤ A|s − t|, |s − t| ≥ 5ϱ^{2}, s < t\}. \end{aligned}

Using Lemmas 7, 8 and 9 one can prove that if (Y, s) ∈ ΓA^{+}(X, t, R) then G(Y, s, ·) satisfies a strong Harnack inequality in C_{R}(X, t) ∩ Ω while if (Y, s) ∈ ΓA^{−}(X, t, R) then G( · , Y, s) satisfies a strong Harnack inequality in C_{R}(X, t) ∩ Ω. Moreover, λ̃ depends only on A once the NTA-constants λ and γ have been chosen. Using the notion of strong Harnack inequality the following two comparison lemmas can be proven.

Lemma 10. Let u, v > 0 be continuous in \overline{C}_{2r}(X, t) ∩ \overline{Ω}, u = v = 0 on Δ(X, t, 2r) and assume that u and v both are solutions either to the heat or the adjoint heat equation in C_{2r}(X, t)∩Ω. If u, v satisfy a strong Harnack inequality in C_{2r}(X, t) ∩ Ω for some λ̃ ≥ 1, then

\frac{u(Y, s)}{v(Y, s)} ≤ c(λ̃)\frac{u(Û)}{v(Û)} in C_{r/2}(X, t) ∩ \overline{Ω}.

Here Û = Ā_{r}(X, t) when u, v are solutions to the heat equation while Û = \underline{A}_{r}(X, t) when u, v are solutions to the adjoint heat equation in Ω ∩ C_{2r}(X, t).

Lemma 11. Under the same hypotheses as in Lemma 10 there exists γ̃ = γ̃(λ̃), 0 < γ̃ ≤ 1/2, and c = c(λ̃) ≥ 1, such that whenever 0 < ϱ ≤ r/2 then

|\frac{u(Z, τ)v(Y, s)}{v(Z, τ)u(Y, s)} − 1| ≤ c (ϱ/r)γ̃ for (Z, τ), (Y, s) ∈ C_{ϱ}(X, t) ∩ Ω.

3.2. Refined estimates. Let Ω ⊂ R^{n+1} be a δ_{0} -Reifenberg flat domain and define Ω^{1} = Ω ⊂ R^{n+1} , Ω^{2} = R^{n+1} \ \overline{Ω}. We assume that δ_{0} > 0 is small. Let (X̂^{i}, t̂^{i}) ∈ Ω^{i}, for i ∈ {1, 2}, t̂^{2} < t̂^{1} and define ω^{1}(X̂^{1}, t̂^{1}, ·) and ω^{2}(X̂^{2}, t̂^{2}, ·) to be the caloric measures defined with respect to Ω^{1} and Ω^{2} , respectively. In the following we will assume that ω^{2}(X̂^{2}, t̂^{2}, ·) is absolutely continuous with respect to ω^{1}(X̂^{1}, t̂^{1}, ·) on ∂Ω and that the Radon–Nikodym derivative k(X̂^{1}, t̂^{1}, X̂^{2}, t̂^{2}, ·) = dω^{2}(X̂^{2}, t̂^{2}, ·)/dω^{1}(X̂^{1}, t̂^{1}, ·) is such that log k(X̂^{1}, t̂^{1}, X̂^{2}, t̂^{2}, ·) is in the space VMO(dω^{1}(X̂^{1}, t̂^{1}, ·)). To properly define the space VMO(dω^{1}(X̂^{1}, t̂^{1}, ·)) we let a = a(Δ(X, t, ϱ), f) denote the average of f = log k(X̂^{1}, t̂^{1}, X̂^{2}, t̂^{2}, ·) on Δ(X, t, ϱ) with respect to dω^{1}(X̂^{1}, t̂^{1}, ·). f is said to be in VMO(dω^{1}(X̂^{1}, t̂^{1}, ·)) provided for each compact K ⊂ ∂Ω ∩ {(Y, s) : s < t̂^{2}},

lim_{r→0} sup_{{(X,t)∈K \atop 0<ϱ≤r}} ω^{1}(X̂^{1}, t̂^{1}, Δ(X, t, ϱ))^{−1} ∫_{Δ(X,t,ϱ)} |f(Y, s) − a| dω^{1}(X̂^{1}, t̂^{1}, ·) = 0.

We start by exploring the information contained in the condition

log k(X̂^{1}, t̂^{1}, X̂^{2}, t̂^{2}, ·) ∈ VMO(dω^{1}(X̂^{1}, t̂^{1}, ·)).

Lemma 12. Let Ω ⊂ R^{n+1} be a δ_{0} -Reifenberg flat domain and define Ω^{1} = Ω ⊂ R^{n+1} , Ω^{2} = R^{n+1} \ \overline{Ω}. Assume furthermore that ω^{2}(X̂^{2}, t̂^{2}, ·) is absolutely continuous with respect to ω^{1}(X̂^{1}, t̂^{1}, ·) on ∂Ω and that the Radon– Nikodym derivative

k(X̂^{1}, t̂^{1}, X̂^{2}, t̂^{2}, ·) = dω^{2}(X̂^{2}, t̂^{2}, ·)/dω^{1}(X̂^{1}, t̂^{1}, ·)

is such that

log k(X̂^{1}, t̂^{1}, X̂^{2}, t̂^{2}, ·) ∈ VMO(dω^{1}(X̂^{1}, t̂^{1}, ·)).

Then there exists α ∈ (0, 1) and a constant C = C(n, δ_{0}, A) such that the following is true. If (X, t) ∈ ∂Ω, r < r_{0} , |X − X̂^{i}|^{2} ≤ A(t̂^{i} − t) for some A ≥ 2 and for i ∈ {1, 2}, min{t̂^{1}, t̂^{2}} − t ≥ 8r^{2} and E ⊂ Δ(X, t, r), then

\frac{ω^{2}(X̂^{2}, t̂^{2}, E)}{ω^{2}(X̂^{2}, t̂^{2}, Δ(X, t, r))} ≤ C(\frac{ω^{1}(X̂^{1}, t̂^{1}, E)}{ω^{1}(X̂^{1}, t̂^{1}, Δ(X, t, r))})^{α}.

Proof. Let E ⊂ Δ(X, t, r) and Δ(X, t, r) be as in the statement of the lemma. Note that the restrictions on the points (X, t) and (X̂^{i}, t̂^{i}) imply uniform bounds on the doubling constants of the caloric measures under consideration. The inequality stated in the lemma is therefore a standard consequence of the fact that if log k(X̂^{1}, t̂^{1}, X̂^{2}, t̂^{2}, ·) ∈ VMO(dω^{1}(X̂^{1}, t̂^{1}, ·)) then k(X̂^{1}, t̂^{1}, X̂^{2}, t̂^{2}, ·) is an A_{∞} weigth with respect to the measure ω^{1}(X̂^{1}, t̂^{1}, ·). For more on the relation between VMO and A_{p} -weights we refer to [S].

Lemma 13. Let Ω be a δ_{0} -Reifenberg flat domain with δ_{0} > 0 small enough. Let (X, t) ∈ ∂Ω, r > 0 and let u > 0 be a solution to either the heat or the adjoint heat equation in C_{2r}(X, t) ∩ Ω, continous in \overline{C}_{2r}(X, t) ∩ \overline{Ω} and such that u = 0 on Δ(X, t, 2r). If u satisfies a strong Harnack inequality in C_{2r}(X, t) ∩ Ω for some λ̃ ≥ 1, then, given ε > 0, there exists δ̂_{0} = δ̂_{0}(n, ε, λ̃) > 0 and a constant C = C(n, ε, λ̃) such that if δ_{0} ≤ δ̂_{0} and r̂ ≤ r then

C^{−1}(\frac{r̂}{r})^{1+ε}u(A_{r}(X, t)) ≤ u(Ar̂(X, t)) ≤ C(\frac{r̂}{r})^{1−ε}u(A_{r}(X, t)).

Proof. By scaling and translation we can without loss of generality assume that r = 1, (X, t) = (0, 0). We can furthermore assume that u(A_{1}(0, 0)) = 1. Based on these simplifications we want to prove that given ε > 0, there exists δ̂_{0} = δ̂_{0}(n, ε, λ̃) > 0 and a constant C = C(n, ε, λ̃) such that if δ_{0} ≤ δ̂_{0} and r̂ ≤ 1 then

C^{−1}r̂^{1+ϵ} ≤ u(Ar̂(0, 0)) ≤ Cr̂^{1−ε}.

