Exceptional Boundary Sets for Solutions of Fully Nonlinear Parabolic PDEs

TL;DR

This paper studies the boundary behavior of solutions to fully nonlinear parabolic PDEs, providing conditions under which supersolutions stay nonnegative despite boundary irregularities.

Contribution

It introduces a sufficient condition based on Hausdorff measure bounds that guarantees supersolutions remain nonnegative even with boundary irregularities.

Findings

A Hausdorff measure condition ensures boundary regularity for supersolutions.

The results extend boundary regularity understanding for fully nonlinear parabolic PDEs.

The condition applies to the exceptional boundary set where boundary data may be irregular.

Abstract

This article investigates the exceptional set of the boundary for the following problem: \begin{equation*} \begin{aligned} -\frac{\partial u}{\partial t} + \mathcal{M}_{\lambda,\Lambda}^+(D^2u) + b(x,t)\cdot Du + c(x,t)u =0 \quad \rm{in} ~ \Omega_{T}, \end{aligned} \end{equation*} We provide a sufficient condition on the exceptional set in terms of the bound of the Hausdorff measure of this boundary portion. This condition ensures that even if the boundary values are not nonnegative on this portion, the supersolution remains nonnegative.