Abstract
We consider the mixed boundary value problem, or Zaremba's problem, for the Laplacian in a bounded Lipschitz domain Ω in Rn, n ≥ 2. We decompose the boundary ∂Ω= D ∪ N with D and N disjoint. The boundary between D and N is assumed to be a Lipschitz surface in ∂Ω. We find an exponent q0 > 1 so that for p between 1 and q0 we may solve the mixed problem for Lp. Thus, if we specify Dirichlet data on D in the Sobolev space W1,p(D) and Neumann data on N in Lp (N), the mixed problem with data fD and fN has a unique solution and the non-tangential maximal function of the gradient lies in Lp(∂Ω). We also obtain results for p = 1 when the data comes from Hardy spaces.
Original language | English |
---|---|
Pages (from-to) | 1333-1364 |
Number of pages | 32 |
Journal | Potential Analysis |
Volume | 38 |
Issue number | 4 |
DOIs | |
State | Published - May 2013 |
Bibliographical note
Funding Information:Research supported, in part, by the National Science Foundation.
Funding Information:
This work was partially supported by a grant from the Simons Foundation (#195075 to Russell Brown).
Keywords
- Laplacian
- Mixed boundary value problem
- Non-smooth domain
ASJC Scopus subject areas
- Analysis