In order to prove this inequality we will make use of a number of auxiliary sets and functions. Recall that by definition of the δ_{0} -Reifenberg flatness there exists an n-dimensional plane P̂ = P̂(0, 0, 2), containing (0, 0) and a line parallel to the t axis, having unit normal n̂ = n̂(0, 0, 2) such that

\begin{matrix}\{(Ẑ , τ̂) + rn̂ ∈ C_{2}(0, 0) : (Ẑ , τ̂) ∈ P̂, r > 2δ_{0}\} ⊂ Ω, \\ \{(Ẑ , τ̂) − rn̂ ∈ C_{2}(0, 0) : (Ẑ , τ̂) ∈ P̂, r > 2δ_{0}\} ⊂ R^{n+1} \ Ω. \end{matrix}

Based on this we introduce

\begin{aligned}S^{−} & = S^{−}(δ_{0}) = \{(Ẑ , τ̂) + rn̂ : (Ẑ , τ̂) ∈ P̂, r > −4δ_{0}\}, \\ S^{+} & = S^{+}(δ_{0}) = \{(Ẑ , τ̂) + rn̂ : (Ẑ , τ̂) ∈ P̂, r > 4δ_{0}\}. \end{aligned}

Then S^{+} ⊂ S^{−} and S^{+} is a translation of the halfspace S^{−}. Consider the sets S^{−} ∩C_{2}(0, 0) and S^{+} ∩C_{2}(0, 0). The parabolic boundary of S^{−} ∩C_{2}(0, 0) consists of two pieces Γ1^{−} and Γ2^{−} where

\begin{aligned}Γ1^{−} & = C_{2}(0, 0) ∩ \{(Ẑ , τ̂) − 4δ_{0}n̂ : (Ẑ , τ̂) ∈ P̂\}, \\ Γ2^{−} & = ∂_{p}C_{2}(0, 0) ∩ \{(Ẑ , τ̂) + rn̂ : (Ẑ , τ̂) ∈ P̂, r > −4δ_{0}\}. \end{aligned}

Here ∂_{p}C_{2}(0, 0) is the parabolic boundary of C_{2}(0, 0). Similarly the parabolic boundary of S^{+} ∩ C_{2}(0, 0) consists of two pieces Γ1^{+} and Γ2^{+} . We define two auxilary functions ũ^{−} and ũ^{+} . ũ^{−} is a caloric function defined in S^{−}(δ_{0})∩C_{2}(0, 0) satisfying ũ^{−}(Y, s) = 0 if (Y, s) ∈ Γ1^{−} and ũ^{−}(Y, s) = 1 if (Y, s) ∈ Γ2^{−}. Similarly ũ^{+} is a caloric function defined in S^{+}(δ_{0}) ∩ C_{2}(0, 0) satisfying ũ^{+}(Y, s) = 0 if (Y, s) ∈ Γ1^{+} and ũ^{+}(Y, s) = 1 if (Y, s) ∈ Γ2^{+} . Finally we also define

ṽ^{−}((Ẑ , τ̂) + rn̂) := r + 4δ_{0}, ṽ^{+}((Ẑ , τ̂) + rn̂) := r − 4δ_{0}

for all (Ẑ , τ̂) ∈ P̂ and r ∈ R. I.e., the last two caloric functions are independent of the time variable and grow linearly in the direction of the normal n̂.

Based on this notation we will now prove the right-hand side inequality. By the maximum principle we have, by construction, that

u(Y, s) ≤ ũ^{−}(Y, s)

for all (Y, s) ∈ Ω ∩ C_{1}(0, 0). Note that if δ_{0} ≤ δ̂_{0} then ũ^{−} will satisfy, as ũ^{−} essentially is the caloric measure in S^{−}(δ_{0})∩C_{2}(0, 0) of the set S^{−}(δ_{0})∩∂_{p}C_{2}(0, 0), a strong Harnack inequality in S^{−}(δ_{0}) ∩ C_{1/2}(0, 0) with a universal λ̃. Let in the following (Ẑ , τ̂) ∈ P̂. For all (Y, s) = (Ẑ , τ̂) + rn̂ ∈ C_{1/2}(0, 0) ∩ Ω, r ∈ R, we get, using Lemma 10, that

ũ^{−}(Y, s) ≤ Cṽ^{−}(Y, s) = C(r + 4δ_{0}).

Therefore if (Y, s) = (Ẑ , τ̂) + rn̂ ∈ C_{θ}(0, 0) ∩ Ω for θ < 1/2 an elementary argument implies that ũ^{−}(Y, s) ≤ C(θ + δ_{0}) and hence u(Y, s) ≤ C(θ + δ_{0}). Iteratively u(Y, s) ≤ [C(θ + δ_{0})]^{k} for (Y, s) = (Ẑ , τ̂) + rn̂ ∈ C_{θ^{k}}(0, 0) ∩ Ω. In particular, we can conclude that if δ_{0} is small enough, then u(Y, s) ≤ [2Cδ_{0}]^{k} for all (Y, s) = (Ẑ , τ̂) + rn̂ ∈ C_{δ_{0}^{k}}(0, 0) ∩ Ω. For given ε > 0 small, let δ_{0} be such that 2Cδ_{0} ≤ δ_{0}^{1−ε}. If we let k be determined through δ_{0}^{k} = r̂, then u(Ar̂(0, 0)) ≤ Cr̂^{1−ε} and the proof is complete in one direction.

Left is to prove the inequality in the other direction. To start with we in this case let û be a caloric function defined in Ω ∩ C_{2}(0, 0) satisfying û(Y, s) = 0 if (Y, s) ∈ ∂Ω ∩ C_{2}(0, 0) and û(Y, s) = 1 if (Y, s) ∈ Ω ∩ ∂_{p}C_{2}(0, 0). Again if δ_{0} ≤ δ̂_{0} then û will satisfy a strong Harnack inequality in Ω ∩ C_{1}(0, 0) with a universal λ̃. Applying Lemma 10 it follows that there exists a constant C such that

C^{−1} \frac{u(A_{1}(0, 0))}{û(A_{1}(0, 0))} ≤ \frac{u(Ar̃(0, 0))}{û(Ar̃(0, 0))} ≤ C \frac{u(A_{1}(0, 0))}{û(A_{1}(0, 0))}

for all 0 < r̃ ≤ 1. Therefore, as u(A_{1}(0, 0)) = 1 ∼ û(A_{1}(0, 0)), we have u(Ar̃(0, 0)) ∼ û(Ar̃(0, 0)) for all 0 < r̃ ≤ 1. Furthermore, by construction, we have by the maximum principle that

ũ^{+}(Y, s) ≤ û(Y, s)

for all (Y, s) ∈ S^{+}(δ_{0}) ∩ C_{1}(0, 0). Applying Lemma 10 once again we can also conclude that

\frac{ũ^{+}(A_{32δ_{0}}(0, 0))}{ṽ^{+}(A_{32δ_{0}}(0, 0))} ∼ \frac{ũ^{+}(A_{1}(0, 0))}{ṽ^{+}(A_{1}(0, 0))} ∼ \frac{ũ^{+}(A_{1}(0, 0))}{1 − 4δ_{0}}.

This implies that

ũ^{+}(A_{32δ_{0}}(0, 0)) ≥ Cδ_{0} \frac{ũ^{+}(A_{1}(0, 0))}{1 − 4δ_{0}} ≥ Cδ_{0}ũ^{+}(A_{1}(0, 0)).

By iteration

ũ^{+}(A_{(32δ_{0})^{k}}(0, 0)) ≥ (Cδ_{0})^{k}ũ^{+}(A_{1}(0, 0)).

Choosing δ_{0} so small that Cδ_{0} ≥ (32δ_{0})^{1+ε} we can conclude that if

r̂ ∈ [(32δ_{0})^{k+1}, (32δ_{0})^{k}]

then by the Harnack principle

\begin{aligned}ũ^{+}(Ar̂(0, 0)) & ≥ Cũ^{+}(A_{(32δ_{0})^{k}}(0, 0)) \\ & ≥ C(32δ_{0})^{k(1+ε)}ũ^{+}(A_{1}(0, 0)) ≥ Cr̂^{(1+ε)}ũ^{+}(A_{1}(0, 0)). \end{aligned}

As ũ^{+}(A_{1}(0, 0)) ∼ û(A_{1}(0, 0)) and as ũ^{+} ≤ û on S^{+}(δ_{0}) ∩ C_{2}(0, 0) we can, by a straightforward comparison argument, conclude that

û(Ar̂(0, 0)) ≥ Cr̂^{1+ε}û(A_{1}(0, 0)).

As û(A_{1}(0, 0)) ∼ u(A_{1}(0, 0)) = 1 we can put all the estimates together and finally conclude that

u(Ar̂(0, 0)) ≥ Cr̂^{1+ε}u(A_{1}(0, 0)) = Cr̂^{1+ε}.

This completes the proof of the lemma.

3.3. The Green function and caloric measure at infinity. In this section we will clarify the notion of Green function with pole at infinity and the associated caloric measure.

Lemma 14. Let Ω be a δ_{0} -Reifenberg flat domain with δ_{0} > 0 small. Then there exists a unique function u (unique modulo a constant) such that u is a nonnegative solution to the adjoint heat in Ω and such that u vanishes continuously on ∂Ω.

In fact a similar result holds for the heat equation. The function u, in the statement of the lemma, should be referred to as the Green function with pole at +infinity. By + we refer to the ‘infinity’ in the positive direction of time.

Proof. There are two steps in the proof, the uniqueness and the existence. We start by proving the existence. We let (X, t) ∈ ∂Ω and let R > 0 be a large positive number. Assume that (X̂, t̂) ∈ ΓA^{+}(X, t, 100R) and let K ⊂ R^{n+1} be a fixed compact set. Assume that R is so large that K ∩ Ω ⊂ C_{R}(X, t). Using Lemma 6, the fact that if (X̂, t̂) ∈ ΓA^{+}(X, t, 100R) then G(X̂, t̂, ·) satisfies a strong Harnack inequality in C_{R}(X, t)∩Ω and Lemma 10 it follows that if (Z, τ) ∈ K ∩Ω, then

G(X̂, t̂, Z, τ) ≤ C_{K,n,A}G(X̂, t̂, A_{1}(X, t)).

In particular this implies that if (X̂, t̂) ∈ ΓA^{+}(X, t, 100R) then

sup_{(Z,τ)∈K∩Ω} \frac{G(X̂, t̂, Z, τ)}{G(X̂, t̂, A_{1}(X, t))} ≤ C_{K,n,A}.

