Revista Matemática Iberoamericana


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Volume 22, Issue 1, 2006, pp. 181–204
DOI: 10.4171/RMI/453

Published online: 2006-04-30

The Poisson’s problem for the Laplacian with Robin boundary condition in non-smooth domains

Loredana Lanzani[1] and Osvaldo Méndez[2]

(1) University of Arkansas, Fayetteville, USA
(2) University of Texas at El Paso, USA

Given a bounded Lipschitz domain $\Omega\subset {\mathbb R}^n$, $n\geq 3$, we prove~that the Poisson's problem for the Laplacian with right-hand side in $L^p_{-t}(\Omega)$, Robin-type boundary datum in the Besov space $B^{1-1/p-t,p}_{p}(\partial \Omega)$ and non-negative, non-everywhere vanishing Robin coefficient $b\in L^{n-1}(\partial \Omega)$, is uniquely solvable in the class $L^p_{2-t}(\Omega)$ for $(t,\frac{1}{p})\in {\mathcal V}_{\epsilon}$, where ${\mathcal V}_{\epsilon}$ ($\epsilon\geq 0$) is an open ($\Omega$,$b$)-dependent plane region and ${\mathcal V}_{0}$ is to be interpreted ad the common (optimal) solvability region for all Lipschitz domains. We prove a similar regularity result for the Poisson's problem for the 3-dimensional Lamé System with traction-type Robin boundary condition. All solutions are expressed as boundary layer potentials.

Keywords: Non-smooth domains, Besov spaces, Triebel-Lizorkin spaces, boundary layer potentials, regularity of PDE’s, Robin condition, Lamé system, Poisson’s problem

Lanzani Loredana, Méndez Osvaldo: The Poisson’s problem for the Laplacian with Robin boundary condition in non-smooth domains. Rev. Mat. Iberoam. 22 (2006), 181-204. doi: 10.4171/RMI/453