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A Class of De Giorgi Type and Hölder Continuity for Some Problems in Musielak-Orlicz-Sobolev Spaces

Hlel Missaoui, Anouar Bahrouni, Hichem Ounaies

TL;DR

This work develops a nonstandard regularity theory for quasilinear elliptic equations with Musielak–Orlicz growth. By introducing the De Giorgi-type class $\mathcal{B}_{G(x,t)}$, it obtains Hölder continuity under general structural assumptions on $G(x,t)$, including critical growth scenarios, and accommodates both Dirichlet and Neumann boundary conditions. The authors establish boundedness of weak solutions and local (plus up-to-boundary) Hölder regularity via a novel De Giorgi iteration adapted to Musielak–Orlicz–Sobolev spaces, unifying and extending variable-exponent and Orlicz-type results. The results have broad applicability to a wide spectrum of nonhomogeneous growth models, including $(p(x),q(x))$-growth and double-phase-like frameworks, and provide a robust toolkit for regularity in nonstandard growth PDEs.

Abstract

In this paper, we introduce a new class of De Giorgi type functions, denoted by \(\mathcal{B}_{G(x,t)}\), and establish the Hölder continuity of its elements under suitable additional assumptions on the generalized \textnormal{N}-function \(G(x,t)\). As an application, we prove the Hölder 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 Hölder 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.

A Class of De Giorgi Type and Hölder Continuity for Some Problems in Musielak-Orlicz-Sobolev Spaces

TL;DR

This work develops a nonstandard regularity theory for quasilinear elliptic equations with Musielak–Orlicz growth. By introducing the De Giorgi-type class , it obtains Hölder continuity under general structural assumptions on , including critical growth scenarios, and accommodates both Dirichlet and Neumann boundary conditions. The authors establish boundedness of weak solutions and local (plus up-to-boundary) Hölder regularity via a novel De Giorgi iteration adapted to Musielak–Orlicz–Sobolev spaces, unifying and extending variable-exponent and Orlicz-type results. The results have broad applicability to a wide spectrum of nonhomogeneous growth models, including -growth and double-phase-like frameworks, and provide a robust toolkit for regularity in nonstandard growth PDEs.

Abstract

In this paper, we introduce a new class of De Giorgi type functions, denoted by \(\mathcal{B}_{G(x,t)}\), and establish the Hölder continuity of its elements under suitable additional assumptions on the generalized \textnormal{N}-function \(G(x,t)\). As an application, we prove the Hölder 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 Hölder 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.

Paper Structure

This paper contains 13 sections, 25 theorems, 267 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. Assumptions and Main Results
  4. Examples
  5. Structure of the paper
  6. Preliminaries
  7. Notations
  8. Musielak-Orlicz-Sobolev space W1G(x,t)
  9. A New Class of De Giorgi-Type Functions $B_{G(x,t)}$
  10. Application to Problems in Musielak--Orlicz--Sobolev Spaces
  11. Boundedness of Weak Solutions to Problems with Dirichlet Boundary Condition
  12. Boundedness of Weak Solutions to Problems with Neumann Boundary Condition
  13. Hölder Continuity of Weak Solutions to Problems with Dirichlet or Neumann Boundary Conditions

Key Result

Theorem 1.2

Suppose that the function $G(x,t)$ satisfies conditions D22--GG1 and GG3. Then the De Giorgi-type class belongs to a Hölder space: where the Hölder exponent $\alpha \in (0,1]$ depends only on the parameters $n, g^-, g^+, F, L:=\exp{L_0}, \gamma$, and $\delta$, and is independent of $M$ and $\gamma_1$.

Theorems & Definitions (67)

  • Definition 1.1
  • Theorem 1.2
  • Definition 1.3: see Sect. 1, Chapter 1 OL
  • Definition 1.4: see Sect. 1, of Chap. I of OL
  • Theorem 1.5
  • Remark 1.6
  • Definition 1.7
  • Definition 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 57 more