On the structure of diffuse measures for parabolic capacities

TL;DR

This paper characterizes diffuse measures related to parabolic p-capacity, showing that measures with a specific decomposition are exactly those that do not charge sets of zero capacity, thus completing a known characterization.

Contribution

It proves the converse of a previous result, establishing that measures with a certain decomposition are precisely the diffuse measures for parabolic p-capacity.

Findings

Measures with the decomposition form are exactly the diffuse measures.

The characterization completes the understanding of measures in relation to parabolic p-capacity.

Abstract

Let $Q=(0,T)\times\Omega$, where $\Omega$ is a bounded open subset of $\mathbb{R}^d$. We consider the parabolic $p$-capacity on $Q$ naturally associated with the usual $p$-Laplacian. Droniou, Porretta and Prignet have shown that if a bounded Radon measure $\mu$ on $Q$ is diffuse, i.e. charges no set of zero $p$-capacity, $p>1$, then it is of the form $\mu=f+\mbox{div}(G)+g_t$ for some $f\in L^1(Q)$, $G\in (L^{p'}(Q))^d$ and $g\in L^p(0,T;W^{1,p}_0(\Omega)\cap L^2(\Omega))$. We show the converse of this result: if $p>1$, then each bounded Radon measure $\mu$ on $Q$ admitting such a decomposition is diffuse.