Variable-coefficient parabolic theory as a high-dimensional limit of elliptic theory

This paper continues the study initiated in Davey (Arch Ration Mech Anal 228:159–196, 2018), where a high-dimensional limiting technique was developed and used to prove certain parabolic theorems from their elliptic counterparts. In this article, we extend these ideas to the variable-coefficient setting. This generalized technique is demonstrated through new proofs of three important theorems for variable-coefficient heat operators, one of which establishes a result that is, to the best of our knowledge, also new. Specifically, we give new proofs of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2 \rightarrow L^2$$\end{document}L2→L2 Carleman estimates and the monotonicity of Almgren-type frequency functions, and we prove a new monotonicity of Alt–Caffarelli–Friedman-type functions. The proofs in this article rely only on their related elliptic theorems and a limiting argument. That is, each parabolic theorem is proved by taking a high-dimensional limit of a related elliptic result.


INTRODUCTION
In this paper, we explore the connections between the elliptic and parabolic theory of partial differential equations, generalizing the work done by the first-named author for constant-coefficient equations in [17]. Specifically, we generalize the ideas from [17] and establish a technique that can be used to prove variablecoefficient parabolic theorems from their appropriate elliptic counterparts. The key idea is that certain parabolic estimates may be obtained by taking high-dimensional limits of their corresponding elliptic results. We obtain information about solutions to div(A∇u)+ ∂ t u = 0 on the parabolic side by analyzing the behavior of solutions to non-homogeneous equations of the form div(κ∇v) = κℓ on the elliptic side. Here, A has a specific structure, v and κ are defined in terms of u and A (see Lemmas 2.1 and 2.2), respectively, and ℓ depends on both u and A. Rewriting our elliptic equation as κ −1 div(κ∇v) = ℓ, we notice that the associated operator is a special type of Witten Laplacian, or weighted Laplacian (see 2.4 in [33], and also [33], [27], [14], [43], [13], [41], [42], and [44]). From this perspective, the ideas in this article show how to obtain results for variable-coefficient parabolic operators from those for the Witten Laplacian.
Perelman first considered parabolic theory as a high-dimensional limit of elliptic theory in [47]. This general principle was discussed in the blog of Tao [52], modified in the coursenotes of Sverak [51], then developed and applied in [17]. In our setting, we follow the ideas from [17]; namely, we use classical probabilistic formulae, essentially going back to Wiener [54], with a slight modification used by Sverak in [51]. However, to account for the presence of variable-coefficients, we have modified (and complicated) the change of variables formula from [17]. Once the general framework has been established, we demonstrate the utility of this technique by establishing three new proofs of theorems regarding variable-coefficient parabolic equations. In comparison to the results of [17] for constant-coefficient equations, the techniques here are substantially modified to account for the variable-coefficients.
At the heart of the high-dimensional limiting process is a sequence of maps F d,n : R d×n → R d × R + which take high-dimensional space points y ∈ R d×n on the elliptic side to space-time points (x,t) ∈ R d × R + on the parabolic side. These maps can be viewed from three perspectives. First, they naturally arise through certain random walk processes. Second, we can use the chain rule to see how these maps connect elliptic and parabolic operators. That is, suppose we are given some parabolic function u = u (x,t) defined on B. D. has been partially supported by the NSF LEAPS-MPS grant DMS-2137743, and a Simons Foundation Collaboration Grant 430198.
M.S.V.G has been partially supported by the NSF grant DMS-2054282. 1 in which the right-hand side of the equation need not vanish at the free boundary. Their main result is not a monotonicity result per se, but rather a clever uniform bound on the monotonicity functional, which is just as useful as monotonicity itself. Later on, Matevosyan and Petrosyan [45] further extended that result, proving an almost monotonicity estimate for non-homogeneous elliptic and parabolic operators with variable coefficients. We refer the interested reader to [48] and [4] for an introduction to the use of the ACF monotonicity formula; [48] deals with the elliptic setting, while [4] addresses both the parabolic and elliptic settings. In the context of almost minimizers of variable-coefficient Bernoulli-type functionals, ACF-type monotonicity formulas have also been used to study the free boundary, see for example [18]. Given that our transformation maps relate solutions to homogeneous parabolic equations to a sequence of solutions to non-homogeneous elliptic equations, our proofs of the monotonicity results have added complexity. More specifically, given that each v n solves an elliptic equations with a right-hand side, we require elliptic monotonicity results that apply to solutions to non-homogeneous equations. Therefore, the first step in each parabolic monotonicity proof is to establish the related underlying variable-coefficient elliptic result with a right-hand side. These new elliptic results, which appear at the beginning of Sections 5 and 6, may be of independent interest. We point out that these elliptic estimates generalize both the constant-coefficient and the homogeneous results.
