On notions of $p$-parabolic capacity and applications

TL;DR

This paper explores various concepts of capacity related to the parabolic p-Laplace equation, establishing their properties, interrelations, and applications to polar sets and removability of supersolutions.

Contribution

It introduces a consistent variational notion of p-parabolic capacity for all 1<p<∞ and analyzes its properties and connections to other capacities and measures.

Findings

Established basic properties of the variational capacity

Connected the variational capacity to other notions and measures

Applied the capacity to study polar sets and removability of supersolutions

Abstract

We consider different notions of capacity related to the parabolic $p$-Laplace equation. Our focus is on a variational notion, which is consistent in the full range $1<p<\infty$. For such a notion we show some basic properties as well as its connection to other notions of capacity presented in the literature, and to a certain parabolic version of the Hausdorff measure. As applications, we use the introduced variational notion of capacity to study polar sets and removability results for supersolutions.