Title: A Variable Coefficient Free Boundary Problem for $L^p$-solvability of Parabolic Dirichlet Problems in Graph Domains Show Chinese title

Title: 一个关于图域中抛物Dirichlet问题的$L^p$可解性的变系数自由边界问题

Abstract: We investigate variable coefficient analogs of a recent work of Bortz, Hofmann, Martell and Nystr\"om [BHMN25]. In particular, we show that if $\Omega$ is the region above the graph of a Lip(1,1/2) (parabolic Lipschitz) function and $L$ is a parabolic operator in divergence form \[L = \partial_t - \text{div} A \nabla\] with $A$ satisfying an $L^1$ Carleson condition on its spatial and time derivatives, then the $L^p$-solvability of the Dirichlet problem for $L$ and $L^*$ implies that the graph function has a half-order time derivative in BMO. Equivalently, the graph is parabolic uniformly rectifiable. In the case of $A$ symmetric, we only require that the Dirichlet problem for $L$ is solvable, which requires us to adapt a clever integration by parts argument by Lewis and Nystr\"om. A feature of the present work is that we must overcome the lack of translation invariance in our equation, which is a fundamental tool in similar works, including [BHMN25]. Show Chinese abstract

Abstract: 我们研究Bortz、Hofmann、Martell和Nyström [BHMN25] 最近工作的一个变量系数类比。 特别是，我们证明如果$\Omega$是一个 Lip(1,1/2)（抛物 Lipschitz）函数图像上方的区域，而$L$是一种散度形式的抛物算子\[L = \partial_t - \text{div} A \nabla\]，其中$A$满足其空间和时间导数的$L^1$Carleson 条件，那么对于$L$和$L^*$的 Dirichlet 问题的$L^p$可解性意味着图像函数在 BMO 中具有半阶时间导数。 等价地，该图是抛物线一致可求长的。 在$A$对称的情况下，我们只需要保证$L$的狄利克雷问题可解，这就要求我们采用 Lewis 和 Nyström 的巧妙分部积分论证。 本工作的一个特点是，我们必须克服方程中缺乏平移不变性，这是类似工作中的一项基本工具，包括 [BHMN25]。