A singular perturbation free boundary problem for elliptic equations in divergence form.

In this paper we study the free boundary problem arising as a limit as ɛ → 0 of the singular perturbation problem $${\textrm{div}(A(x)\nabla u) = \Gamma(x) \beta_\varepsilon(u)}$$ , where A = A( x) is Holder continuous, β _ɛ converges to the Dirac delta δ₀. By studying some suitable level sets of u _ɛ, uniform geometric properties are obtained and show to hold for the free boundary of the limit function. A detailed analysis of the free boundary condition is also done. At last, using very recent results of Salsa and Ferrari, we prove that if A and Γ are Lipschitz continuous, the free boundary is a C ^1,γ surface around $${\mathcal{H}^{N-1}}$$ a.e. point on the free boundary. [ABSTRACT FROM AUTHOR]