Cavity problems in discontinuous media

We study cavitation type equations, $$\text{div}$(a_{ij}(X) \nabla u) \sim \delta_0(u)$,forbounded,measurableellipticmedia$a_{ij}(X)$. 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, $\mathrm{\partial }\{u>0\}$. 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 $0<\varsigma \le 1$.