Uniform W-1,W-p estimates for an elliptic operator with Robin boundary condition in a C-1 domain

Abstract

We consider the Robin boundary value problem div(A del u) = div f + F in Omega, a C-1 domain, with (A del u - f) . n + alpha u = g on Gamma, where the matrix A belongs to VMO(R-3), and discover the uniform estimates on parallel to u parallel to (W1,p(Omega)), with 1 < p < infinity, independent of alpha. At the difference with the case p = 2, which is simpler, we call here the weak reverse Holder inequality. This estimates show that the solution of the Robin problem converges strongly to the solution of the Dirichlet (resp. Neumann) problem in corresponding spaces when the parameter alpha tends to infinity (resp. 0).