On the Two Obstacles Problem in Orlicz-Sobolev Spaces and Applications

TL;DR

This paper establishes Lewy-Stampacchia inequalities for the two obstacles problem in Orlicz-Sobolev spaces, leading to new regularity and existence results for related quasi-linear elliptic and variational problems.

Contribution

It extends obstacle problem analysis to Orlicz-Sobolev spaces and derives new regularity and existence results for complex elliptic and variational problems.

Findings

Proved Lewy-Stampacchia inequalities in Orlicz-Sobolev spaces.

Established $C^{1,eta}$ regularity for solutions of quasi-linear elliptic operators.

Proved existence of solutions for quasi-variational problems in generalized Orlicz-Sobolev spaces.

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 $C^{1,\alpha}$ 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.