Let (X̂_{j}, t̂_{j}) ∈ ΓA^{+}(X, t, 2^{j}R) for j = 1, 2, … and define for (Z, τ) ∈ C_{R}(X, t) ∩ Ω

u_{j}(Z, τ) = \frac{G(X̂_{j}, t̂_{j}, Z, τ)}{G(X̂_{j}, t̂_{j}, A_{1}(X, t))}.

Then {u_{j}} is a set of positive adjoint caloric functions in C_{R}(X, t) ∩ Ω vanishing on ∂Ω. Furthermore, we can assume that {u_{j}} is a uniformly bounded set of functions on Ω ∩ C_{R}(X, t). By the Arzela–Ascoli theorem there exists a subsequence {j̃_{k}} such that {uj̃_{k}} converges to a non-negative solution ũ = ũ_{R} to the adjoint heat equation in Ω ∩ C_{R}(X, t). If we choose a sequence of numbers R_{i} such that R_{i} → ∞ and pick a diagonal subsequence we can conclude that there exists a subsequence {j_{k}} such that {u_{j_{k}}} converges to a non-negative solution u_{∞} to the adjoint heat equation in Ω, uniformly on compact sets of Ω. Furthermore, u_{∞} vanishes continuously on ∂Ω and u_{∞}(A_{1}(X, t)) = 1.

Left is to prove uniqueness. Let u and v be two functions fulfilling the statement of the lemma and assume that u(A_{1}(X, t)) = v(A_{1}(X, t)) for some point (X, t) ∈ ∂Ω. Under these assumptions we want to prove that u ≡ v. Let ϱ and R be fixed numbers such that 0 < ϱ ≤ R/2. Using the same argument as in the proof of Lemma 8 (see the proof of Lemma 3.11 in [HLN2]) one can prove that u and v satisfy a strong Harnack inequality in C_{R}(X, t) ∩ Ω with a constant λ̃ which only depends on the Reifenberg constant δ_{0} and the dimension n. Using Lemma 11 we therefore get that whenever (Z, τ), (Y, s) ∈ C_{ϱ}(X, t) ∩ Ω then

|\frac{u(Z, τ)v(Y, s)}{v(Z, τ)u(Y, s)} − 1| ≤ c (ϱ/R)^{γ}.

Hence if we put (Y, s) = A_{1}(X, t) then

|\frac{u(Z, τ)}{v(Z, τ)} − 1| ≤ C(ϱ/R)^{γ}

whenever (Z, τ) ∈ C_{ϱ}(X, t) ∩ Ω. Letting R → ∞ completes the proof.

Lemma 15. Let Ω be a δ_{0} -Reifenberg flat domain with δ_{0} > 0 small. Let (X, t) ∈ ∂Ω. Then there exists a unique doubling Radon measure ω such that ω(Δ(X, t, 1)) = 1 and a non-negative solution u to the adjoint heat in Ω vanishing continuously on ∂Ω such that for all ϕ ∈ C_{0}^{∞}(R^{n+1})

∫_{∂Ω} ϕ(Y, s) dω(Y, s) = ∫_{Ω} u(Y, s)(Δ − ∂_{s})ϕ(Y, s) dY ds.

ω is referred to as the caloric measure for Ω at + infinity and normalized at (X, t).

Proof. Again there are two steps in the proof, the uniqueness and the existence. In this case we start by proving the uniqueness. I.e., we assume that there exist two measures ω_{1} and ω_{2} as in the statement of the lemma and such that ω_{1}(Δ(X, t, 1)) = ω_{2}(Δ(X, t, 1)) = 1 for a point (X, t) ∈ ∂Ω. We want to prove that ω_{1} ≡ ω_{2} . Let u_{1} and u_{2} be related to ω_{1} respectively ω_{2} according to the statement of the lemma. Using Lemma 14 we can conclude that there exist constants α_{1} and α_{2} as well as a function u such that u_{i} = α_{i}u. Here u is a non-negative solution to the adjoint heat in Ω such that u vanishes continuously on ∂Ω. I.e., for all ϕ ∈ C_{0}^{∞}(R^{n+1})

∫_{∂Ω} ϕ(Y, s) dω_{i}(Y, s) = α_{i} ∫_{Ω} u(Y, s)(Δ − ∂_{s})ϕ(Y, s) dY ds.

From this we can we conclude that

α_{1}^{−1} ∫_{∂Ω} ϕ(Y, s) dω_{1}(Y, s) = α_{2}^{−1} ∫_{∂Ω} ϕ(Y, s) dω_{2}(Y, s).

Choosing ϕ as the indicator function of Δ(X, t, 1) and using the normalization of ω_{1} and ω_{2} we get that α_{1} = α_{2} . Therefore u_{1} ≡ u_{2} and ω_{1} ≡ ω_{2} .

To prove the existence we argue as in the proof of Lemma 14. We let (X, t) ∈ ∂Ω and define R > 0 to be a large positive number. Let (X̂_{j}, t̂_{j}) ∈ ΓA^{+}(X, t, 2^{j}R) for j = 1, 2, … and define for (Z, τ) ∈ C_{R}(X, t) ∩ Ω

u_{j}(Z, τ) = \frac{G(X̂_{j}, t̂_{j}, Z, τ)}{G(X̂_{j}, t̂_{j}, A_{1}(X, t))}.

Let ϕ ∈ C_{0}^{∞}(C_{R}(X, t)) and let as usual ω(X̂_{j}, t̂_{j}, ·) be the caloric measure defined with respect to (X̂_{j}, t̂_{j}). Then

\begin{matrix}∫_{∂Ω} ϕ(Z, τ) & G(X̂_{j}, t̂_{j}, A_{1}(X, t))^{−1} dω(X̂_{j}, t̂_{j}, Z, τ) \\ = ∫_{Ω} u_{j}(Z, τ)(Δ − ∂_{τ})ϕ(Z, τ) dZ dτ. \end{matrix}

Defining measures

dμ_{j}(Z, τ) = G(X̂_{j}, t̂_{j}, A_{1}(X, t))^{−1} dω(X̂_{j}, t̂_{j}, Z, τ)

we can conclude that

∫_{∂Ω} ϕ(Z, τ) dμ_{j}(Z, τ) = ∫_{Ω} u_{j}(Z, τ)(Δ − ∂_{τ})ϕ(Z, τ) dZ dτ

for all ϕ ∈ C_{0}^{∞}(C_{R}(X, t)). Using Lemma 7 and the fact that G(X̂_{j}, t̂_{j}, ·) satisfies a strong Harnack inequality in C_{R}(X, t) ∩ Ω we have that

\begin{aligned}μ_{j}(Δ(X, t, R)) & = \frac{ω(X̂_{j}, t̂_{j}, Δ(X, t, R))}{G(X̂_{j}, t̂_{j}, A_{1}(X, t))} ∼ \frac{R^{n}G(X̂_{j}, t̂_{j}, A_{R}(X, t))}{G(X̂_{j}, t̂_{j}, A_{1}(X, t))} \\ & ∼ R^{n}u_{j}(Z, τ) \end{aligned}

for all (Z, τ) ∈ C_{R}(X, t) ∩ Ω. As in the proof of Lemma 14, {u_{j}} is a uniformly bounded set of functions on Ω∩C_{R}(X, t). Hence the sequence {μ_{j}} is a uniformly bounded set of measures on C_{R}(X, t)∩∂Ω and therefore there exists a subsequence {μj̃_{k}} and a Radon measure μ such that,

∫_{∂Ω} ϕ(Z, τ) dμ_{j_{k}}(Z, τ) → ∫_{∂Ω} ϕ(Z, τ) dμ(Z, τ),

for all ϕ ∈ C_{0}^{∞}(C_{R}(X, t)) as k → ∞. If we again choose a sequence of numbers R_{i} such that R_{i} → ∞ and pick a diagonal subsequence we can therefore conclude that there exists a subsequence {j_{k}} such that {μ_{j_{k}}} converges to a Radon measure μ such that for all ϕ ∈ C_{0}^{∞}(R^{n+1}),

∫_{∂Ω} ϕ(Z, τ) dμ_{j_{k}}(Z, τ) → ∫_{∂Ω} ϕ(Z, τ) dμ(Z, τ),

as k → ∞. Repeating the argument of Lemma 14 we can also conclude that {u_{j_{k}}} converges, uniformly on compacts subsets, to a non-negative solution u_{∞} and for all ϕ ∈ C_{0}^{∞}(R^{n+1})

∫_{∂Ω} ϕ(Z, τ) dμ(Z, τ) = ∫_{Ω} u_{∞}(Z, τ)(Δ − ∂_{τ})ϕ dZdτ.

Define ω_{∞} = μ/μ(Δ(X, t, 1)) and u_{∞} = u_{∞}/μ(Δ(X, t, 1)). Then for all ϕ ∈ C_{0}^{∞}(R^{n+1})

∫_{∂Ω} ϕ(Z, τ) dω_{∞}(Z, τ) = ∫_{Ω} u_{∞}(Z, τ)(Δ − ∂_{τ})ϕ dZ dτ.

