Uniqueness and the maximum principle for quasilinear elliptic equations with quadratic growth conditions
type: Item
instance of: scholarly article
title: Uniqueness and the maximum principle for quasilinear elliptic equations with quadratic growth conditions
zbMATH Open document ID: 0859.35031
DOI: 10.1007/BF00375351
author: François Murat
author: Guy Barles
published in: Archive for Rational Mechanics and Analysis
publication date: 1997-04-09T00:00:00Z
review text: This paper is devoted to the study of uniqueness properties for quasilinear uniformly elliptic equations with quadratic growth conditions: \[ -\text{div}(a(x,v,\nabla v))+ b(x,v,\nabla v)=f(x)\quad\text{in }\Omega, \] where \(f\in H^{-1}(\Omega)\) and \(\Omega\) is a bounded domain in \(\mathbb{R}^N\). In order to prove the uniqueness of the solution of the boundary value problems the authors establish the comparison principle. Under some structure condition of the equation involving the first Dirichlet eigenvalue of \(-\Delta\), the authors prove the comparison principle for the solutions \(v\in H^1(\Omega)\cap L^\infty(\Omega)\) satisfying \(a(x,v,\nabla v)\in (L^2(\Omega))^N\) and \(b(x,v,\nabla v)\in L^1(\Omega)\). Therefore, if the natural growth conditions on \(a\) and \(b\), namely, \[ |a(x,v,\xi)|\leq C(|v|)(1+|\xi|)\quad\text{and}\quad |b(x,v,\xi)|\leq C(|v|)(1+|\xi|^2) \] are assumed, the comparison principle holds for the solutions \(v\in H^1(\Omega)\cap L^\infty(\Omega)\). As its applications, there are two typical comparison results: One is the comparison principle for the solutions \(u\in H^1(\Omega)\cap L^\infty(\Omega)\) of the equation \[ -\Delta u+H(x,u,\nabla u)=0\quad\text{in}\quad\Omega \] with \({\partial H\over \partial u}(x,u,p)\geq \alpha_0\geq 0\) and \(|{\partial H\over \partial p}(x,u,p)|\leq C(1+|p|)\), and the other is the comparison principle for the solutions \(u\in H^1(\Omega)\) with \(g(u)|\nabla u|^2\in L^1(\Omega)\) of the equation \[ -\Delta u+g(u)|\nabla u|^2=0\quad\text{in}\quad\Omega \] with \(f\in H^{-1}(\Omega)\), \(g'(\cdot)>0\), and \(g(0)=0\). In both cases, the method of proof consists in making a certain change of function \(u=\varphi(v)\) and in proving that the transformed equation satisfies the structure condition of the main comparison principle. Existence of such solutions has been obtained in \textit{L. Boccardo}, the second author and \textit{J. P. Puel} [Port. Math. 41, 507-534 (1982; Zbl 0544.35040)] and in \te…
reviewed by: Shigeru Sakaguchi
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35B50
Mathematics Subject Classification ID: 35B05
zbMATH DE Number: 880580
zbMATH Keywords: uniqueness
zbMATH Keywords: natural growth conditions
MaRDI profile type: Publication
cites work: On a nonlinear partial differential equation having natural growth terms and unbounded solution
cites work: Q3037899
cites work: Elliptic Partial Differential Equations of Second Order
cites work: Maximum principles for linear, non-uniformly elliptic operators with measurable coefficients
full work available at URL: https://doi.org/10.1007/bf00375351
OpenAlex ID: W2042188069
description: scientific article; zbMATH DE number 880580