On the structure of diffuse measures for parabolic capacities

Abstract

Let Q=(0,T)×Ω, where Ω is a bounded open subset of Rd. 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 μ on Q is diffuse, i.e. charges no set of zero p-capacity, p>1, then it is of the form μ=f+div(G)+gt for some f∈L1(Q), G∈(Lp′(Q))d and g∈Lp(0,T;W01,p(Ω)∩L2(Ω)). We show the converse of this result: if p>1, then each bounded Radon measure μ on Q admitting such a decomposition is diffuse.