Stratification of free boundary points for a two-phase variational problem

Abstract

In this paper we prove the local Lipschitz regularity of the minimizers of the two-phase Bernoulli type free boundary problem arising from the minimization of the functionalJ(u):=∫Ω|∇u|p+λ+pχ{u>0}+λ−pχ{u≤0},1<p<∞. Here Ω⊂RN is a bounded smooth domain and λ± are positive constants such that λ+p−λ−p>0. Furthermore, we show that for p>1 the free boundary has locally finite perimeter and the set of non-smooth points of the free boundary is of zero (N−1)-dimensional Hausdorff measure. For this, our approach is new even for the classical case p=2.