The Lewy–Stampacchia inequality for the obstacle problem with quadratic growth in the gradient

Abstract

In this paper we prove that at least one solution of the obstacle problem for a linear elliptic operator perturbed by a nonlinearity having quadratic growth in the gradient satisfies the Lewy-Stampacchia inequality.Résumé. Nous démontrons dans cet article qu'au moins une solution du problème de l'obstacle pour un opérateur elliptique linéaire perturbé par une non linéarité à croissance quadratique par rapport au gradient vérifie l'inégalité de Lewy-Stampacchia.

Lewy–Stampacchia’s inequality for a Leray–Lions operator with natural growth in the gradient

“…Finally in [7] we considered the case of problem (1.1) with a nonlinear perturbation H , but only in the case where the Leray-Lions operator is actually the linear operator −div A(x)Du; note that in this case p = 2 and a(x, s, ξ) = A(x)ξ is independent of s. The present paper is therefore a natural generalization of [7].…”

Strong Solutions of Semilinear Parabolic Equations with Measure Data and Generalized Backward Stochastic Differential Equations

“…i.e. the Lewy-Stampacchia inequality for solutions of the problem (5.2), (5.3) (see [5,20] and [16] for the inequality for solutions of elliptic equations).…”

Nonlinear Elliptic Inclusions with Unilateral Constraint and Dependence on the Gradient

“…All these papers consider problems with no unilateral constraint (that is, ϕ = 0) and the reaction term F is single-valued. Variational inequalities (that is, problems where ϕ is the indicator function of a closed, convex set), were investigated by Arcoya, Carmona and Martinez Aparicio [2], Matzeu and Servadei [15], Mokrane and Murat [17]. All have single valued source term.…”