A Class of De Giorgi Type and H\"older Continuity for Some Problems in Musielak-Orlicz-Sobolev Spaces

Abstract

In this paper, we introduce a new class of De Giorgi type functions, denoted by \(\mathcal{B}_{G(x,t)}\), and establish the H\"older continuity of its elements under suitable additional assumptions on the generalized \textnormal{N}-function \(G(x,t)\). As an application, we prove the H\"older continuity of solutions to quasilinear equations whose principal part is in divergence form with \(G(x,t)\)-growth conditions, including both critical and standard growth cases. The novelty of our work lies in the generalization of the H\"older continuity results previously known for variable exponent \cite[X, Fan and D. Zhao]{Fan1999} and Orlicz \cite[G. M. Lieberman]{Li1991} problems. Moreover, our results encompass a wide variety of quasilinear equations.