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

Hlel Missaoui

TL;DR

This work extends elliptic regularity theory to divergence-form equations with Musielak–Orlicz growth, proving global $C^{1,\alpha}$-regularity for bounded generalized solutions of $-$div$ A(x,u,Du)=B(x,u,Du)$ under Dirichlet or Neumann conditions. The authors leverage Gehring-type higher integrability in Musielak–Orlicz spaces, compare with frozen-coefficient problems, and apply Campanato–Morrey estimates, together with boundary treatments, to obtain interior and boundary $C^{1,\alpha}$ regularity. The results generalize and unify variable-exponent, Orlicz, and double-phase growth frameworks, and they establish quantitative Hölder continuity of the gradient that depends on structural data and boundary smoothness. This provides a robust framework for nonstandard growth elliptic PDEs, including spatially dependent and highly nonuniform growth, with potential applications to heterogeneous media modeling.

Abstract

In this paper, we establish global $C^{1,α}$-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.

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

TL;DR

This work extends elliptic regularity theory to divergence-form equations with Musielak–Orlicz growth, proving global -regularity for bounded generalized solutions of div under Dirichlet or Neumann conditions. The authors leverage Gehring-type higher integrability in Musielak–Orlicz spaces, compare with frozen-coefficient problems, and apply Campanato–Morrey estimates, together with boundary treatments, to obtain interior and boundary regularity. The results generalize and unify variable-exponent, Orlicz, and double-phase growth frameworks, and they establish quantitative Hölder continuity of the gradient that depends on structural data and boundary smoothness. This provides a robust framework for nonstandard growth elliptic PDEs, including spatially dependent and highly nonuniform growth, with potential applications to heterogeneous media modeling.

Abstract

In this paper, we establish global -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 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.
Paper Structure (12 sections, 22 theorems, 135 equations)

This paper contains 12 sections, 22 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. Main Results
  4. Paper Structure
  5. Notations and Preliminaries
  6. Notations
  7. Musielak-Orlicz-Sobolev space W1G(x,t)
  8. Technical Tools
  9. Proofs of main results
  10. Proof of Theorem \ref{['Thm1.1']}
  11. Proof of Theorem \ref{['Thm1.2']}
  12. Proof of Theorem \ref{['Thm1.3']}

Key Result

Theorem 1.4

Under assumptions 6.31--7.41, 6.51, D22--GG3, and H111--YoungTriple1, if $u \in W^{1,G(x,t)}(\Omega) \cap L^{\infty}(\Omega)$ is a bounded generalized solution of problem P satisfying M, then $u \in C^{1,\alpha}_{\mathrm{loc}}(\Omega)$. The Hölder exponent $\alpha$ depends only on $a_1,a_2, a_3, a_4

Theorems & Definitions (54)

  • Definition 1.1: see Sect. 1, Chap. I of OL
  • Remark 1.2
  • Example 1.1
  • Definition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Remark 1.7
  • Definition 2.1
  • Definition 2.2
  • ...and 44 more