The monotonicity of frequency functions is often proved by showing that the derivative of the frequency function has a fixed sign. In our proofs of the monotonicity of variable-coefficient Almgren-type and ACFtype frequency functions, we show that each parabolic frequency function may be described as a pointwise limit of a sequence of elliptic frequency functions. While we know that each such elliptic frequency function is differentiable and almost monotonic, that is not sufficient to guarantee the differentiability of the corresponding parabolic frequency function. Therefore, our proofs rely on a more delicate analysis that allows us to conclude directly that our parabolic frequency functions are monotonic. These ideas are described at the ends of the proofs of Theorems 5.3 and 6.4.
We work with time-independent variable-coefficient operators for which the coefficient matrix has a specific structure. That is, we consider symmetric matrices of the form A = GG T , where G is the Jacobian of some invertible map g : R d → R d . Clearly, there are bounded, elliptic coefficient matrices A whose structure is not of this form. However, if A = I, then A is associated to the identity map g(z) = z with Jacobian G = I, showing that this structural condition on A is a reasonable generalization to constant-coefficient operators. In subsequent work, we will explore operators with even more general coefficient matrices.
The article is organized as follows. In Sections 2 and 3, we develop the framework that connects the elliptic and parabolic theory. That is, we introduce and examine the maps F d,n : R d×n → R d × R + that take points in the high-dimensional (elliptic) space to space-time (parabolic) points. Section 2 takes the perspective of random walks to introduce these maps, then uses the chain rule to explore how these maps relate elliptic and parabolic operators. Section 3 examines these maps from a measure theoretic perspective. In particular, we present the pushforward computations and describe how the integrals on the elliptic side are related to those on the parabolic side. These two sections contain a collection of calculations and statements that will be referred to throughout the article.
The L 2 → L 2 variable-coefficient parabolic Carleman estimate is proved in Section 4. The Almgren-type frequency function theorem for variable-coefficient heat operators is presented in Section 5. Finally, Section 6 contains the monotonicity result for Alt-Caffarelli-Friendman-type energies associated with variablecoefficient heat operators.
Acknowledgements. We thank Stefan Steinerberger for interesting discussions.

ELLIPTIC-TO-PARABOLIC TRANSFORMATIONS
In this section, we construct the transformations that connect so-called parabolic functions u = u(x,t) defined on R d × (0, T ) to a sequence of elliptic functions v n = v n (y) defined in R d×n for all n ∈ N. More specifically, for each n ∈ N, we construct a mapping of the form Given a function u = u(x,t) defined on a space-time domain, i.e. a subset of R d × R + , we use F d,n to define a function v n = v n (y) on the space R d×n by setting v n (y) = u(F d,n (y)). As we show below via the chain rule, if u is a solution to a backward parabolic equation, then each v n is a solution to some non-homogeneous elliptic equation. This observation explains why we think of u as a parabolic function and of each v n as an elliptic function. Moreover, as n becomes large, the function v n behaves (heuristically) more and more like a solution to a homogeneous elliptic equation. As such, the transformation F d,n becomes more useful to our purposes as n → ∞, thereby illuminating why the notion of a high-dimensional limit is relevant here. From another perspective, when we use F d,n to pushforward measures on spheres and balls in R d×n , we produce measures in space-time that are weighted by approximations to generalized Gaussians. This perspective is explored in the next section, Section 3.
We can think of the transformations F d,n : R d×n → R d × R + in a number of ways. On one hand, as we show in this section, these maps are constructed so that each v n solves an elliptic equation whenever u solves a parabolic equation. This viewpoint is purely computational as these relationships are illuminated by the chain rule. This perspective is perhaps the most (easily) checkable, but it is somewhat mysterious. Random walks are the underlying mechanism in defining these mappings, so we begin with that perspective.
Let y = (y 1,1 , y 1,2 , . . . , y 1,n , . . . , y d,1 , y d,2 , . . . , y d,n ) ∈ R d×n denote the variables that play the role of the "random steps" in our random walk. For some t > 0, assume that y satisfies In this setting, we do not fix the step size but simply assume that y is uniformly distributed over the sphere of radius √ 2dt. Define so that z = (z 1 , . . . , z d ) ∈ R d . In the notation from [17], we define f d,n : R d×n → R d so that Now let and then since z = h(x) we have For the moment, assume that g is sufficiently regular for the computations below to hold in a weak sense. When further regularity is needed, we specify it. 4 The Jacobian of g = (g 1 , . . . , g d ) is a d × d invertible matrix function whose inverse matrix is the Jacobian of h = (h 1 , . . . , h d ). Let G and H denote the Jacobian matrices of g and h, respectively. That is, Let γ(z) = det G(z) and η(x) = det H(x). From (5), we obtain ).
