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

摘要

In this paper we study the free boundary problem arising as a limit as epsilon -> 0 of the singular perturbation problem div(A(x)del u) = Gamma(x)beta(epsilon)(u), where A = A(x) is Holder continuous, beta(epsilon)(u) converges to the Dirac delta delta(0). By studying some suitable level sets of u(epsilon), 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 Gamma are Lipschitz continuous, the free boundary is a C-1,C-gamma surface around H(N-1)a.e. point on the free boundary.