Continuity estimates for porous medium type equations with measure data

Abstract

We consider parabolic equations of porous medium type of the formut−divA(x,t,u,Du)=μinET, in some space time cylinder ET. The most prominent example covered by our assumptions is the classical porous medium equationut−Δum=μinET, with m≥1. We establish a sufficient condition for the continuity of u in terms of a natural Riesz potential of the right-hand side measure μ. As an application we come up with a borderline condition ensuring the continuity of u: more precisely, if μ∈L(N+22,1), then u is continuous in ET.