Poisson problems for semilinear Brinkman systems on Lipschitz domains in \(\mathbb{R}^n\)
type: Item
instance of: scholarly article
title: Poisson problems for semilinear Brinkman systems on Lipschitz domains in \(\mathbb{R}^n\)
zbMATH Open document ID: 1323.35031
author: Mirela Kohr
author: Massimo Lanza de Cristoforis
author: W. L. Wendland
published in: ZAMP. Zeitschrift für angewandte Mathematik und Physik
publication date: 2015-07-23T00:00:00Z
review text: The paper studies the Dirichlet and the Robin problem for the Brinkman system on a bounded domain \(\Omega \subset\mathbb R^n\) with connected Lipschitz boundary. First it is proved unique solvability of the Dirichlet and the Robin problem for a linear Brinkman system \(\Delta u -Pu-\nabla \pi =f\), \(\nabla \cdot u=g\) in Sobolev-Slobodetskij spaces \(L_s^2(\Omega ,\mathbb R^n) \times L_{s-1}^2(\Omega )\) with \(1/2<s<3/2\). Here \(P\in L^\infty (\Omega ,\mathbb R^n\times\mathbb R^n)\) is a matrix-valued function satisfying \(P(x)\geq 0\) in \(\Omega \). Then the authors prove the existence of a solution of the Dirichlet problem and the Robin problem for a semilinear Brinkman system \(\Delta u-P(x,u(x),\pi (x))u-\nabla \pi =f\) in \(\Omega \), \(\nabla \cdot u=g\) in \(\Omega \), where \(P\in L^\infty (\Omega \times\mathbb R^n\times\mathbb R,\mathbb R^n\times\mathbb R^n)\) satisfies the Carathéodory condition and \(P\geq 0\).
reviewed by: Dagmar Medková
Mathematics Subject Classification ID: 35J57
Mathematics Subject Classification ID: 76D99
Mathematics Subject Classification ID: 76M99
zbMATH DE Number: 6463627
zbMATH Keywords: semilinear Brinkman system
zbMATH Keywords: Lipschitz domain
zbMATH Keywords: Poisson problem
zbMATH Keywords: layer potential operators
zbMATH Keywords: Sobolev spaces
zbMATH Keywords: fixed point theorem
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: Robin problem
MaRDI profile type: Publication
full work available at URL: s00033-014-0439-0
OpenAlex ID: W2002920589
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DOI: 10.1007/S00033-014-0439-0
description: scientific article