On the two obstacles problem in Orlicz–Sobolev spaces and applications

Abstract

We prove the Lewy-Stampacchia inequalities for the two obstacles problem in abstract form for T-monotone operators. As a consequence for a general class of quasi-linear elliptic operators of Ladyzhenskaya-Uraltseva type, including p(x)-Laplacian type operators, we derive new results of C 1,α regularity for the solution. We also apply those inequalities to obtain new results to the N-membranes problem and the regularity and monotonicity properties to obtain the existence of a solution to a quasi-variational problem in (generalized) Orlicz-Sobolev spaces.

Homogenization of obstacle problems in Orlicz–Sobolev spaces

“…and letting ϕ → 1, we obtain (3.24). Since there are Lewy-Stampacchia inequalities also for the two obstacles problem (see [19]), the Theorem 3.1 can be extended for two obstacles problems with similar assumptions.…”

“…Our approach is a development of the classical methods [6,11] (see also [20,21,24]) combined with the Lewy-Stampacchia inequalities in the Orlicz-Sobolev framework, in accordance with [19], which then allows the use of a Rellich-Kondrachov compactness argument.…”

A two-obstacle problem with variable exponent and measure data

“…As n tends to infinity, we have the following convergence result for the solution sequence {u n } (the existence and uniqueness of the solution u n to (2.2) for each n follows from standard results for monotone, coercive operators; see [14,20]). …”

“…More recently, in [20] Rodrigues and Teymurazyanin studied a double-obstacle problem, which included problem (1.1) as a special case. The existence and uniqueness result was obtained when the data involved were regular enough.…”

The abstract Lewy–Stampacchia inequality and applications

“…harmonic and hence ∆u = 0, while where {u = ϕ} we have ∆u = ∆ϕ (the precise derivation of (1.1) -which we do not discuss in this introduction -must take into account what happens at the boundary of the set {u = ϕ}). Over time, inequality (1.1) has been generalized in several different directions, among others we mention [34] as a general reference for general linear operators and boundary values, [25] for nonlinear Leray-Lions operators, [29] for nonlinear p-Laplacian type operators, and [30] for the fractional Laplacian and the Laplacian in the Heisenberg group (see also [27] for this latter setting).…”