Cavity problems in discontinuous media

Abstract

We study cavitation type equations, $\text{div}(a_{ij}(X) \nabla u) \sim \delta_0(u)$, for bounded, measurable elliptic media $a_{ij}(X)$. De Giorgi-Nash-Moser theory assures that solutions are $\alpha$-H\"older continuous within its set of positivity, $\{u>0\}$, for some exponent $\alpha$ strictly less than one. Notwithstanding, the key, main result proven in this paper provides a sharp Lipschitz regularity estimate for such solutions along their free boundaries, $\partial \{u>0 \}$. Such a sharp estimate implies geometric-measure constrains for the free boundary. In particular, we show that the non-coincidence $\{u>0\}$ set has uniform positive density and that the free boundary has finite $(n- \varsigma )$-Hausdorff measure, for a universal number $0< \varsigma \le 1$.