Local porosity of the free boundary in a minimum problem

Abstract

Given certain set $ \mathcal{K} $ and functions $ q $ and $ h $, we study geometric properties of the set $ \partial\{x\in\Omega:u(x) > 0\} $ for non-negative minimizers of the functional $ \mathcal{J} (u) = \int_{\Omega }^{} \, \left(\frac{1}{p}| \nabla u| ^p+q(u^+)^\gamma +hu\right)\text{d}x $ over $ \mathcal{K} $, where $ {\Omega \subset} \mathbb{R} ^n(n\geq 2) $ is an open bounded domain, $ p\in(1, +\infty) $ and $ \gamma \in (0, 1] $ are constants, $ u^+ $ is the positive part of $ u $ and $ \partial\{x\in\Omega :u(x) > 0\} $ is the so-called free boundary. Such a minimum problem arises in physics and chemistry for $ \gamma = 1 $ and $ \gamma \in(0, 1) $, respectively. Using the comparison principle of $ p $-Laplacian equations, we establish first the non-degeneracy of non-negative minimizers near the free boundary, then prove the local porosity of the free boundary.