Um problema de fronteira livre para equações elípticas na forma divergente

Abstract

Based on the article Cavity Problems in Discontinuous Media, we investigate regularity properties along the free boundary for minimizers of the functional associated with the cavity problem. Our main goal is to show that such functions are Lipschitz continuous on this set. The approach adopted consists in establishing this property for uniform limits of sequences of minimizers of regularized functionals, so that the result for arbitrary minimizers follows from an analogous argument. To this end, we obtain a regularity estimate that also implies geometric properties of the free boundary of the limit functions.