Orlicz-Sobolev versus H\"older local minimizer for nonlinear Robin problems

TL;DR

This paper proves boundedness and regularity of weak solutions for Robin problems involving Orlicz g-Laplacian operators, and shows that local minimizers in C^1 are also minimizers in the Orlicz Sobolev space.

Contribution

It establishes new regularity results for solutions of Robin problems with Orlicz g-Laplacian and links classical and Orlicz minimizers.

Findings

Weak solutions are bounded.

C^1 local minimizers are also W^{1,G} local minimizers.

Regularity results extend to Robin-Orlicz problems.

Abstract

In this paper, we establish a regularity results for weak solutions of Robin problems driven by the well-known Orlicz $g$-Laplacian operator. Precisely, by using a suitable variation of the Moser iteration technique, we prove that every weak solution of our problem is bounded. Moreover, we combine this result with the Lieberman regularity theorem, to show that every $C^1(\overline{\Omega})$-local minimizer is also a $W^{1,G}(\Omega)$-local minimizer for the corresponding energy functional of Robin-Orlicz problem.