Cavity problems in discontinuous media

TL;DR

This paper establishes sharp Lipschitz regularity estimates for solutions to cavitation equations in discontinuous media, leading to geometric constraints on the free boundary such as positive density and finite Hausdorff measure.

Contribution

It provides the first sharp Lipschitz regularity result for solutions to cavitation problems in discontinuous elliptic media, with implications for free boundary geometry.

Findings

Solutions are Lipschitz continuous up to the free boundary.

The free boundary has finite Hausdorff measure.

The positivity set has uniform positive density.

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 $0< \varsigma \le 1$.