\(W^{1,1}(\omega)\) solutions of nonlinear problems with nonhomogeneous Neumann boundary conditions
type: Item
instance of: scholarly article
title: \(W^{1,1}(\omega)\) solutions of nonlinear problems with nonhomogeneous Neumann boundary conditions
zbMATH Open document ID: 1336.35163
author: Lourdes Moreno-Mérida
author: Lucio Boccardo
published in: Milan Journal of Mathematics
publication date: 2016-01-06T00:00:00Z
review text: The paper starts with an introduction summarizing some results on nonlinear problems with Dirichlet boundary conditions. The new result here is, however, some existence result for a Neumann type problem, namely \(-\operatorname{div}(a(x,u,\nabla u))+|u|^{s-1}u=0\) on some domain \(\Omega\subset{\mathbb R}^N\) with the boundary condition \(a(x,u,\nabla u)\cdot\eta=\psi\) on \(\partial\Omega\). Here \(\eta\) is the unit outward normal on \(\partial\Omega\), \(\psi\in L^m(\partial\Omega)\) for some \(m>1\), and \(a:\Omega\times{\mathbb R}\times{\mathbb R}^N\to{\mathbb R}^N\) is a Caratheodory function with the usual \(p\)-growth conditions. The main result is the existence of a \(W^{1,1}\) distributional solution to the problem in the borderline case \(m={N-1\over N(p-1)}\), with \(1<p<2-{1\over N}\) and \(0<s<{1+N(p-1)\over N-1}\). The proof uses approximation by problems with \(L^\infty\) data.
reviewed by: Andreas Gastel
Mathematics Subject Classification ID: 35J66
zbMATH DE Number: 6526831
zbMATH Keywords: nonlinear problems
zbMATH Keywords: Neumann boundary condition
zbMATH Keywords: distributional solution
MaRDI profile type: Publication
full work available at URL: s00032-015-0235-0
OpenAlex ID: W2002620049
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DOI: 10.1007/S00032-015-0235-0
description: scientific article; zbMATH DE number 6526831