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 $\mathbb{R}^{n+1}$, $n \geq 2$, 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 $\mathbb{R}^{n+1}$, $n \geq 2$, 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.