论文标题
及时对热量量的爆炸量变化的域和应用到两相问题
Blow-ups of caloric measure in time varying domains and applications to two-phase problems
论文作者
论文摘要
我们开发了一种方法来研究$ \ mathbb {r}^{n+1} $，$ n \ geq 2 $的脱节边界的结构，并可能在$ \ mathbb {r}^{r}^{r}^{r}^{r}^{r}^{r}^{n+geq 2 $中，在其各自的热量度量的相互绝对连续性下。我们的一组技术基于抛物线切线措施，使我们能够解决以下问题：1）让$ω_1$和$ω_2$在$ \ mathbb {r}^{r}^{n+1} $，$ n \ egeq 2 $中均不符合热量方程式和常规方程，并在$ \ mathbb {r}^{r}^{n+1} $中脱节域，并且相关热量测量的集合绝对连续性的集合的假设$ω_i$与POL $ \ bar {p} _i =（p_i，t_i）\inΩ_i$，$ i = 1,2 $。然后，我们获得了Kenig，Preiss和Toro的结果的抛物线类似物，即，我们表明抛物线寄生虫的Hausdorff尺寸为$ω_1| _e $是$ n+1 $，切线度量为$ω_1$ $ω_1$ in $ω_1$ -a.e.e。 $ e $的点等于抛物线$（n+1）$ - Hausdorff度量的常数倍数限于包含与时间轴平行的线路的超平面。 2）如果此外，如果$ω_1$和$ω_2$两倍，$ \ log \ frac {dmoc__2 | _e} {dmoc__1 | _e} \在vmo（ω_1| _e）$ in vmo（ω_1| _e）$，以及$ e $的$ e $，则$ e $的$ eys $ eye $ e当$之间{与伴随热量多项式相关的度量。作为推论，我们可以在补充$δ$ -REIFENBERG平坦域中获得，如果$δ$足够小，并且$ \ log \ frac {dΩ_2} {dΩ_1} \ in VMO（ω_1）$，则是$ω_1_1\ cap \ cap \ cap \ cap \ t <t <t_2 \} $这概括了Kenig和Toro的Laplacian的结果。 3）我们建立了Tsirelson定理的抛物线版本，围绕三分点进行谐波测量。
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 $Ω_1$ and $Ω_2$ 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 $ω_i$ with poles at $\bar{p}_i=(p_i,t_i)\inΩ_i$, $i=1,2$. Then, we obtain a parabolic analogue of the results of Kenig, Preiss, and Toro, i.e., we show that the parabolic Hausdorff dimension of $ω_1|_E$ is $n+1$ and the tangent measures of $ω_1$ at $ω_1$-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, $ω_1$ and $ω_2$ are doubling, $\log \frac{dω_2|_E}{dω_1|_E} \in VMO(ω_1|_E)$, and $E$ is relatively open in the support of $ω_1$, 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 $δ$-Reifenberg flat domains, if $δ$ is small enough and $\log \frac{dω_2}{dω_1} \in VMO(ω_1)$, then $Ω_1 \cap \{t<t_2\}$ 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.