Now we collect some observations that follow from the Chain Rule. For the first lemma, we describe how these determinants and their derivatives transform through the map g • f d,n . .
Proof. Since ∂ z k ∂ y i, j = δ k,i , then an application of Jacobi's formula shows that We then get The next set of observations shows how the derivatives of some parabolic function u : R d × (0, T ) → R can be related to those of its associated elliptic functions v n : B √ 2dT ⊂ R d×n → R. That is, given a parabolic function u = u(x,t), we define the elliptic functions v n = v n (y) = u(F d,n (y)), where The following set of results justifies why we refer to u as parabolic and each v n as elliptic.
Moreover, with κ n as in (6), where the expression on the right depends on x and t.
The proof of this result relies on the chain rule and can be found in Appendix A. When u is a solution to a variable-coefficient backwards heat equation, we immediately reach the following consequence.
In summary, we have constructed a sequence of maps F d,n , given in (8), that serve as the connection between the elliptic and parabolic settings. The following table describes these relationships and the notation that we use to describe our elliptic and parabolic settings. Note that F d,n is the connection between the elliptic column and the parabolic column.

Elliptic
Parabolic Going forward, we write ∇u to indicate ∇ x u, unless explicitly indicated otherwise. Similarly, we write ∇v n to indicate ∇ y v n , unless explicitly indicated otherwise. That is, u is understood to be a function of x and t, while v n is a function of y, and all derivatives are interpreted appropriately.
The results above will be used extensively when we prove our parabolic theorems. Therefore, we work with variable-coefficient operators for which the coefficient matrix has the specific structure that is described by the previous two results. That is, we consider A(x) = B(z) where B = GG T and G is the Jacobian of some invertible map g : R d → R d . Clearly, there are bounded, elliptic coefficient matrices A whose structure is not of this form. However, if A = I, then A is associated to the identity map g(z) = z with Jacobian G = I, showing that this structural condition on A is a reasonable generalization to constant-coefficient operators.
Before concluding this section, we examine some examples of non-trivial coefficient matrices that satisfy our structural condition. For simplicity, we assume that d = 1.
Given a bounded, elliptic function λ ≤ C(x) ≤ Λ, we construct a function g such that the corresponding function A(x) coincides with C(x). Recall that .
where F(x|m) is the elliptic integral of the first kind with parameter m = k 2 . Again, we define g(z) = h −1 (z).

INTEGRAL RELATIONSHIPS
In what follows, we examine how integrals of functions φ (x,t) defined in space-time R d × R + can be related to integrals of φ (F d,n (y)) in high-dimensional space R d×n . We start from a probabilistic viewpoint. That is if y ∈ R d×n is uniformly distributed over a fixed sphere, we seek to determine the probability distribution of x ∈ R d , where (x,t) = F d,n (y). As we show below, understanding this probability distribution on the parabolic side reduces to a pushforward computation which we carry out explicitly. We then collect the consequences that will be used in our elliptic-to-parabolic proofs.
Let S n t denote the sphere of radius √ 2dt in R d×n . That is, Let σ t dn−1 denote the canonical surface measure on this sphere S n t . We make the assumption that the vectors y = (y 1,1 , y 1,2 , . . . , y d,1 , . . . , y d,n ) ∈ R d×n are uniformly distributed over S n t with respect to σ t dn−1 . Set µ t d,n to be the normalized surface measure on the sphere S n t ; that is, Our goal is to find the probability distribution of x in the limit as n → ∞, where x = g(z) = g( f d,n (y)) and z = f d,n (y) is as in (3). We first define an intermediate measure on z ∈ R d for each fixed t via the pushforward as ω t d,n = f d,n #µ t d,n . This is the pushforward from [17] and it is shown there that 7 where we have introduced the notation for the ball of radius √ 2dnt in R d . The probability distribution of x is the push-forward of µ t d,n by g • f d,n : ν t d,n := (g • f d,n )#(µ t d,n ). This implies, in particular, that for all ϕ ∈ C 0 (R d ), computing the pushforward here is the same as carrying out a change of variables. By definition, we have that As shown in [17,Lemma 6], where the limit is pointwise. A much stronger version of convergence holds for this sequence, stated as follows.
The proof of this result can be found in Appendix A. In fact, the arguments there, in combination the proof of [17, Lemma 1], show that there exists C d so that for every n ∈ N and every (x,t) ∈ R d × R + , This bound will be used in many of our subsequent arguments. One might wonder how our transformed heat kernel relates to the standard one. The following lemma, proved in Appendix A, shows that K(h(x),t) is not a solution to the associated homogeneous variablecoefficient heat equation.
