Cavity problems in discontinuous media

Abstract

We study cavitation type equations, div(a(i j) (X)del u) similar to delta(0)(u), for bounded, measurable elliptic media a(i j) (X). De Giorgi-Nash-Moser theory assures that solutions are alpha-Holder 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 derivative{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 - zeta)-Hausdorff measure, for a universal number 0 < zeta <= 1.