H\"{o}lder continuity for nonlinear elliptic problem in Musielak-Orlicz-Sobolev space

TL;DR

This paper establishes Hölder continuity for solutions of fully nonlinear elliptic problems within Musielak-Orlicz-Sobolev spaces, extending regularity results to this generalized functional framework.

Contribution

It introduces a De Giorgi process adapted to Musielak-Orlicz-Sobolev spaces, proving Hölder continuity for minimizers and weak solutions of nonlinear elliptic equations.

Findings

Hölder continuity of minimizers in Musielak-Orlicz-Sobolev spaces

Hölder continuity of weak solutions for nonlinear elliptic equations

Extension of De Giorgi process to Musielak-Orlicz framework

Abstract

Under appropriate assumptions on the $N(\Omega)$-fucntion, the De Giorgi process is presented in the framework of Musielak-Orlicz-Sobolev space to prove the H\"{o}lder continuity of fully nonlinear elliptic problems. As the applications, the H\"{o}lder continuity of the minimizers for a class of the energy functionals in Musielak-Orlicz-Sobolev spaces is proved; and furthermore, the H\"{o}lder continuity of the weak solutions for a class of fully nonlinear elliptic equations is provided.