Summarizing our pushforward computations, we have established the following result: Lemma 3.3 (cf. Lemma 2 and Lemma 3 in [17]). Let S n t , K t,n and K t be as given in (10), (12) and (13), respectively. If ϕ : R d → R is integrable with respect to K t (h(x)) dx, then for every n ∈ N, ϕ is integrable with respect to K t,n (h(x)) dx and it holds that where κ n is as defined in (6).
Remark 3.4. Going forward, we often use the notation dσ or dσ (y) in place of dσ t dn−1 . It is also helpful to interpret the measures ν t d,n as time slices of a space-time object, which comes from the projection of some global measure in the space y ∈ R d×n onto the space-time. To do so, we project a measure µ d,n on R d×n to the space-time R d × R + by the map F d,n as given in (8). With m denoting Lebesgue measure, define The "slices" of the measure µ d,n by the spheres S t n project by F d,n onto the measures ν t d,n ; that is, Integrating the equality from Lemma 3.3 leads to the following result.
Lemma 3.5 (cf. Lemma 4 in [17]). Let B n t , K n and K be as given in (15), (12) and (13), respectively. If φ : R d × (0, T ) → R is integrable with respect to K (h(x)) dx dt, then for every n ∈ N, φ is integrable with respect to K n (h(x)) dx dt and for every t ∈ (0, T ), it holds that where κ n is as defined in (6).
Lemmas 3.3 and 3.5 will be used many times in our proofs below. Lemma 3.3 transforms a κ n -weighted integral over a sphere in R d×n to a K t,n -weighted integral at a particular slice in space-time. Similarly, Lemma 3.5 transforms a weighted integral over a ball in R d×n to a K t,n -weighted integral in space-time. Of note, the elliptic weight in Lemma 3.5 is κ n times the fundamental solutions of the Laplacian, while by (13), the parabolic weight is an approximation to a transformed heat kernel. As dimension-limiting arguments will be crucial to our proofs below, we need to understand what happens when n → ∞. By (13) and 14 in combination with the Dominated Convergence Theorem, the parabolic integrals in both of these lemmas converge to K(h(x),t)-weighted integrals.
For the remainder of this section, we introduce some definitions and draw some conclusions based on Lemmas 3.3 and 3.5.
With this definition and Lemma 3.5, we reach the following result.
Proof. An application of Lemma 3.5 shows that where we have applied (14) and that dn ≥ 2d ≥ d.
Now we introduce some function classes for parabolic and elliptic functions that will be used below.
We say that such a function u has moderate h-growth at infinity if H, D, and T belong to Moreover, if both v and ∇v ∈ L 2 (B R , κ(y) dy), then we say that v belongs to the κ-weighted Sobolev space and write v ∈ W 1,2 (B R , κ(y) dy).
Now we may connect these two function classes.
Lemma 3.9 (Function class relationship). If u : R d × (0, T ) → R has moderate h-growth at infinity, then for v n : B n T ⊂ R d×n → R defined by v n (y) = u (F d,n (y)) and κ n defined by (6), it holds that v n ∈ W 1,2 (B n T , κ n (y)dy), whenever n ≥ 2.
Proof. An application of Lemma 3.5 shows that where we have applied (14). Since u has moderate h-growth at infinity, then For the gradient term, we use Lemma 3.5 in combination with Lemma 2.2 to get where the last step uses Cauchy-Schwarz. Since |h(x)| 2 ≤ 2dns on the support of K n (h(x)), then where we have again used that u has moderate h-growth at infinity.
The following elliptic Carleman estimate is the L 2 → L 2 case of Theorem 1 from [3]. The original theorem was used to establish unique continuation properties of functions that satisfy partial differential inequalities of the form |∆v| ≤ |V | |v|, where Remark 4.2. In order for this theorem to be meaningful, we choose τ ∈ R so that τ − N 2 ∈ Z ≥0 . Remark 4.3. Other versions of this theorem hold with more general norms. More specifically, [3, Theorem 1] establishes that for any 1 ≤ q ≤ 2 ≤ p < ∞ and µ : However, since the condition that µ > 0 is equivalent to N < 2pq p−q , we must have that p and q are both very close to 2 for large N. In particular, when N → ∞, p, q → 2. Since we use the high-dimensional limit of this elliptic Carleman estimate to establish its parabolic counterpart, this explains why we restrict to the L 2 → L 2 Carleman estimate.
The following parabolic Carleman estimate is a variable-coefficient L 2 → L 2 version of Theorem 1 from [20], and it resembles [23,Theorem 4]. For a much more general result, we refer the reader to [38,Theorem 3]. The original theorem was used to prove strong unique continuation of solutions to the heat equation. As such, this version could be used to establish unique continuation results for solutions to variable-coefficient heat equations. .
