Iñigo U. Erneta - Lipschitz regularity for the Poisson equation involving surface measures

The Poisson equation with measure data has been studied since the beginnings of classical potential theory and is by now well understood. For general measures, the gradient of solutions is only in the weak-L^{1^{\star}} space, which is optimal in light of the fundamental solution. A natural question is to identify conditions on the measure under which the gradient is bounded. Classical potential theory shows that this regularity holds when a suitable Riesz potential of the measure is bounded.
 
In this talk, I will focus on the case of surface measures, where the above criterion becomes critical. Our main result establishes that, for C^1 Dini surfaces with Dini-continuous density, solutions are differentiable up to the surface from both sides. As a consequence, we deduce the Lipschitz regularity of solutions. I will also present counterexamples that illustrate the sharpness of these assumptions.
 
This is joint work with María Soria-Carro.