This completes the existence part of the proof. Left is to prove that ω_{∞} is a doubling measure. But if (X̃ , t̃) ∈ ∂Ω and r > 0 it follows from Lemma 9 that

ω_{∞}(Δ(X̃ , t̃, 2r)) ≤ lim inf_{j_{k}→∞} \frac{ω(X̂_{j_{k}}, t̂_{j_{k}}, Δ(X̃ , t̃, 2r))}{μ(Δ(X, t, 1))G(X̂_{j_{k}}, t̂_{j_{k}}, A_{1}(X, t))}

≤ C lim inf_{j_{k}→∞} \frac{ω(X̂_{j_{k}}, t̂_{j_{k}}, Δ(X̃ , t̃, r/2))}{μ(Δ(X, t, 1))G(X̂_{j_{k}}, t̂_{j_{k}}, A_{1}(X, t))}

≤ Cω_{∞}(Δ(X̃ , t̃, r)).

This completes the proof.

4. A blow-up argument and the classification of global solutions

In the following we let Ω be δ_{0} -Reifenberg flat. Let (X_{j}, t_{j}) ∈ ∂Ω → (X̃ , t̃) ∈ ∂Ω and assume that (X̃ , t̃) = (0, 0). For a sequence {r_{j}} of real numbers tending to zero we define,

Ω_{j} = {(r_{j}^{−1}(X − X_{j}), r_{j}^{−2}(t − t_{j})) : (X, t) ∈ Ω}.

This section is devoted to the analysis of these blow-ups by making use of our assumption on the kernel k(X̂^{1}, t̂^{1}, X̂^{2}, t̂^{2}, ·) and therefore we will also assume that t_{j} < t̂^{2} < t̂^{1} for all j.

4.1. Blow-ups. Recall that the parabolic distance between the two sets F_{1}, F_{2} is defined as

d(F_{1}, F_{2}) = inf{|X − Y | + |s − t|^{1/2} : (X, t) ∈ F_{1}, (Y, s) ∈ F_{2}}.

Based on this we introduce the parabolic Hausdorff distance between two sets F_{1} , F_{2} as

D(F_{1}, F_{2}) = sup{d(x, F_{2}) : x ∈ F_{1}} + sup{d(F_{1}, y) : y ∈ F_{2}}.

In the following we will consider uniform Hausdorff convergence (in the metric induced by the parabolic Hausdorff distance) on compact sets. To define this properly we consider a sequence of closed sets {F_{j}}_{j}, F_{j} ⊂ R^{n+1} . We say that F_{j} converges to a closed set F ⊂ R^{n+1} in the parabolic Hausdorff distance sense, uniformly on compact subsets of R^{n+1} , if for any compact set K ⊂ R^{n+1} and any ε > 0 there exists j_{0} ≥ 1 so that if j ≥ j_{0} then

D(F_{j} ∩ K, F ∩ K) < ε.

Furthermore a sequence of open sets {E_{j}}_{j}, E_{j} ⊂ R^{n+1} , is said to converge to an open set E ⊂ R^{n+1} in the parabolic Hausdorff distance sense uniformly on compact subsets of R^{n+1} if R^{n+1} \ E_{j} → R^{n+1} \ E in the parabolic Hausdorff distance sense uniformly on compact subsets of R^{n+1} .

Let ω(X̂, t̂, ·) and G(X̂, t̂, · , ·) be the caloric measure and Green function defined with Ω as domain of reference. For arbitrary Borel sets E ⊂ R^{n+1} we define,

ω_{j}(E) = \frac{ω(X̂, t̂, \{(Z, τ) ∈ ∂Ω : ((Z − X_{j})/r_{j}, (τ − t_{j})/r_{j}^{2}) ∈ E\})}{ω(X̂, t̂, Δ(X_{j}, t_{j}, r_{j}))}.

We furthermore define

u_{j}(Z, τ) = \frac{G(X̂, t̂, X_{j} + r_{j}Z, t_{j} + r_{j}^{2}τ)}{ω(X̂, t̂, Δ(X_{j}, t_{j}, r_{j}))}r_{j}^{n}

whenever (Z, τ) ∈ Ω_{j}. Then u_{j} is adjoint caloric in Ω_{j} outside of its pole and it is zero on ∂Ω_{j}.

We will start by proving the following two lemmas.

Lemma 16. Let Ω be δ_{0} -Reifenberg flat domain with δ_{0} > 0 small. Let (X_{j}, t_{j}) ∈ ∂Ω → (X̃ , t̃) ∈ ∂Ω and assume that (X̃ , t̃) = (0, 0). For a sequence {r_{j}} of real numbers tending to zero we define,

Ω_{j} = {(r_{j}^{−1}(X − X_{j}), r_{j}^{−2}(t − t_{j})) : (X, t) ∈ Ω}.

Then Ω_{j} → Ω_{∞} and ∂Ω_{j} → ∂Ω_{∞} in the parabolic Hausdorff distance sense uniformly on compact subsets of R^{n+1} as j → ∞. Furthermore, Ω_{∞} is a 4δ_{0} - Reifenberg flat domain.

Lemma 17. Let Ω_{j} and Ω_{∞} be as in Lemma 16. Then u_{j} → u_{∞} uniformly on compact subsets, u_{∞} is a positive adjoint caloric function in Ω_{∞} and u_{∞} = 0 on ∂Ω_{∞}. Moreover ω_{j} → ω_{∞} weakly as Radon measures and for all ϕ ∈ C_{0}^{∞}(R^{n+1})

∫_{∂Ω} ϕ(Y, s) dω_{∞}(Y, s) = ∫_{Ω} u_{∞}(Y, s)(Δ − ∂_{s})ϕ(Y, s) dY ds.

ω_{∞} is the caloric measure of Ω_{∞} at infinity.

We start by proving Lemma 16.

Proof. Note that for each j ≥ 1, (0, 0) ∈ ∂Ω_{j} and C_{1}(0, 0) ∩ Ω_{j} ≠ ∅. Using this we can conclude that given a compact set K ⊂ R^{n+1} there exists a subsequence {j̃_{m}}_{m} such that K ∩ ∂Ωj̃_{m} and K ∩ Ωj̃_{m} converge in the parabolic Hausdorff distance sense. We can therefore exhaust R^{n+1} by a sequence of compact sets in order to ensure that there exists a subsequence {j_{m}}_{m} such that ∂Ω_{j_{m}} and Ω_{j_{m}} converge in the parabolic Hausdorff distance sense uniformly on compact sets. Hence by an appropriate relabeling we can conclude that as j → ∞, Ω_{j} → Ω_{∞}, ∂Ω_{j} → Σ_{∞} in the parabolic Hausdorff distance sense uniformly on compact subsets of R^{n+1} . In analogy with the proof of Theorem 4.1 in [KT2] we want to prove that ∂Ω_{∞} = Σ_{∞} and that ∂Ω_{∞} is 4δ_{0} -Reifenberg flat.

Since ω is a doubling measure we have that for any compact set K ⊂ R^{n+1} , sup_{j≥1} ω_{j}(K) ≤ C_{K}. Hence arguing as in the proof of Lemma 15 there exists a subsequence (which we relabel) such that ω_{j} → ω_{∞} in the sense that

∫ ϕ dω_{j} → ∫ ϕ dω_{∞}

for all ϕ ∈ C_{0}^{∞}(R^{n+1}). We start by proving that the support of ω_{∞} equals Σ_{∞}. To do this we let (Ẑ , τ̂) ∈ Σ_{∞}. By construction there exists a sequence (Z_{j}, τ_{j}) such that (Z_{j}, τ_{j}) ∈ ∂Ω and

(Z̃_{j}, τ̃_{j}) := ((Z_{j} − X_{j})/r_{j}, (τ_{j} − t_{j})/r_{j}^{2}) → (Ẑ , τ̂).

Furthermore for every r ∈ (0, 1) there exists j_{0} ≥ 1 such that for j ≥ j_{0} , d((Z̃_{j}, τ̃_{j}), (Ẑ , τ̂)) < r/2 and d((Z_{j}, τ_{j}), (X_{j}, t_{j})) < Cr_{j} for some large C = C((Ẑ , τ̂)). Hence we have that

\begin{aligned}ω_{j}(C_{r}(Ẑ , τ̂)) & = \frac{ω(X̂, t̂, \{(Z, τ) ∈ ∂Ω : ((Z − X_{j})/r_{j}, (τ − t_{j})/r_{j}^{2}) ∈ C_{r}(Ẑ , τ̂)\})}{ω(X̂, t̂, Δ(X_{j}, t_{j}, r_{j}))} \\ & ≥ \frac{ω(X̂, t̂, C_{rr_{j}/2}(Z_{j}, τ_{j}))}{ω(X̂, t̂, Δ(X_{j}, t_{j}, r_{j}))} ≥ \frac{ω(X̂, t̂, C_{r_{j}/2}(Z_{j}, τ_{j}))}{ω(X̂, t̂, C_{2Cr_{j}}(Z_{j}, τ_{j}))} ≥C̃(r, C) \end{aligned}

as ω is a doubling measure. Hence this implies that if (Ẑ , τ̂) ∈ Σ_{∞} then (Ẑ , τ̂) is in the support of ω_{∞}. Left is to prove the other inclusion. I.e., in this case we start by assuming that (Ẑ , τ̂) is in the support of ω_{∞}. We want to prove that there exists (Z_{j}, τ_{j}) ∈ ∂Ω such that

(Z̃_{j}, τ̃_{j}) := ((Z_{j} − X_{j})/r_{j}, (τ_{j} − t_{j})/r_{j}^{2}) → (Ẑ , τ̂).

