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.