On the Lipschitz regularity of solutions of a minimum problem with free boundary

Abstract

In this article under assumption of “small” density for negativity set, we prove local Lipschitz regularity for the one phase minimization problem with free boundary for the functional

where ${\lambda }_1,{\lambda }_2$ are positive constants so that $\Lambda ={\lambda }_1^p-{\lambda }_2^p<0$, ${\chi }_D$ is the characteristic function of set $D$, $\Omega \subset {\mathbb{R}}^n$ is (smooth) domain and minimum is taken over a suitable subspace of $W^{1,p}(\Omega )$.