Global Hölder Solvability of parabolic equations on domains with capacity density conditions

TL;DR

This work proves global Hölder solvability for the Cauchy-Dirichlet problem $\mathcal{H}u = \partial_t u - \text{div}(A\nabla u)=f$ in domains $D$ satisfying a capacity-density condition, with zero boundary and initial data. The authors introduce an Ancona-type barrier $s_{\Gamma}$ that encodes boundary geometry and yields a sharp parabolic boundary regularity estimate, enabling global Hölder continuity for weak solutions even when $f$ has wall-like singularities $\sim d_{\partial D}^{-2}$. The core method combines barrier construction with the comparison principle, producing a robust framework for parabolic regularity on non-smooth domains and extending to non-homogeneous boundary data with explicit Hölder bounds. These results broaden the class of domains and forcing terms for which global Hölder solvability holds, with potential impact on the analysis of parabolic PDEs in rough geometries. Key contributions include the CDC-based barrier, quantitative boundary regularity, and Hölder estimates that scale with the inradius and boundary distance $\delta$.

Abstract

We investigate the Cauchy-Dirichlet problem for linear parabolic equations in divergence form. Under mild assumptions on the source term and the domain, we prove the existence of globally Hölder continuous solutions. Notably, our results accommodate data exhibiting singularities nearly as critical as the inverse square of the distance from the boundary.

Theorems & Definitions (32)

• Example 2.1
• Definition 3.1
• Definition 3.2
• Lemma 3.3
• Lemma 3.4
• Lemma 3.5
• Lemma 3.6
• Lemma 3.7
• Lemma 3.8
• Lemma 3.9