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 R n+1 , n ≥ 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 Ω1 and Ω2 be disjoint domains in R n+1 , n ≥ 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 ⊂ ∂Ω1 ∩ ∂Ω2 of mutual absolute continuity of the associated caloric measures ωi with poles at pi = (pi, ti) ∈ Ωi, 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 ω1|E is n + 1 and the tangent measures of ω1 at ω1-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, ω1 and ω2 are doubling, log, and E is relatively open in the support of ω1, then their tangent measures at every point of E are caloric measures associated with adjoint caloric polynomials. As a corollary we obtain that if Ω1 is a δ-Reifenberg flat domain for δ small enough and Ω2 = R n+1 \ Ω1, and log dω 2 dω 1 ∈ VMO(ω1), then Ω1 ∩ {t < t2} is vanishing Reifenberg flat. This generalizes results of Kenig and Toro for the Laplacian.(3) Finally, we establish a parabolic version of a theorem of Tsirelson about triple-points for harmonic measure. Assuming that Ωi, 1 ≤ i ≤ 3, are quasi-regular domains for both the heat and the adjoint heat equations, the set of points on ∩ 3 i=1 ∂Ωi, where the three caloric measures are mutually absolutely continuous has null caloric measure. In the course of proving our main theorems, we obtain new results on heat potential theory, parabolic geometric measure theory, and nodal sets of caloric functions, that may be of independent interest.