On the structure of diffuse measures for parabolic capacities

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 $μ=f+\mbox{div}(G)+g_t$ for some $f\in L^1(Q)$, $G\in (L^{p^{\prime }}(Q){)}^d$ and $g\in L^p(0,T;W_0^{1,p}(\Omega )\cap L^2(\Omega ))$. We show the converse of this result: if $p>1$, then each bounded Radon measure $\mu$ on $Q$ admitting such a decomposition is diffuse.