L('p) regularity of solutions of the mixed boundary problem for Laplace's equation on a Lipschitz graph domain.

摘要

In this dissertation, we study the boundary regularity of solutions of the mixed problem for Laplace's equation in a Lipschitz graph domain W ; i.e., W is the region above the graph of a Lipschitz function. We assume that the boundary of W is decomposed as 6W=N∪D , where N∩D=empty . Given functions f on D and g on N, we wish to find a function u which is harmonic in W , which satisfies u = f on D, and which satisfies 6u6n=g on N, where 6u6n denotes the outer normal derivative on 6W .;In particular, we consider domains which satisfy an additional condition which means, roughly, that the sets N and D meet at an angle strictly less than p . For such a domain, we will show that if the Neumann data g is in LpN and if the Dirichlet data f is in the Sobolev space Lp,1D , for 1 < p < 2, then the mixed boundary problem has a unique solution u for which N1u∈Lp 6W , where N1u is the non-tangential maximal function of the gradient of u. Our process adapts the techniques of Dahlberg and Kenig in their study of the Neumann and Dirichlet problems in Lipschitz domains. We first use the asymptotic expansion of Serrin and Weinberger to prove an L1 regularity result for solutions of the mixed problem with data in atomic Hardy spaces. This is then interpolated with the known theory for p = 2 to produce the desired results. Uniqueness of solutions is proven using limiting arguments, with an appeal to the theory of conjugate harmonic functions.