The weak Dirichlet and Neumann problem for the Laplacian in Lq for bounded and exterior domains. Applications.

The purpose of these lectures is to present a rather elementary and selfcontained approach to the weak first and second boundary value problem for the Laplacian in L q where 1 < q < ∞. These problems are basic for a lot of applications in mathematical physics, like as e.g. Stokes’ problem. From the viewpoint of applications it is necessary to consider as well bounded as exterior domains. Our approach rests on two variational inequalities in Lq and a type of regularity argument. The results presented here are part of a joint work with H. Sohr (Paderborn/FRG) [9].