Regularity results for elliptic equations in Lipschitz domains

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