If this is not the case then there exists ε > 0 and j_{0} such that for any sequence (Z_{j}, τ_{j}) ∈ ∂Ω as above d((Z̃_{j}, τ̃_{j}), (Ẑ , τ̂)) ≥ ε if j ≥ j_{0} . In particular in this case C_{ε/2}(Ẑ , τ̂) ∩ ∂Ω_{j} = ∅. If we take ϕ ∈ C_{0}^{∞}(C_{ε/2}(Ẑ , τ̂)) we then get as ω_{j} → ω_{∞} that

0 = ∫ ϕ(Y, s) dω_{j}(Y, s) → ∫ ϕ(Y, s) dω_{∞}(Y, s).

I.e.,

∫ ϕ(Y, s) dω_{∞}(Y, s) = 0

for all ϕ ∈ C_{0}^{∞}(C_{ε/2}(Ẑ , τ̂)). This contradicts the assumption that (Ẑ , τ̂) is in the support of ω_{∞}. I.e., (Ẑ , τ̂) ∈ Σ_{∞} and we can conclude that the support of ω_{∞} coincides with Σ_{∞}.

We now prove that ∂Ω_{∞} ⊂ Σ_{∞}. To do this we let (Z, τ) ∈ ∂Ω_{∞} = \overline{Ω_{∞}} ∩ \overline{R^{n+1} \ Ω_{∞}} and note that given ε > 0, there exist (Y, s) ∈ Ω_{∞} ∩ C_{ε}(Z, τ) and (Ŷ, ŝ) ∈ [R^{n+1} \ \overline{Ω_{∞}}] ∩ C_{ε}(Z, τ). There also exist sequences of points (Y_{j}, s_{j}) ∈ Ω and (Ŷ_{j}, ŝ_{j}) ∈ [R^{n+1} \ \overline{Ω}] such that

(Y, s) = ((Y_{j} − X_{j})/r_{j}, (s_{j} − t_{j})/r_{j}^{2}), (Ŷ, ŝ) = ((Ŷ_{j} − X_{j})/r_{j}, (ŝ_{j} − t_{j})/r_{j}^{2}).

Let l_{j} be the parabolic line connecting (Y_{j}, s_{j}) and (Ŷ_{j}, ŝ_{j}) and pick (Z_{j}, τ_{j}) ∈ l_{j} ∩∂Ω. As ∂Ω separates R^{n+1} at least one such point exists. As {(Z_{j} −X_{j})/r_{j}}_{j} as well as {(τ_{j} −t_{j})/r_{j}^{2}}_{j} are bounded sequences there exists a subsequence (which we relabel) such that

((Z_{j} − X_{j})/r_{j}, (τ_{j} − t_{j})/r_{j}^{2}) → (Ẑ , τ̂) ∈ Σ_{∞}.

Furthermore as

d (((Z_{j} − X_{j})/r_{j}, (τ_{j} − t_{j})/r_{j}^{2}), ((Y_{j} − X_{j})/r_{j}, (s_{j} − t_{j})/r_{j}^{2}))

can be bounded by r_{j}^{−1}d((Y_{j}, s_{j}), (Ŷ_{j}, ŝ_{j})), we can conclude, by letting j → ∞, that

d((Y, s), (Ẑ , τ̂)) ≤ d((Y, s), (Ŷ, ŝ)) ≤ Cε

for a universal constant C. By the same line of thought

\begin{aligned}d((Z, τ), (Ẑ , τ̂)) & ≤ d((Z, τ), (Ŷ, ŝ)) + d((Ẑ , τ̂), (Ŷ, ŝ)) \\ & ≤ d((Z, τ), (Ŷ, ŝ)) + d((Y, s), (Ŷ, ŝ)) ≤C̃ε. \end{aligned}

In total we have proved that for any (Z, τ) ∈ ∂Ω_{∞} and for any ε > 0 there exists (Ẑ , τ̂) ∈ Σ_{∞} such that d((Z, τ), (Ẑ , τ̂)) ≤ ε. This argument proves that (Z, τ) is in the closure of the set Σ_{∞}. But the closure of the set Σ_{∞} equals, as we have proven above, the closure of the support of ω_{∞}. The latter equals the support of ω_{∞} as the support is closed. Based on this we can conclude that (Z, τ) ∈ Σ_{∞} and that ∂Ω_{∞} ⊂ Σ_{∞}.

Left is to prove that Σ_{∞} ⊂ ∂Ω_{∞}. We let (Ẑ , τ̂) ∈ Σ_{∞}. By construction there exists a sequence (Z_{j}, τ_{j}) such that (Z_{j}, τ_{j}) ∈ ∂Ω and

(Z̃_{j}, τ̃_{j}) := ((Z_{j} − X_{j})/r_{j}, (τ_{j} − t_{j})/r_{j}^{2}) → (Ẑ , τ̂).

In order to argue as in the proof of Theorem 4.1 in [KT2] we start by fixing M and by considering arbitrary ϱ > 0. Let n̂(Z_{j}, τ_{j}, ϱr_{j}) be the normal of the plane through (Z_{j}, τ_{j}), associated with the scale ϱr_{j}, which appears in the definition of Reifenberg flatness in Definition 1. We define

\begin{aligned}A_{ϱr_{j}}(Z_{j}, τ_{j}) & = (Z_{j} + ϱr_{j}n̂(Z_{j}, τ_{j}, ϱr_{j}), τ_{j}), \\ Ã_{ϱr_{j}}(Z_{j}, τ_{j}) & = (Z_{j} − ϱr_{j}n̂(Z_{j}, τ_{j}, ϱr_{j}), τ_{j}) \end{aligned}

where we assume that A_{ϱr_{j}}(Z_{j}, τ_{j}) ∈ Ω and Ã_{ϱr_{j}}(Z_{j}, τ_{j}) ∈ R^{n+1} \ \overline{Ω}. Choosing M large enough we can conclude that

\begin{aligned}C_{ϱr_{j}/M}(A_{ϱr_{j}}(Z_{j}, τ_{j})) & ⊂ Ω, d(A_{ϱr_{j}}(Z_{j}, τ_{j}), (Z_{j}, τ_{j})) ≤ ϱr_{j}, \\ C_{ϱr_{j}/M}(Ã_{ϱr_{j}}(Z_{j}, τ_{j})) & ⊂ R^{n+1} \ \overline{Ω}, d(Ã_{ϱr_{j}}(Z_{j}, τ_{j}), (Z_{j}, τ_{j})) ≤ ϱr_{j}. \end{aligned}

We also define

\begin{aligned}A_{j}(ϱ) & = ((Z_{j} + ϱr_{j}n̂(Z_{j}, τ_{j}, ϱr_{j}) − X_{j})r_{j}^{−1}, (τ_{j} − t_{j})r_{j}^{−2}), \\ Ã_{j}(ϱ) & = ((Z_{j} − ϱr_{j}n̂(Z_{j}, τ_{j}, ϱr_{j}) − X_{j})r_{j}^{−1}, (τ_{j} − t_{j})r_{j}^{−2}). \end{aligned}

Then

\begin{aligned}C_{ϱ/M}(A_{j}(ϱ)) & ⊂ Ω_{j}, d(A_{j}(ϱ), (Z̃_{j}, τ̃_{j})) ≤ ϱ, \\ C_{ϱ/M}(Ã_{j}(ϱ)) & ⊂ R^{n+1} \ \overline{Ω}_{j}, d(Ã_{j}(ϱ), (Z̃_{j}, τ̃_{j})) ≤ ϱ. \end{aligned}

Going to the limit we can conclude that, for every ϱ > 0, there exist points A_{∞}(ϱ) ∈ Ω_{∞} and Ã_{∞}(ϱ) ∈ R^{n+1} \ \overline{Ω}_{∞} such that

\begin{aligned}C_{ϱ/M}(A_{∞}(ϱ)) & ⊂ Ω_{∞}, d(A_{∞}(ϱ), (Ẑ , τ̂)) ≤ ϱ, \\ C_{ϱ/M}(Ã_{∞}(ϱ)) & ⊂ R^{n+1} \ \overline{Ω}_{∞}, d(Ã_{∞}(ϱ), (Ẑ , τ̂)) ≤ ϱ. \end{aligned}

If we let ϱ → 0 we can therefore conclude that (Ẑ , τ̂) ∈ ∂Ω_{∞} and hence that Σ_{∞} ⊂ ∂Ω_{∞}. In total we have proven that Σ_{∞} = ∂Ω_{∞}.

Left is to estimate the Reifenberg constant of ∂Ω_{∞}. To do this we again let (Ẑ , τ̂) ∈ ∂Ω_{∞} and consider r > 0. By construction there exists a sequence (Z_{j}, τ_{j}) such that (Z_{j}, τ_{j}) ∈ ∂Ω and

(Z̃_{j}, τ̃_{j}) := ((Z_{j} − X_{j})/r_{j}, (τ_{j} − t_{j})/r_{j}^{2}) → (Ẑ , τ̂).

Let ε > 0. As ∂Ω is δ_{0} -Reifenberg flat there exists, for each j, an n-dimensional plane P̂_{j} = P̂(Z_{j}, τ_{j}, r_{j}r), containing (Z_{j}, τ_{j}) and a line parallel to the t axis, having unit normal n̂_{j} = n̂(Z_{j}, τ_{j}, r_{j}r) such that

D(∂Ω ∩ C_{r_{j}r}(Z_{j}, τ_{j}), P̂_{j} ∩ C_{r_{j}r}(Z_{j}, τ_{j})) ≤ r_{j}r(δ_{0} + ε).

