Pseudo-monotonicity and the Leray-Lions condition
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, for this class of operators, as both necessary and sufficient for pseudo-monotonicity. The proof in Section 2 uses a technique which is standard in the calculus of variations to derive necessary conditions for weak lower semi-continuity. The proof in Section 3 involves a combination of Minty's trick from the theory of monotone operators, Leray-Lions' original argument and a lemma by Landes.
scientific article; zbMATH DE number 219693