The mixed problem in Lipschitz domains with general decompositions of the boundary

TL;DR

This paper investigates the solvability of the mixed boundary value problem for the Laplacian on Lipschitz domains with complex boundary decompositions, establishing existence, uniqueness, and regularity results under geometric and functional conditions.

Contribution

It extends the theory of mixed boundary problems by considering general boundary decompositions and establishing solvability in various function spaces based on geometric criteria.

Findings

Existence and uniqueness of solutions for p in (1, p_0)

Results for boundary data in Hardy spaces at p=1

Solutions with boundary data in weighted Sobolev spaces

Abstract

This paper continues the study of the mixed problem for the Laplacian. We consider a bounded Lipschitz domain $\Omega\subset \reals^n$, $n\geq2$, with boundary that is decomposed as $\partial\Omega=D\cup N$, $D$ and $N$ disjoint. We let $\Lambda$ denote the boundary of $D$ (relative to $\partial\Omega$) and impose conditions on the dimension and shape of $\Lambda$ and the sets $N$ and $D$. Under these geometric criteria, we show that there exists $p_0>1$ depending on the domain $\Omega$ such that for $p$ in the interval $(1,p_0)$, the mixed problem with Neumann data in the space $L^p(N)$ and Dirichlet data in the Sobolev space $W^ {1,p}(D) $ has a unique solution with the non-tangential maximal function of the gradient of the solution in $L^p(\partial\Omega)$. We also obtain results for $p=1$ when the Dirichlet and Neumann data comes from Hardy spaces, and a result when the boundary data comes from weighted Sobolev spaces.