Pseudo-monotonicity and the Leray-Lions condition
type: Item
instance of: scholarly article
title: Pseudo-monotonicity and the Leray-Lions condition
zbMATH Open document ID: 0786.35055
published in: Differential and Integral Equations
publication date: 1993-06-29T00:00:00Z
review text: Let \(\Omega\) be a bounded open subset of \(\mathbb{R}^ N\), let \(1<p<\infty\) and let \(T\) be the mapping from \(W_ 0^{1,p}(\Omega)\) to \(W^{- 1,p'}(\Omega)\) which is generated by the differential operator \[ -\sum^ N_{i=1}{\partial \over \partial x_ i}a_ i(x,u,\nabla u)+a_ 0(x,u,\nabla u), \tag{1.1} \] where the functions \(a_ i(x,\eta,\zeta)\) and \(a_ 0 (x,\eta, \zeta)\) satisfy suitable regularity and growth assumptions. The standard condition which guarantees that \(T\) is pseudo- monotone is the so-called Leray-Lions condition: \[ \sum^ N_{i=1} (a_ i(x,\eta,\zeta)-a_ i(x,\eta,\overline\zeta))\;(\zeta_ i- \overline\zeta_ i)>0 \tag{1.2} \] for almost every \(x \in \Omega\), all \(\eta \in \mathbb{R}\) and all \(\zeta\), \(\overline\zeta \in \mathbb{R}^ N\) with \(\zeta \neq \overline\zeta\). In this paper we are interested in the necessity of condition (1.2) as well as in the sufficiency of a related weaker condition. It is known that the pseudo-monotonicity of \(T\) implies that \[ \sum^ N_{i=1}(a_ i(x,\eta,\zeta)-a_ i(x,\eta, \overline\zeta))\;(\zeta_ i-\overline\zeta_ i) \geq 0 \tag{1.3} \] for almost every \(x \in \Omega\), all \(\eta \in \mathbb{R}\) and all \(\zeta\), \(\overline\zeta \in \mathbb{R}^ N\). In Section 2 we study the necessity of the strict condition (1.2). We restrict ourselves there to operators where the top order terms \(a_ i\) only depend on \(\nabla u\). We show that if, for any lower order term \(a_ 0\), the corresponding mapping \(T\) is pseudo-monotone, then the top order terms \(a_ i\) satisfy (1.2). Condition (1.2) thus appears, for this class of operators, as both necessary and sufficient for pseudo-monotonicity under any lower order perturbation. In Section 3 we study the sufficiency of the nonstrict condition (1.3). We show in particular that (1.3) implies the pseudo- monotonicity of \(T\) when \(a_ 0\) does not depend on the gradient (or more generally is an affine function of the gradient). Condition (1.3) thus appears, …
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 47H05
zbMATH DE Number: 219693
zbMATH Keywords: quasilinear elliptic operator
author: J.-P. Gossez
author: Vesa Mustonen
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 219693