Also as ∂Ω_{j} → ∂Ω_{∞} in the parabolic Hausdorff distance sense there exists j_{0} ≥ 1 such that if j ≥ j_{0} then

D(∂Ω_{j} ∩ C_{r}(Ẑ , τ̂), ∂Ω_{∞} ∩ C_{r}(Ẑ , τ̂)) < εr, d((Z̃_{j}, τ̃_{j}), (Ẑ , τ̂)) < εr.

We define a new plane Λ_{j} := P̂(Z_{j}, τ_{j}, r_{j}r) − (Z_{j}, τ_{j}) + (Ẑ , τ̂). Note that this is a plane containing (Ẑ , τ̂). Then,

\begin{aligned}D(∂Ω_{∞} ∩ C_{r}(Ẑ , τ̂), Λ_{j} ∩ C_{r}(Ẑ , τ̂)) & ≤ D(∂Ω_{∞} ∩ C_{r}(Ẑ , τ̂), ∂Ω_{j} ∩ C_{r}(Ẑ , τ̂)) \\ & + D(∂Ω_{j} ∩ C_{r}(Ẑ , τ̂), Λ_{j} ∩ C_{r}(Ẑ , τ̂)). \end{aligned}

Left is therefore to estimate D(∂Ω_{j} ∩ C_{r}(Ẑ , τ̂), Λ_{j} ∩ C_{r}(Ẑ , τ̂)). As

d((Z̃_{j}, τ̃_{j}), (Ẑ , τ̂)) < εr,

we have

∂Ω_{j} ∩ C_{r}(Ẑ , τ̂) ⊂ ∂Ω_{j} ∩ C_{r(1+ε)}(Z̃_{j}, τ̃_{j}).

But by construction of the plane Λ_{j} we get

\begin{aligned}D(∂Ω_{j} ∩ C_{r(1+ε)}(Z̃_{j}, τ̃_{j}), P̂(Z̃_{j}, τ̃_{j}, r) ∩ C_{r(1+ε)}(Z̃_{j}, τ̃_{j})) & < r(1 + ε)(2δ_{0} + ε) \\ & < 4δ_{0}r + 2εr. \end{aligned}

Summing up we can conclude that

D(∂Ω_{∞} ∩ C_{r}(Ẑ , τ̂), Λ_{j} ∩ C_{r}(Ẑ , τ̂)) ≤ 4δ_{0}r + 2εr.

As ε is arbitrary this completes the proof.

To continue we proceed with the proof of Lemma 17.

Proof. Recall that from the argument of the proof of Lemma 16 it follows that ω_{j} → ω_{∞} and that the support of ω_{∞} coincides with ∂Ω_{∞}. Let ϕ ∈ C_{0}^{∞}(R^{n+1} \ (X̂, t̂)) and define ϕ_{j}(Y, s) = ϕ(r_{j}^{−1}(Y − X_{j}), r_{j}^{−2}(s − t_{j})). By the Riesz representation formula we have

∫_{∂Ω} ϕ_{j}(Y, s) dω(X̂, t̂, Y, s) = ∫_{Ω} G(X̂, t̂, Z, τ)(Δϕ_{j} − ∂_{τ} ϕ_{j}) dZ dτ.

If we let (X̂_{j}, t̂_{j}) = (r_{j}^{−1}(X̂ − X_{j}), r_{j}^{−2}(t̂− t_{j})), then by a change of variables,

∫_{∂Ω_{j}} ϕ(Y, s) dω_{j}(X̂_{j}, t̂_{j}, Y, s) = ∫_{Ω_{j}} u_{j}(Z, τ)(Δϕ − ∂_{τ} ϕ) dZ dτ

where ω_{j} and u_{j} were introduced above the statement of the lemma. Defining u_{j} ≡ 0 on the complement of Ω_{j} we can conclude, using the same argument as in the proof of Lemma 14, that {u_{j}} is a uniformly bounded sequence on compacts. By the Arzela–Ascoli theorem, u_{j} → u_{∞} uniformly on compact subsets, and u_{∞} is a positive adjoint caloric function in Ω_{∞} such that u_{∞} = 0 on ∂Ω_{∞}. By weak convergence we can therefore conclude that

∫_{∂Ω_{∞}} ϕ(Y, s) dω_{∞}(Y, s) = ∫_{Ω_{∞}} u_{∞}(Y, s)(Δ − ∂_{s})ϕ(Y, s) dY ds

for all ϕ ∈ C_{0}^{∞}(R^{n+1}).

Finally we will now explore the information contained in the condition

log k(X̂^{1}, t̂^{1}, X̂^{2}, t̂^{2}, ·) ∈ VMO(dω^{1}(X̂^{1}, t̂^{1}, ·)).

Lemma 18. Let Ω be a δ_{0} -Reifenberg flat domain with δ_{0} > 0 small and define Ω^{1} = Ω and Ω^{2} = R^{n+1} \ \overline{Ω}. We also let (X̂^{i}, t̂^{i}) ∈ Ω^{i}, for i ∈ \{1, 2\}, and define ω^{i}(X̂^{i}, t̂^{i}, ·) to be the caloric measure defined with respect to Ω^{i}. Assume that ω^{2}(X̂^{2}, t̂^{2}, ·) is absolutely continuous with respect to ω^{1}(X̂^{1}, t̂^{1}, ·) on ∂Ω and that the Radon–Nikodym derivative k(X̂^{1}, t̂^{1}, X̂^{2}, t̂^{2}, ·) = dω^{2}(X̂^{2}, t̂^{2}, ·)/dω^{1}(X̂^{1}, t̂^{1}, ·) is such that log k(X̂^{1}, t̂^{1}, X̂^{2}, t̂^{2}, ·) ∈ VMO(dω^{1}(X̂^{1}, t̂^{1}, ·)). If, using the notation of Lemma 17, ω_{j}^{i} → ω_{∞}^{i} for i ∈ {1, 2} then

ω_{∞}^{1} ≡ ω_{∞}^{2}.

Proof. To prove the lemma we prove that for any ϕ ∈ C_{0}^{∞}(R^{n+1}), ϕ ≥ 0, we have

∫_{∂Ω_{∞}} ϕ dω_{∞}^{1} = ∫_{∂Ω_{∞}} ϕ dω_{∞}^{2}.

In the following we write k(Y, s) = k(X̂^{1}, t̂^{1}, X̂^{2}, t̂^{2}, Y, s), ω^{1}(E) = ω^{1}(X̂^{1}, t̂^{1}, E) and ω^{2}(E) = ω^{2}(X̂^{2}, t̂^{2}, E). Applying Lemma 12 there exists α ∈ (0, 1) and a constant C = C(n, δ_{0}, A) such that for all (X, t) ∈ ∂Ω, r < r_{0} , |X −X̂^{i}|^{2} ≤ A(t̂^{i}− t) for some A ≥ 2 and for i ∈ {1, 2}, min{t̂^{1}, t̂^{2}} − t ≥ 8r^{2} and E ⊂ Δ(X, t, r),

\frac{ω^{2}(E)}{ω^{2}(Δ(X, t, r))} ≤ C(\frac{ω^{1}(E)}{ω^{1}(Δ(X, t, r))})^{α}.

Let ϕ ∈ C_{0}^{∞}(R^{n+1}) and recall that in our blow-up argument we considered a sequence of points (X_{j}, t_{j}) ∈ ∂Ω → (X̃ , t̃) ∈ ∂Ω and a sequence of scales {r_{j}}, r_{j} → 0, and we assumed for simplicity that (X̃ , t̃) = (0, 0). Let Δ_{j} = Δ(X_{j}, t_{j}, r_{j}). In the following we will assume that supp ϕ ⊂ C_{M}(0, 0) for some M > 1 and that ϕ ≥ 0. Let ε > 0 be given. As log k( · , ·) ∈ VMO(dω^{1}) there exists, by the John–Nirenberg inequality, j_{0} such that for j ≥ j_{0} , there exists G_{j} ⊂ Δ(X_{j}, t_{j}, Mr_{j}) := Δ̃_{j}, ω^{1}(Δ̃_{j}) ≤ (1 + ε)ω^{1}(G_{j}) and such that for every (Y, s) ∈ G_{j}

(1 − ε)\frac{1}{ω^{1}(Δ̃_{j})} ∫Δ̃_{j} k dω^{1} ≤ k(Y, s) ≤ (1 + ε)\frac{1}{ω^{1}(Δ̃_{j})} ∫Δ̃_{j} k dω^{1}.

Using this inequality we have,

(1 − ε)ω^{1}(Δ_{j} ∩ G_{j}) ≤ \frac{ω^{2}(Δ_{j} ∩ G_{j})ω^{1}(Δ̃_{j})}{ω^{2}(Δ̃_{j})} ≤ (1 + ε)ω^{1}(Δ_{j} ∩ G_{j}).

