Blow-ups of caloric measure in time varying domains and applications to two-phase problems

Abstract

We develop a method to study the structure of the common part of the boundaries of disjoint and possibly non-complementary time-varying domains in ${\mathbb{R}}^{n+1}$, $n\ge 2$, at the points of mutual absolute continuity of their respective caloric measures. Our set of techniques, which is based on parabolic tangent measures, allows us to tackle the following problems: 1) Let ${\Omega }_1$ and ${\Omega }_2$ be disjoint domains in ${\mathbb{R}}^{n+1}$, $n\ge 2$, which are quasi-regular for the heat equation and regular for the adjoint heat equation, and their complements satisfy a mild non-degeneracy hypothesis on the set $E$ of mutual absolute continuity of the associated caloric measures ${\omega }_i$ with poles at ${\bar{p}}_i=(p_i,t_i)\in {\Omega }_i$, $i=1,2$. Then, we obtain a parabolic analogue of the results of Kenig, Preiss, and Toro, i.e., we show that the parabolic Hausdorff dimension of ${\omega }_1{|}_E$ is $n+1$ and the tangent measures of ${\omega }_1$ at ${\omega }_1$-a.e. point of $E$ are equal to a constant multiple of the parabolic $(n+1)$-Hausdorff measure restricted to hyperplanes containing a line parallel to the time-axis. 2) If, additionally, ${\omega }_1$ and ${\omega }_2$ are doubling, $\log \frac{d{\omega }_2{|}_E}{d{\omega }_1{|}_E}\in VMO({\omega }_1{|}_E)$, and $E$ is relatively open in the support of ${\omega }_1$, then their tangent measures at {\it every} point of $E$ are caloric measures associated with adjoint caloric polynomials. As a corollary we obtain that in complementary $\delta$-Reifenberg flat domains, if $\delta$ is small enough and $\log \frac{d{\omega }_2}{d{\omega }_1}\in VMO({\omega }_1)$, then ${\Omega }_1\cap \{t<t_2\}$ is vanishing Reifenberg flat. This generalizes results of Kenig and Toro for the Laplacian. 3) We establish a parabolic version of a theorem of Tsirelson about triple-points for harmonic measure.

In v.3 we made a few minor changes in the proof of Lemma 4.13 and fixed some typos. Accepted for publication in Journal de Mathématiques Pures et Appliquées