Global H\"{o}lder Solvability of parabolic equations on domains with capacity density conditions

TL;DR

This paper proves the existence of globally Hölder continuous solutions for linear parabolic equations in divergence form on domains with capacity density conditions, even with nearly critical singular data.

Contribution

It establishes global Hölder solvability under mild assumptions, extending previous results to include nearly critical boundary singularities.

Findings

Existence of globally Hölder solutions under capacity density conditions

Solutions accommodate data with near-critical boundary singularities

Results apply to a broad class of linear parabolic equations

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 H\"{o}lder continuous solutions. Notably, our results accommodate data exhibiting singularities nearly as critical as the inverse square of the distance from the boundary.