Then there is a constant C, depending only on ε and δ , such that for every where h is some invertible function and We show that Theorem 4.4 follows from the elliptic result, Proposition 4.1, Lemma 3.5, and the results of Section 2. More specifically, given u, we define a sequence of functions {v n } and we then apply Proposition 4.1 to each one. Applications of Lemma 3.5 allow us to transform the integral inequalities for v n into integral inequalities for u. By taking a limit of this sequence of inequalities and using (13), we arrive at our conclusion.
where F d,n is as defined in (8). Note that each v n is compactly supported in B n .
then for δ > 0 as given, where C δ = 1 + δ −1 and c δ = 1 + δ . An application of Theorem 4.1 with v = √ κ n v n and τ = τ n shows that Since dn 2 −1−τ n satisfies the hypotheses of Lemma 3.5 and it follows that 13 Because div (A∇u) dn 2 +1−τ n also satisfies the hypotheses of Lemma 3.5, then another application of Lemma 3.5 shows that where we have used (9) from Lemma 2.2 and the triangle inequality to reach the last line.
since (9) with v n = κ −1/2 n gives |y| 2 κ n (y) div y κ n (y)∇ y κ n (y) which belongs to L ∞ R d by assumption. Substituting (19), (20), and (21) into (18) and simplifying shows that 14 where we have used the definition of c δ and that 4T 0 Observe that since we assumed that 2α − d 2 − 3 > 0. Returning to (22), the last term on the right may be absorbed into the left to get . We now take the limit as n → ∞. An application of Lemma 3.1 shows that as required.

ALMGREN MONOTONICITY FORMULA
In this section, we show that the Almgren-type frequency function associated with the parabolic operator div(A∇) + ∂ t is monotonically non-decreasing. When A = I, a corresponding result goes back to Poon, [49]. In that paper, the monotonicity was key in the proof of unique continuation results for caloric functions. A version of this result was later proved in the context of the parabolic, constant-coefficient Signorini problem in [16], where the authors used the monotonicity to establish the optimal regularity of solutions and study the free boundary.
To establish our parabolic result through the high-dimensional limiting technique, we require a similar result for solutions to non-homogeneous variable-coefficient elliptic equations. Many similar results for the homogenous setting have been previously established. For example, Almgren-type monotonicity formulas for variable-coefficient operators have also been extensively used to study a wide variety of free boundary problems, as in [34], [32], [31], [30], [35], [8], [19]. The following result is crucial to our upcoming parabolic proof, but it may also be of independent interest.
Proposition 5.1. For some R > 0, let B R ⊂ R N . Assume that for κ : B R → R + it holds that ∇ log κ · y ∈ L ∞ (B R ). Let v ∈ W 1,2 (B R , κdy) be a weak solution to div (κ∇v) = κℓ in B R , where ℓ is integrable with respect to both κ v and κ∇v · y on each B r , for r ∈ (0, R). For every r ∈ (0, R), assuming that each v| ∂ B r is 15 non-trivial, define . Then for all r ∈ (0, R), it holds that Remark 5.2. Notice that if v is a solution to the homogeneous equation div (κ∇v) = 0 in B R , i.e. ℓ = 0, then L(r) is non-decreasing in r. Moreover, if κ = 1, we recover the non-homogenous elliptic result from [17, Corollary 1], which is the non-homogeneous version of Poon's result, [49]. In particular, we recover the expected monotonicity formula for solutions to elliptic equations.
Proof. Observe first that since wheren indicates the outer unit normal. Moreover, integration by parts and the equation for v shows that Now differentiating and integrating by parts shows that 16 Therefore, by putting it all together, we see that

It follows that
or rather, Now we use Lemma 3.3 in combination with Proposition 5.1 to establish its parabolic counterpart. Before stating the result, we discuss the kinds of solutions that we work with.
Let u : R d × (0, T ) → R have moderate h-growth at infinity, as described in Definition 3.8. For every t ∈ (0, T ), assume first that u is sufficiently regular to define the functionals where and all derivatives are interpreted in the weak sense. Then we say that such a function u belongs to the function class A R d × (0, T ) , h if u has moderate h-growth at infinity (so is consequently continuous), and for every t 0 ∈ (0, T ), there exists ε ∈ (0,t) so that and there exists p > 1 so that This is the class of functions that we consider in our result. We remark that this may not be the weakest set of conditions under which our proof holds, but it is far less restrictive than assuming that our solutions are smooth and compactly supported, for example.
