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

Abstract

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, βɛ converges to the Dirac delta δ0. By studying some suitable level sets of uɛ, 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 C1,γ surface around \({\mathcal{H}^{N-1}}\) a.e. point on the free boundary.