Regularity Results for Elliptic Equations in Lipschitz Domains

We develop a simple variational argument based on the usual Nirenberg difference quotient technique to deal with the regularity of the solutions of Dirichlet and Neumann problems for some linear and quasilinear elliptic equation in Lipschitz domains. We obtain optimal regularity results in the natural family of Sobolev spaces associated with the variational structure of the equations. In the linear case, we obtain in a completely different way some of the results of D. Jerison and C. E. Kenig about the Laplace equation