JournalsrmiVol. 22, No. 1pp. 181–204

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

  • Loredana Lanzani

    University of Arkansas, Fayetteville, USA
  • Osvaldo Méndez

    University of Texas at El Paso, USA
The Poisson’s problem for the Laplacian with Robin boundary condition in non-smooth domains cover
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Abstract

Given a bounded Lipschitz domain ΩRn\Omega\subset {\mathbb R}^n, n3n\geq 3, we prove~that the Poisson's problem for the Laplacian with right-hand side in Ltp(Ω)L^p_{-t}(\Omega), Robin-type boundary datum in the Besov space Bp11/pt,p(Ω)B^{1-1/p-t,p}_{p}(\partial \Omega) and non-negative, non-everywhere vanishing Robin coefficient bLn1(Ω)b\in L^{n-1}(\partial \Omega), is uniquely solvable in the class L2tp(Ω)L^p_{2-t}(\Omega) for (t,1p)Vϵ(t,\frac{1}{p})\in {\mathcal V}_{\epsilon}, where Vϵ{\mathcal V}_{\epsilon} (ϵ0\epsilon\geq 0) is an open (Ω\Omega,bb)-dependent plane region and V0{\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.

Cite this article

Loredana Lanzani, Osvaldo Méndez, The Poisson’s problem for the Laplacian with Robin boundary condition in non-smooth domains. Rev. Mat. Iberoam. 22 (2006), no. 1, pp. 181–204

DOI 10.4171/RMI/453