Global $C^{1,\alpha}$-Regularity for Musielak-Orlicz Equations in Divergence Form

TL;DR

This paper proves that solutions to certain elliptic equations with complex growth conditions are smoothly differentiable up to the boundary, extending previous regularity results to more general Musielak-Orlicz settings.

Contribution

It establishes global $C^{1,eta}$ regularity for solutions of divergence form equations with Musielak-Orlicz growth, generalizing prior results to broader non-standard growth conditions.

Findings

Solutions are globally $C^{1,eta}$ regular under Musielak-Orlicz growth.

Extends regularity results to variable exponent and Orlicz spaces.

Identifies new boundary behavior conditions for generalized growth.

Abstract

In this paper, we establish global $C^{1,\alpha}$-regularity for bounded generalized solutions of elliptic equations in divergence form with Musielak-Orlicz growth and subject to Dirichlet or Neumann boundary conditions. In fact, our findings extend and generalize several important regularity results in cases of special attention such as variable exponent spaces, Orlicz spaces, and some $(p,q)$ situations. We also point out new conditions in the analysis that focus on the interplay between non-standard growth conditions and the boundary behavior in such generalized examples.