In the following C_{M} will denote constants which depend on M and other parameters but are independent of j. Using these inequalities and the consequence of the VMO condition stated above, the constant A can be chosen uniformly and independent of j as r_{j} → 0 and as the sequence (X_{j}, t_{j}) converges to a point located below (X̂^{1}, t̂^{1}) as well as (X̂^{2}, t̂^{2}), we have

\frac{ω^{2}(Δ_{j})}{ω^{1}(Δ_{j})} = \frac{ω^{2}(Δ_{j} ∩ G_{j})}{ω^{1}(Δ_{j})} + \frac{ω^{2}(Δ_{j} \ G_{j})}{ω^{1}(Δ_{j})}

≤ (1 + ε)\frac{ω^{1}(Δ_{j} ∩ G_{j})}{ω^{1}(Δ_{j})} \frac{ω^{2}(Δ̃_{j})}{ω^{1}(Δ̃_{j})} + \frac{ω^{2}(Δ_{j} \ G_{j})}{ω^{1}(Δ_{j})}

≤ (1 + ε)\frac{ω^{2}(Δ̃_{j})}{ω^{1}(Δ̃_{j})} + \frac{ω^{2}(Δ_{j})}{ω^{1}(Δ_{j})}(\frac{ω^{1}(Δ_{j} \ G_{j})}{ω^{1}(Δ_{j})})^{α}

≤ (1 + ε)\frac{ω^{2}(Δ̃_{j})}{ω^{1}(Δ̃_{j})} + C_{M}ε^{α} \frac{ω^{2}(Δ_{j})}{ω^{1}(Δ_{j})}.

By similar deductions

\frac{ω^{2}(Δ_{j})}{ω^{1}(Δ_{j})} ≥ \frac{ω^{2}(Δ_{j} ∩ G_{j})}{ω^{1}(Δ_{j})} ≥ (1 − ε)\frac{ω^{1}(Δ_{j} ∩ G_{j})}{ω^{1}(Δ_{j})} \frac{ω^{2}(Δ̃_{j})}{ω^{1}(Δ̃_{j})}

= (1 − ε)(1 − \frac{ω^{1}(Δ_{j} \ G_{j})}{ω^{1}(Δ_{j})})\frac{ω^{2}(Δ̃_{j})}{ω^{1}(Δ̃_{j})}

≥ (1 − ε)(1 − C_{M}ε)\frac{ω^{2}(Δ̃_{j})}{ω^{1}(Δ̃_{j})}.

To continue we define Ĝ_{j} = {(r_{j}^{−1}(Y − X_{j}), r_{j}^{−2}(s − t_{j})) : (Y, s) ∈ G_{j}}, Ê_{j} = Δ(0, 0, M) \ Ĝ_{j}. Using this notation we can conclude that for (Y, s) ∈ Ĝ_{j},

(1 − ε)I_{j} ≤ k_{j}(Y, s) ≤ (1 + ε)I_{j}, I_{j} := \frac{ω^{1}(Δ_{j})}{ω^{1}(Δ̃_{j})} \frac{ω^{2}(Δ̃_{j})}{ω^{2}(Δ_{j})}.

432 Kaj Nyström Based on the deductions above we can conclude that

I_{j}^{−1} ≤ \frac{ω^{1}(Δ̃_{j})}{ω^{2}(Δ̃_{j})}[(1 + ε)\frac{ω^{2}(Δ̃_{j})}{ω^{1}(Δ̃_{j})} + C_{M}ε^{α} \frac{ω^{2}(Δ_{j})}{ω^{1}(Δ_{j})}] ≤ (1 + C_{M}ε^{α})

and

I_{j} ≤ (1 − ε)^{−1}(1 − C_{M}ε)^{−1}.

In total it follows there exist two functions A(ε) and B(ε) such that if j ≥ j_{0} then

A(ε) ≤ I_{j} ≤ B(ε).

Furthermore, A(ε) → 1 and B(ε) → 1 as ε → 0. Continuing we have that

(1 − ε)I_{j} ∫_{G}_{j} ϕ dω_{j}^{1} ≤ ∫_{G}_{j} ϕk_{j} dω_{j}^{1} ≤ (1 + ε)I_{j} ∫_{G}_{j} ϕ dω_{j}^{1}.

Note that

∫_{∂Ω_{j}} ϕ dω_{j}^{2} = ∫_{∂Ω_{j}} ϕk_{j} dω_{j}^{1} = ∫_{G}_{j} ϕk_{j} dω_{j}^{1} + ∫_{E}_{j} ϕk_{j} dω_{j}^{1}

≤ (1 + ε)I_{j} ∫_{G}_{j} ϕ dω_{j}^{1} + ∥ϕ∥_{∞}ω_{j}^{2}(Ê_{j})

≤ (1 + ε)I_{j} ∫_{∂Ω_{j}} ϕ dω_{j}^{1} + C_{M}ε^{α}∥ϕ∥_{∞}.

Similarly

∫_{∂Ω_{j}} ϕ dω_{j}^{2} ≥ ∫_{G}_{j} ϕk_{j} dω_{j}^{1} ≥ (1 − ε)I_{j} ∫_{G}_{j} ϕ dω_{j}^{1}

= (1 − ε)I_{j} ∫_{∂Ω_{j}} ϕ dω_{j}^{1} − (1 − ε)I_{j} ∫_{E}_{j} ϕ dω_{j}^{1}

≥ (1 − ε)I_{j} ∫_{∂Ω_{j}} ϕ dω_{j}^{1} − C_{M}(1 − ε)I_{j}ε∥ϕ∥_{∞}.

Based on this we can conclude that

\begin{aligned}∫_{∂Ω_{j}} ϕ dω_{j}^{2} & ≤ (1 + ε)B(ε) ∫_{∂Ω_{j}} ϕ dω_{j}^{1} + C_{M}ε^{α}∥ϕ∥_{∞}, \\ ∫_{∂Ω_{j}} ϕ dω_{j}^{2} & ≥ (1 − ε)A(ε) ∫_{∂Ω_{j}} ϕ dω_{j}^{1} − C_{M}(1 − ε)B(ε)ε∥ϕ∥_{∞}. \end{aligned}

Hence

∫_{∂Ω_{∞}} ϕ dω_{∞}^{2} = lim_{j→∞} ∫_{∂Ω_{j}} ϕ dω_{j}^{2} = lim_{j→∞} ∫_{∂Ω_{j}} ϕ dω_{j}^{1} = ∫_{∂Ω_{∞}} ϕ dω_{∞}^{2}.

This completes the proof of the lemma.

4.2. Classification of global solutions and the proof of Theorem 1

Lemma 19. Assume that the original domain Ω is δ_{0} -Reifenberg flat with δ_{0} small enough. Assume furthermore that the assumptions in Lemma 18 are fulfilled and, using the notation of Lemma 17, u_{j}^{i} → u_{∞}^{i} for i ∈ {1, 2}. Define for (Y, s) ∈ R^{n+1} , u_{∞}(Y, s) = u_{∞}^{1}(Y, s) − u_{∞}^{2}(Y, s), where u_{∞}^{1}(Y, s) ≡ 0 in Ω_{∞}^{2} , u_{∞}^{2}(Y, s) ≡ 0 in Ω_{∞}^{1} . Then u_{∞} is a linear function in the space variables and Ω_{∞} is a half space containing a line parallel to the time-axis.

Proof. Applying Lemma 18 we can conclude that ω_{∞}^{1} ≡ ω_{∞}^{2} and that for all ϕ ∈ C_{0}^{∞}(R^{n+1})

∫_{∂Ω_{∞}^{i}} ϕ(Y, s) dω_{∞}^{i}(Y, s) = ∫_{Ω_{∞}^{i}} u_{∞}^{i}(Y, s)(Δ − ∂_{s})ϕ(Y, s) dY ds.

Hence

∫_{R^{n+1}} u_{∞}(Y, s)(Δ − ∂_{s})ϕ(Y, s) dY ds = ∫_{Ω_{∞}^{1}} u_{∞}^{1}(Y, s)(Δ − ∂_{s})ϕ(Y, s) dY ds
− ∫_{Ω_{∞}^{2}} u_{∞}^{2}(Y, s)(Δ − ∂_{s})ϕ(Y, s) dY ds
= 0.

As u_{∞} is continous, it is weakly adjoint caloric in R^{n+1} and therefore adjoint caloric in R^{n+1} . By a change of the time direction we can assume that u_{∞} is caloric in R^{n+1} . We also note that u_{∞}(0, 0) = 0. By standard estimates for the heat equation we have that

max_{C_{r/2}(0,0)} |D_{Z}^{k}D_{τ}^{l} u_{∞}(Z, τ)| ≤ \frac{C_{kl}}{r^{k+2l}} max_{C_{r}(0,0)} |u_{∞}(Z, τ)|.

As, according to Lemma 16, ∂Ω_{∞} is Reifenberg flat we define for (X, t) ∈ ∂Ω, r > 0

A_{r}^{1}(X, t) = (X + rn̂, t) ∈ Ω_{∞}, A_{r}^{2}(X, t) = (X − rn̂, t) ∈ R^{n+1} \ \overline{Ω}_{∞}.

Here n̂ = n̂(X, t, r). Using this notation we have by the backward in time Harnack principle in Lemma 8 that

max_{C_{r}(0,0)} |u_{∞}(Z, τ)| ≤ C max{u_{∞}^{1}(A_{r}^{1}(0, 0)), u_{∞}^{2}(A_{r}^{2}(0, 0))}.

Using Lemmas 7 and 9 we have

\frac{r^{n}u_{∞}^{1}(A_{r}^{1}(0, 0))}{ω_{∞}^{1}(Δ(0, 0, r))} ∼ \frac{r^{n}u_{∞}^{2}(A_{r}^{2}(0, 0))}{ω_{∞}^{2}(Δ(0, 0, r))} ∼ 1.

