Revista Matemática Iberoamericana

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

Loredana Lanzani and Osvaldo Méndez

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Abstract

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.

Article information

Source
Rev. Mat. Iberoamericana Volume 22, Number 1 (2006), 181-204.

Dates
First available in Project Euclid: 24 May 2006

Permanent link to this document
https://projecteuclid.org/euclid.rmi/1148492180

Mathematical Reviews number (MathSciNet)
MR2268117

Zentralblatt MATH identifier
1160.35389

Subjects
Primary: 45E99: None of the above, but in this section 47G10: Integral operators [See also 45P05] 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems

Keywords
non-smooth domains Besov spaces Triebel-Lizorkin spaces boundary layer potentials regularity of PDE's Robin condition Lamé system Poisson's problem

Citation

Lanzani , Loredana; Méndez , Osvaldo. The Poisson's problem for the Laplacian with Robin boundary condition in non-smooth domains. Rev. Mat. Iberoamericana 22 (2006), no. 1, 181--204. https://projecteuclid.org/euclid.rmi/1148492180.


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