Proof. We first check that with κ n and v n defined through the transformation maps F d,n , the hypothesis of Lemma 5.1 are satisfied. Recall that B n T ⊂ R d×n is given by (15). Let κ n be as in Lemma 2.1, which shows that In particular, ∇ log κ n · y ∈ L ∞ (B n T ) for each n. Let u ∈ A R d × (0, T ) , h be a non-trivial solution to div (A∇u) + ∂ t u = 0 in R d × (0, T ). For every n ∈ N ≥2 , let v n : B n T → R satisfy v n (y) = u (F d,n (y)) .
Since u has moderate h-growth at infinity, then Lemma 3.9 shows that each v n belongs to W 1,2 (B n T , κ n (y) dy). 18 An application of Corollary 2.3 shows that where J is defined in (25) and does not depend on n. For every n, define ℓ n : B n T → R so that ℓ n (y) = J (F d,n (y)) and then div (κ n ∇v n ) = κ n ℓ n .
Since u ∈ A R d × (0, T ) , h , then (27) implies that for every t ∈ (0, T ), J belongs to L 1 [0,t] , s − d 2 ds . Corollary 3.7 then shows that ℓ n v n is integrable with respect to κ n dy in each B n t . Similarly, because K belongs to L 1 [0,t] , s − d 2 ds , then Lemma 2.2 and Corollary 3.7 show that ℓ n ∇v n ·y is integrable with respect to κ n dy in each B n t . Rephrased, this means that ℓ n is integrable with respect to both κ n v n and κ n ∇v n · y on each B n t . By backward uniqueness of heat equations, (as in [46], for example), u (·,t) is a non-trivial function of x for each t ∈ (0, T ).
Since v n | ∂ B r = u ·, r 2 2d is non-trivial for each r ∈ 0, √ 2dT , then all of the assumptions from Proposition 5.1 hold. Therefore, we may apply Proposition 5.1 to each v n on any ball of radius √ 2dt for t < T . First we compute the frequency function associated to v n on the ball of radius √ 2dt. By Lemma 3.3,

Lemmas 3.3 and 2.2 imply that
while lemma 3.5 implies that Using the expression (23) along with (29) and (30), we see that , s) dx ds. 19 We remark that since u ∈ A R d × (0, T ) , then Lemmas 3.3 and 3.5 guarantee that the integrals in (28), (29), and (30) are all well-defined. Therefore, where we use lim (13), the bound (14), and that for every s ∈ (0,t) along with the Dominated Convergence Theorem.
and we deduce that Define L n (t) = t ϒ n L n (t) for some ϒ n ≥ ∇ log κ n · y L ∞ (B n T ) . Lemma 2.1 shows that where L is as given in Proposition 5.1. An application of the chain rule shows that By Proposition 5.1, it follows that ´S n t κ n v n y · ∇v n ´B n t κ n ℓ n v n ´S n t κ n |v n | 2 2 −´B n t κ n ℓ n y · ∇v ń S n t κ n |v n | 2    .
By Lemmas 3.5 and 2.2 Therefore, substituting this along with the expressions from (28), (29) and (30) into the previous inequality shows that Estimate (14) along with (51) and (52) show that where I, J, and K are defined in (24) and we have introduced To show that L is monotone non-decreasing, it suffices to show that given any t 0 ∈ (0, T ], there exists δ ∈ (0,t 0 ) so that F n converges uniformly to 0 on [t 0 − δ ,t 0 ]. Indeed, since d dt L n (t) ≥ F n (t), then for any t ∈ [t 0 − δ ,t 0 ], it holds that L n (t 0 ) − L n (t) ≥ˆt 0 t F n (s)ds.
By definition and (31), L n (t) = t ϒ L n (t) converges pointwise to L(t) = t ϒ L(t; u, h), from which it follows that Assuming the local uniform convergence of F n to 0 on [t 0 − δ ,t 0 ] ⊃ [t,t 0 ], we see that It remains to justify the local uniform convergence of F n to 0, as described above. Let t 0 ∈ (0, T ] and recall that since u is non-trivial, then backward uniqueness ensures thatˆR is well-defined. 21 We first consider the terms in the denominator of F n , defined as H n (t) above. Observe that for any n ∈ N, we have from (33) that where have used a Taylor expansion to show that if |h(x)| 2 ≤ dnt, then Fix some µ < min t 0 2 , T − t 0 . Since u is continuous and K M × [t 1 − µ,t 1 + µ] is compact, then there exists δ ∈ (0, µ] so that whenever x ∈ K M and |t − t 0 | ≤ δ , it holds that In particular, if t ∈ [t 0 − δ ,t 0 ], then It follows that for any n ≥ N and any t ∈ [t 0 − δ ,t 0 ], we have from (34) that In particular, we have a uniform lower bound on all H n (t) for n ≥ N and all t ∈ [t 0 − δ ,t 0 ]. Since u ∈ A R d × (0, T ) , h , then there exists p > 1 so that (27) holds. Returning to the expression (32), an application of Hölder's inequality shows that . 22 In particular, and an identical argument holds with K in place of J. Assuming that δ ≤ ε from (26) (which holds by possibly redefining δ ), we put (26), (35) and (36) together in (32) to see that whenever n ≥ N, The required version of uniform convergence follows from this bound and completes the proof.

