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

TL;DR

This work extends the modern understanding of parabolic Dirichlet problems to variable-coefficient operators with $L^1$ Carleson oscillation, showing that $L^p$ solvability for $L$ (and $L^*$) forces geometric regularity on the boundary graph $\psi$ of Lip$(1,1/2)$ type. The authors develop a novel corona-decomposition framework adapted to the parabolic setting, including a local square-function estimate for the Green function and an integration-by-parts scheme that handles variable coefficients. By constructing regular Lip$(1,1/2)$ graphs $\psi_{\mathbf{S}}$ that approximate the boundary in corona regions and proving $\mathcal{D}_t\psi_{\mathbf{S}}\in\mathrm{BMO_P}$, they show the graph is parabolic uniformly rectifiable, i.e. boundary regularity is equivalent to$L^p$ solvability. The approach overcomes the lack of translation invariance and uses a detailed analysis of parabolic measure, Green's functions, and dyadic-corpus corona techniques, contributing to the broader understanding of the parabolic $A_\infty$ framework and boundary regularity in time-varying domains. The results have potential implications for stability analyses of parabolic PDEs with rough coefficients and for further study of parabolic singular integrals on irregular graphs.

Abstract

We investigate variable coefficient analogs of a recent work of Bortz, Hofmann, Martell and Nyström [BHMN25]. In particular, we show that if $Ω$ 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 Nyström. 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].

Figures (4)

• Figure 1: A (not-to-scale) depiction of elongated cubes and corkscrews associated to a surface ball $\Delta_R({\mathbf{X}}) = {\mathbf{\Psi}}(Q_R({\mathbf{X}}))$. We always think that time flows from left to right, and $x_0$ is the vertical direction. One should imagine the other $n-1$ spatial directions as being perpendicular to both $x_0$ and $t$, but we will not reflect those in our 2D pictures for simplicity.
• Figure 2: A sketch of how a sawtooth $\Omega_{{\bf S}}$ is comprised of several (fattened) Whitney regions $U_{Q_0}$. If the cubes in ${\bf S}$ get smaller, the sawtooth region $\Omega_{\bf S}$ gets closer to the boundary.
• Figure 3: A (not-to-scale) depiction of a "continuous" sawtooth $\mathcal{S}(Q_0)$ in ${\mathbb{R}}^{n+1}$ (the left figure) associated with the dyadic sawtooth $\Omega_{{\bf S}(Q_0)}$ inside $\Omega$ (the right figure). The map $\psi(r; \cdot)$ takes $\mathcal{S}(Q_0)$ inside $\Omega_{{\bf S}(Q_0)}$, twisting it to adapt to the shape of $\Omega$.
• Figure 4: The picture in the left represents Case I: we show (in red) a possible chain of cubes in $\widetilde{D}_{k_0}(Q)$ connecting the fixed $\widetilde{Q}$ with some other $Q' \in \widetilde{D}_{k_0}(Q)$. The picture in the right represents Case II: to connect $Q'$ to $\widetilde{Q}$, we need to avoid the cubes in $\mathcal{F}_1$ (in blue), which surround $Q({\bf S})$, and may "disconnect" $Q'$ and $\widetilde{Q}$ within $Q$; but there will always be space for a chain within $10Q$.

Theorems & Definitions (84)

• Theorem 1.3
• Theorem 1.4
• Remark 1.5
• Lemma 2.9
• proof
• Remark 2.13
• Remark 2.14
• Definition 2.17: (Semi-)coherent stopping times
• Lemma 2.18: BHHLN-Corona
• Lemma 2.19: DS1