Semilinear Poisson problems in Sobolev-Besov spaces on Lipschitz domains.
type: Item
instance of: scholarly article
title: Semilinear Poisson problems in Sobolev-Besov spaces on Lipschitz domains.
zbMATH Open document ID: 1058.35073
DOI: 10.5565/PUBLMAT_46202_03
author: Martin Dindos
author: Marius Mitrea
published in: Publicacions Matemàtiques
publication date: 2002-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/41457
review text: Let \(M\) be a smooth connected compact boundaryless manifold of real dimension equipped with a Riemann metric tensor. Let us consider the Laplace-Beltrami operator on \(M\): \[ \Delta u \equiv \text{ div(grad } u) , \] and the semilinear elliptic PDE: \[ \Delta u - N(x,u)= F(x), \] associated either to Dirichlet or to Neumann boundary conditions, in a Lipschitz domain \(\Omega \subset M.\) In the case of Dirichlet boundary conditions the authors consider \(u \in L^p_{s+1/p}(\Omega),\) \(F \in L^p_{s+1/p-2}(\Omega),\) \(0<s<1,\) \(1<p<\infty\) and either \(N(x,u)\) has sublinear growth in \(u,\) or, if \(N \in C^1,\) the PDE become \[ \Delta u- a(x,u)\cdot u = f \] where the term \(a\) has an admissible polynomial growth. The case of Neumann boundary condition is similar. The main goal is to study the maximal range of indices \(s, p\) for which the semilinear elliptic PDE is solvable in the context of Lipschitz domains on the Sobolev-Besov scales. The obtained results are sharp in the sense that they are reduced to an optimal linear theory in the absence of nonlinearities.
reviewed by: Maria Alessandra Ragusa
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 31B10
zbMATH DE Number: 1876875
zbMATH Keywords: nonlinear equations
zbMATH Keywords: Lipschitz domains
zbMATH Keywords: elliptic PDE's
zbMATH Keywords: boundary value problems
zbMATH Keywords: Sobolev-Besov spaces.
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1876875