The mixed problem in L^p for some two-dimensional Lipschitz domains

Details The mixed problem in L^p for some two-dimensional Lipschitz domains The mixed problem in L^p for some two-dimensional Lipschitz domains

2005-05-23

arxiv.org/abs/math/0505461v1

Math. Ann. 342 (2008), 91-124

Mathematics

Analysis of PDEs

Scientific paper

10.1007/s00208-008-0223-6

We consider the mixed problem for the Laplace operator in a class of Lipschitz graph domains in two dimensions with Lipschitz constant at most 1. The boundary of the domain is decomposed into two disjoint sets D and N. We suppose the Dirichlet data, f_D has one derivative in L^p(D) of the boundary and the Neumann data is in L^p(N). We find conditions on the domain and the sets D and N so that there is a p_0>1 so that for p in the interval (1,p_0), we may find a unique solution to the mixed problem and the gradient of the solution lies in L^p.

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