The Poisson problem with mixed boundary conditions in Sobolev and Besov spaces in non-smooth domains

Abstract

We introduce certain Sobolev-Besov spaces which are particularly well adapted for measuring the smoothness of data and solutions of mixed boundary value problems in Lipschitz domains. In particular, these are used to obtain sharp well-posedness results for the Poisson problem for the Laplacian with mixed boundary conditions on bounded Lipschitz domains which satisfy a suitable geometric condition introduced by R. Brown in (1994). In this context, we obtain results which generalize those by D. Jerison and C. Kenig (1995) as well as E. Fabes, O. Mendez and M. Mitrea (1998). Applications to Hodge theory and the regularity of Green operators are also presented.