Carleson-type removability for $p$-parabolic equations

TL;DR

The paper provides a sharp, intrinsic characterization of removable sets for Hölder continuous solutions to degenerate parabolic equations with p-growth. It develops a new obstacle-problem–driven approach to establish sufficiency, using decay estimates for the Riesz measure instead of oscillation arguments, and proves necessity by constructing measure-data solutions via a parabolic Frostman framework. The results extend removability theory to a broad parabolic setting, including general operators beyond the classical parabolic p-Laplace equation, and offer a parabolic counterpart to elliptic removability criteria. Overall, the work advances understanding of how geometric size, measured by intrinsic parabolic Hausdorff measures, governs removability and regularity for nonlinear parabolic PDEs.

Abstract

We characterize removable sets for Hölder continuous solutions to degenerate parabolic equations of $p$-growth. A sufficient and necessary condition for a set to be removable is given in terms of an intrinsic parabolic Hausdorff measure, which depends on the considered Hölder exponent. We present a new method to prove the sufficient condition, which relies only on fundamental properties of the obstacle problem and supersolutions, and applies to a general class of operators. For the necessity of the condition, we establish the Hölder continuity of solutions with measure data, provided the measure satisfies a suitable decay property. The techniques developed in this article provide a new point of view even in the case $p=2$.

Theorems & Definitions (32)

• Theorem 1
• Theorem 2
• Definition 2.1
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Lemma 2.6
• Lemma 2.7