The Mixed Boundary Problem in L p and Hardy spaces for Laplace ’ s Equation on a Lipschitz Domain

  title={The Mixed Boundary Problem in L p and Hardy spaces for Laplace ’ s Equation on a Lipschitz Domain},
  author={Jeffery D. Sykes and Russell M. Brown},
We study the boundary regularity of solutions of the mixed problem for Laplace’s equation in a Lipschitz graph domain Ω whose boundary is decomposed as ∂Ω = N∪D, where N∩D = ∅. For a subclass of these domains, we show that if the Neumann data g is in Lp(N) and if the Dirichlet data f is in the Sobolev space Lp,1(D), for 1 < p < 2, then the mixed boundary problem has a unique solution u for which N(∇u) ∈ Lp(∂Ω), where N(∇u) is the non-tangential maximal function of the gradient of u. 

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