Boundary Value Problems on Lipschitz Domains in R or C

The purpose of this note is to bring update results on boundary value problems on Lipschitz domains in R or C. We first discuss the Dirichlet problem, the Neumann problem and the d-Neumann problem in a bounded domain in R. These problems are the prototypes of coercive (or elliptic ) boundary value problems when the boundary of the domain is smooth. When the domain is only Lipschitz, solutions to such problems are not necessarily smooth. The study of these boundary value problems on Lipschitz domains in R over the past thirty years has played important role in singular integrals, harmonic analysis and partial differential equations. In contrast, its counterpart in C is the ∂̄-Neumann problem on pseudoconvex domains, which has not been completely understood. Our goal is to provide an overall view of these problems and review the main results with emphasis on L Sobolev estimates. We first give some basic properties of Lipschitz domains in Chapter 0. These simple and useful facts do not seem to have been systematically treated. In Chapter 1 we review the L theory for boundary value problems in Lipschitz domains in R. For the Dirichlet and Neumann boundary value problems, these results are mainly obtained by Dahlberg [Da], Jerison-Kenig [JK]. The reader should consult the book by Kenig [Ke] for a more detailed proof and treatment of these topics. For the d-Neumann problem, the results are due to Mitrea-Mitrea [MM] and Mitrea-Taylor [MT]. The monograph by Mitrea-Mitrea-Taylor [MMT]) gives a detailed account of the best results in this direction. Our goal here is to introduce the reader to the real d-Neumann problem, a subject of great importance in view of the Hodge theory on manifolds with boundary (see Morrey [Mo] for domains with smooth boundary). In the second part, we survey the results of the ∂̄-Neumann problems on Lipschitz domains in C.