Cavity problems in discontinuous media

We study cavitation type equations, $\text{div}(a_{ij}(X) \nabla u) \sim δ_0(u)$, for bounded, measurable elliptic media $a_{ij}(X)$. De Giorgi-Nash-Moser theory assures that solutions are $α$-Hölder continuous within its set of positivity, $\{u>0\}$, for some exponent $α$ 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 $(n- ς)$-Hausdorff measure, for a universal number $0< ς\le 1$.

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