Hence as ω_{∞}^{1} ≡ ω_{∞}^{2} we can conclude that u_{∞}^{1}(A_{r}^{1}(0, 0)) ∼ u_{∞}^{2}(A_{r}^{2}(0, 0)). As, according to Lemma 16, Ω_{∞} is 4δ_{0} -Reifenberg flat we can apply Lemma 13 to appropriate functions defined in Ω_{∞}. In particular according to Lemma 13, given ε > 0, there exists δ̂_{0} = δ̂_{0}(n, ε) > 0 and a constant C = C(n, ε) such that if δ_{0} ≤ δ̂_{0} and r̂ ≤ r then

C^{−1}(\frac{r̂}{r})^{1+ε}u_{∞}^{1}(A_{r}^{1}(0, 0)) ≤ u_{∞}^{1}(A1 ; r̂(0, 0)) ≤ C(\frac{r̂}{r})^{1−ε}u_{∞}^{1}(A_{r}^{1}(0, 0)).

If we choose r̂ = 1 we can conclude that

u_{∞}^{1}(A_{r}^{1}(0, 0)) ≤ Cu_{∞}^{1}(A_{1}^{1}(0, 0))r^{1+ε}.

In total we can conclude that

max_{C_{r/2}(0,0)} |D_{Z}^{k}D_{τ}^{l} u_{∞}(Z, τ)| ≤ \frac{C_{kl}}{r^{k+2l}} u_{∞}^{1}(A_{r}^{1}(0, 0)) ≤ \frac{C_{kl}}{r^{k+2l−1−ε}} u_{∞}^{1}(A_{1}^{1}(0, 0)).

By letting r → ∞ we get D_{Z}^{k}D_{τ}^{l} u_{∞}(Z, τ) = 0 for all (Z, τ) ∈ R^{n+1} and all (k, l) such that k + 2l − 1 − ε > 0. We can therefore conclude that u_{∞} is in fact a linear function in the space variables and Ω_{∞} is a half space containing a line parallel to the time-axis.

We can now prove Theorem 1 using Lemma 19. According to Definition 1, if ∂Ω is δ_{0} Reifenberg flat, then given any (X̃ , t̃) ∈ ∂Ω, R > 0, there exists an n-dimensional plane P̂ = P̂(X̃ , t̃, R), containing (X̃ , t̃) and a line parallel to the t-axis, having unit normal n̂ = n̂(X̃ , t̃, R) such that

\begin{matrix}\{(Y, s) + rn̂ ∈ C_{R}(X̃ , t̃) : (Y, s) ∈ P̂, r > δ_{0}R\} ⊂ Ω, \\ \{(Y, s) − rn̂ ∈ C_{R}(X̃ , t̃) : (Y, s) ∈ P̂, r > δ_{0}R\} ⊂ R^{n+1} \ Ω. \end{matrix}

We therefore introduce the quantity

Θ(X̃ , t̃, R) := \frac{1}{R} inf ; _{P} D[∂Ω ∩ C_{R}(X̃ , t̃), P̂ ∩ C_{R}(X̃ , t̃)]

where the infimum is taken over all n-dimensional planes P̂ = P̂(X̃ , t̃, R), containing (X̃ , t̃) and a line parallel to the t-axis. For any compact set K ⊂ R^{n+1} we also introduce

Θ_{K}(R) := sup_{(_{X} ,t̃)∈K} Θ(X̃ , t̃, R).

If (X, t) ∈ ∂Ω, r > 0, then the statement that C_{r}(X, t) ∩ ∂Ω is Reifenberg flat with vanishing constant in the parabolic sense is equivalent to the statement that

lim_{r̂→0} Θ_{C_{r}(X,t)∩∂Ω}(r̂) = 0.

To prove Theorem 1 we assume, using the notation of the theorem, that (X, t) ∈ ∂Ω, t̂^{2} > t + 4r^{2} and that

lim_{r̂→0} Θ_{C_{r}(X,t)∩∂Ω}(r̂) = β

for some β > 0. We intend to prove that this is impossible and that β = 0. Let (X_{j}, t_{j}) ∈ C_{r}(X, t) ∩ ∂Ω, (X_{j}, t_{j}) → (X̂, t̂) ∈ C_{r}(X, t) ∩ ∂Ω and r_{j} be a sequence of real numbers tending to zero such that

lim_{j→∞} Θ(X_{j}, t_{j}, r_{j}) = β.

By a translation argument we can without loss of generality assume that (X̂, t̂) = (0, 0) and that (0, 0) ∈ C_{r}(X, t) ∩ ∂Ω. Define the domains,

Ω_{j}^{i} = {(r_{j}^{−1}(X − X_{j}), r_{j}^{−2}(t − t_{j})) : (X, t) ∈ Ω^{i}}.

Then according to Lemma 16 we can assume that Ω_{j}^{i} → Ω_{∞}^{i}, ∂Ω_{j}^{i} → ∂Ω_{∞}^{i} in the parabolic Hausdorff distance sense uniformly on compact subsets of R^{n+1} . Furthermore, Ω_{∞} = Ω_{∞}^{1} and Ω_{∞}^{2} = R^{n+1} \ \overline{Ω}_{∞} are Reifenberg flat domains. We can furthermore apply Lemmas 18 and 19 in order to conclude that Ω_{∞} is a half space containing a line parallel to the time-axis. Still our assumption above gives at hand that

Θ_{∞}(0, 0, 1) := β > 0

where Θ_{∞}(0, 0, 1) is defined with respect to ∂Ω_{∞}. Clearly this is a contradiction and we can conclude that C_{r}(X, t)∩∂Ω is Reifenberg flat with vanishing constant.

References

[AC] Alt, H. W., and L. Caffarelli: Existence and regularity for a minimum problem with free boundary. - J. Reine Angew. Math. 325, 1981, 105–144.

[ACS] Athanasopoulos, I., L. Caffarelli, and S. Salsa: Regularity of the free boundary in parabolic phase transition problems. - Acta Math. 176, 1996, 245–282.

[ACS1] Athanasopoulos, I., L. Caffarelli, and S. Salsa: Caloric functions in Lipschitz domains and the regularity of solutions to phase transition problems. - Ann. of Math.

(1) 143, 1996, 463–484.

[B] Bishop, C.: A characterization of Poissonian domains. - Ark. Mat. 29, 1991, 1–24.

[BCGJ] Bishop, C., L. Carleson, J. Garnett, and P. Jones: Harmonic measure supported on curves. - Pacific J. Math. 138, 1989, 233–236.

436 Kaj Nyström

[BJ] Bishop, C., and P. Jones: Harmonic measure and arc length. - Ann. of Math. 132, 1990, 511–547.

[D] Dahlberg, B.: On estimates of harmonic measure. - Arch. Ration. Mech. Anal. 65, 1977, 275–288.

[Do] Doob, J. L.: Classical Potential Theory and Its Probabilistic Counterpart. - Grundlehren Math. Wiss. 262, Springer-Verlag, 1984.

[FS] Fabes, E., and M. Safonov: Behaviour near the boundary of positive solutions of second order parabolic equations. - J. Fourier Anal. Appl. 3, 1997, 871–882.

[FSY] Fabes, E., M. Safonov, and Y. Yuan: Behaviour near the boundary of positive solutions of second order parabolic equations II. - Trans. Amer. Math. Soc. 351, 1999, 4947–4961.

[HL] Hofmann, S., and J. Lewis: Solvability and representation by caloric layer potentials in time-varying domains. - Ann. of Math. 144, 1996, 349–420.

[HLN1] Hofmann, S., J. Lewis, and K. Nyström: Existence of big pieces of graphs for parabolic problems. - Ann. Acad. Sci. Fenn. Math. 28, 2003, 355–384.

[HLN2] Hofmann, S., J. Lewis, and K. Nyström: Caloric measure in parabolic flat domains. - Duke Math. J. 122(2), 2004, 281–345.

[J] Jerison, D.: Regularity of the Poisson kernel and free boundary problems. - Colloq. Math. 60–61, 1990, 547–567.

[JK] Jerison, D., and C. Kenig: The logarithm of the Poisson kernel in a C^{1} domain has vanishing mean oscillation. - Trans. Amer. Math. Soc. 273, 1982, 781–794.

[KT] Kenig, C., and T. Toro: Harmonic measure on locally flat domains. - Duke Math. J. 87, 1997, 501–551.

[KT1] Kenig, C., and T. Toro: Free boundary regularity for harmonic measure and Poisson kernels. - Ann. of Math. 150, 1999 369–454.

[KT2] Kenig, C., and T. Toro: Poisson kernel characterization of Reifenberg flat chord arc domains. - Ann. Sci. École Norm. Sup. (to appear).

[KT3] Kenig, C., and T. Toro: Free boundary regularity below the continous threshold: 2phase problems. - Preprint, 2004.

[KW] Kaufmann, R., and J. M. Wu: Parabolic measure on domains of class Lip_{1/2} . - Compos. Math. 65, 1988, 201–207.

[LM] Lewis, J., and M. Murray: The method of layer potentials for the heat equation in time-varying domains. - Mem. Amer. Math. Soc. 114, 1995, no. 545.

[LS] Lewis, J., and J. Silver: Parabolic measure and the Dirichlet problem for the heat equation in two dimensions. - Indiana Univ. Math. J. 37, 1988, 801–839.

[N] Nyström, K.: The Dirichlet problem for second order parabolic operators. - Indiana Univ. Math. J. 46, 1996, 183–245.

[S] Stein, E.: Harmonic Analysis. - Princeton Math. Ser. 43, Princeton University Press, 1993.

Received 19 May 2005