ALT-CAFFARELLI-FRIEDMAN MONOTONICITY FORMULA
In the groundbreaking work of Alt-Caffarelli-Friedman [2], the authors study two-phase elliptic free boundary problems. The monotonicity formula described in Proposition 6.1, which we refer to as ACF, is a crucial tool in their work since it is used to establish Lipschitz continuity of minimizers, and study the regularity of the free boundary.
is monotonically non-decreasing in r.
Different versions of this formula were proved in [9] by Caffarelli, and by Caffarelli and Kenig in [40] to show the regularity of solutions to parabolic equations. Caffarelli, Jerison, and Kenig in [10] further extended these ideas, proving a powerful uniform bound on the monotonicity functional, instead of a monotonicity result. Later on, Matevosyan and Petrosyan [45] proved another such uniform bound for nonhomogeneous elliptic and parabolic operators with variable-coefficients. ACF-type monotonicity formulas have also been used to study almost minimizers of variable-coefficient Bernoulli-type functionals, see for example [18].
In this section, we prove a parabolic version of theorem 6.1, given below in Theorem 6.4. To the best of our knowledge, this result is also new.
As in the previous section, to employ the high-dimensional limiting technique to prove this parabolic result, we first need a version of it for solutions to non-homogeneous variable-coefficient elliptic equations. As similar results for homogeneous variable-coefficient elliptic equations have found numerous applications, this monotonicity result could be interesting in its own right. Once we have the suitable elliptic result in hand, we employ techniques similar to those in the previous section to establish our parabolic ACF result. Corollary 6.2. For some R > 0, let B R ⊂ R N . For each i = 1, 2, we make the following assumptions: Let κ i : B R → R + be bounded, elliptic, and regular in the sense that Assume further that for every r < R, Γ i,r := suppv i ∩ S r := suppv i ∩ ∂ B r has non-zero measure. Finally, assume that N ≥ max {ϒ 1 , ϒ 2 } + 2 and that v 1 v 2 ≡ 0. Then for all r < R, we define φ (r) = φ (r; v 1 , v 2 , κ 1 , κ 2 ) as φ (r) = 1 r 4+ϒ 1 +ϒ 2 ˆB r κ 1 (y)|∇v 1 (y)| 2 |y| 2+ϒ 1 −N dy ˆB r κ 2 (y)|∇v 2 (y)| 2 |y| 2+ϒ 2 −N dy . 23 If we set φ (r) = r µ φ (r), where µ = 4 Λ−λ Λ + ϒ 1 + ϒ 2 , then it holds that Remark 6.3. Notice that the main difference between our φ (r) and the one defined in (37) is the introduction of ϒ 1 and ϒ 2 , which depend on ∇κ 1 and ∇κ 2 , respectively, appearing in the powers on |y|. In fact, if we set κ i = 1, we recover a monotonicity formula very similar to [17,Corollary 2], which generalizes Proposition 6.1 to the non-homogeneous setting. If each ℓ i = 0, the right-hand side of (38) vanishes and we reach a true monotonicity result.
Proof. Observe that for a.e. r, Therefore, for a.e. r, We want to estimate this derivative. Throughout this part of the proof, we suppress subscripts for i = 1, 2 on all functions and exponents. That is, in place of v i , κ i , ℓ i , ϒ i , we write v, κ, ℓ, and ϒ. Since div(κ∇v) ≥ κ ℓ in B R , then ∆v ≥ l, where we define l := − (ℓ − + ∇ log κ · ∇v). Define v m and l m to be the mollifications of v and l, respectively, at scale m −1 . Let A r,ε = B r \ B ε for some ε ∈ (0, r). Integration by parts shows that κ v m ∇v m · y |y| ϒ−N dy + ε 1+ϒ−NˆS ε κ |v m | 2 dσ (y). 24 Moreover, By rearrangement, that A r,ε ⊂ B r and ϒ ≥ 0, it follows that Since ∇v m is bounded and ϒ ≥ 0, then I ε → 0 as ε → 0. In particular, the right-hand side of the previous inequality is bounded independent of ε. Accordingly, we may take ε → 0, eliminate I ε on the right-hand side, and replace A r,ε with B r on the left. Now we integrate with respect to r, r 0 < r < r 0 + δ , divide through by δ , then take m → ∞ to get 25 By taking δ → 0, it follows that for a.e. r 0 > 0 After reintroducing the subscripts, we see that for a.e. r > 0, By rescaling, we may assume that r = 1. Let ∇ θ w denote the gradient of function w on S N−1 , the unit sphere. Let Γ i denote the support of v i on S N−1 for i = 1, 2. By assumption, the measures of Γ 1 and Γ 2 are non-zero. For i = 1, 2, define Observe that for any β i ∈ (0, 1), If we set for i = 1, 2, then by combining (40) with the last two inequalities and using that λ ≤ κ i (y) ≤ Λ, we havê . 26 Substituting these inequalities into (39) with r = 1 gives The relation (41) is satisfied when As a function that acts on subsets of S N−1 , γ was studied in [26] and it was shown that γ (E) ≥ ψ |E| |S N−1 | , where ψ is the decreasing, convex function defined by We use the notation γ i = γ (Γ i ) for i = 1, 2. With s i = |Γ i | |S N−1 | , it follows from convexity that Moreover, since each v i ∈ H 1,2 0 (Γ i ), then
Let Γ i,r denote the support of v i on S r and set s i,r = With ℓ i,r (y) = rℓ i (ry), we have div(κ i,r ∇v i,r ) ≥ κ i,r ℓ i,r . Applying the derivative estimates above to the pair v 1,r , v 2,r then rescaling leads to That is, with µ = 4 Λ−λ Λ + ϒ 1 + ϒ 2 , the conclusion described by (38) follows.
