Discontinuous implicit elliptic boundary value problems.
type: Item
instance of: scholarly article
title: Discontinuous implicit elliptic boundary value problems.
zbMATH Open document ID: 1064.35509
author: S. Carl
author: S. Heikkilä
published in: Differential and Integral Equations
publication date: 2003-01-26T00:00:00Z
review text: An implicity given elliptic differential equation with homogeneous Dirichlet boundary conditions on a bounded domain \(\Omega \subset \mathbb R^N\) with a smooth boundary is considered. The problem in question is described by the equation \(f(x,u,Lu)=0\), where \(f\) can be discontinuous in all its arguments, \(L\) is a semilinear elliptic operator of the form \(Lu = -\Delta u + a(x,u)\) and \(a\) is nondecreasing in the last argument. For \(u,v \in W^{2,2} (\Omega )\) the authors first prove the following comparison principle: if \(Lu \leq Lv\) in \(\Omega \) and \(u\leq v\) on \(\partial \Omega \), then \(u\leq v\) in \(\Omega \). The main theorem establishes the existence of the extremal solutions. A one-dimensional illustrative example is presented.
reviewed by: Michal Křížek
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 47J05
Mathematics Subject Classification ID: 47N20
zbMATH DE Number: 1858727
zbMATH Keywords: nonlinear elliptic operators
zbMATH Keywords: extremal solutions
zbMATH Keywords: partial ordering
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1858727