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.

Keywords:

• quasi-linear elliptic operators
• obstacle problems
• variable growth condition
• Orlicz–Sobolev spaces
• N-membranes problem
• elliptic quasi-variational inequalities

AMS Subject Classifications::

• 35J87
• 35J99
• 35R35
• 49J40
• 46E35