In what follows, we introduce our new parabolic ACF-type result for variable-coefficient operators. We prove this result using only Corollary 6.2 and the tools and ideas that have been developed so far in this paper. As in the previous section, we first discuss the kinds of solutions that we work with.
Let u : R d × (0, T ) → R have moderate h-growth at infinity, as described in Definition 3.8. For every t ∈ (0, T ), assume first that u is sufficiently regular to define the functionals D (t; u, h) and T (t; u, h) from (16) as well as where J(x,t) = J(x,t; u, h) is as defined in (25) and all derivatives are interpreted in the weak sense. Then we say that such a function u belongs to the function class C R d × (0, T ) , h if u has moderate h-growth at infinity (so is consequently continuous), and for every t 0 ∈ (0, T ), there exists ε ∈ (0,t 0 ) so that 28 and This is the class of functions that we consider in our result. As before, this may not be the weakest setting in which our proof holds.
Theorem 6.4. For each i = 1, 2, we make the following assumptions: Let tr where h i , H i , and G i are described by (4), (2), and (5).
Proof. Recall that B n T ⊂ R d×n is given by (15). For each i = 1, 2, let κ i,n be as defined in Lemma 2.1. By definition, λ ≤ κ i,n ≤ Λ for every n ∈ N. As Define each ϒ i,n := ϒ i to be independent of n. Let u 1 , u 2 be as in the statement of the theorem. For each i = 1, 2 and n ∈ N ≥N , define v i,n : B n T ⊂ R d×n → R so that v i,n (y) = u i (F d,n (y)) .
Since each u i has moderate h i -growth at infinity, then by Lemma 3.9, v i,n ∈ C 0 (B n T ) ∩ W 1,2 (B n T , κ i,n dy). Moreover, each v i,n is non-negative and v 1,n v 2,n ≡ 0. Assuming without loss of generality that g(0) = 0, we obtain v i,n (0) = u i (0, 0) = 0.
An application of Lemma 2.2 shows that div(κ i,n ∇v i,n ) where J i = J(u i ) is defined in (25) and does not depend on n. For every n, define ℓ i,n : B n T → R so that ℓ i,n (y) = J i (F d,n (y)) and then div (κ i,n ∇v i,n ) ≥ κ i,n ℓ i,n .
Since each u i ∈ C R d × (0, T ) , h i , then (43) in combination with Corollary 3.7 shows that ℓ − i,n v i,n is integrable with respect to κ n dy in each B n t . In other words, each ℓ − i,n is integrable with respect to κ i,n v i,n |y| 2+ϒ i −dn dy on each B n t . Let Γ i,n,t = supp v i,n ∩ S n t . For each t, the measure of supp u i (·,t) vanishes if and only if the measure of Γ i,n,t vanishes for every n. We assume first that for every t, the measures of supp u 1 (·,t) and supp u 2 (·,t) are non-vanishing. Therefore, for every i, n and t, Γ i,n,t has non-zero measure. Thus, we may apply Corollary 6.2 to each pair v 1,n , v 2,n on any ball B n t for t ∈ (0, T ). 29 Define Φ n (t) = 4 (n|S dn−1 |) 2 φ √ 2dt; v 1,n , v 2,n , κ 1,n , κ 2,n , where φ (r) is given in Corollary 6.
We first consider the terms in the denominator of F n . For brevity, we set H Assumption (43) shows that