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

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.

💡 Research Summary

This paper establishes global C¹,α regularity for bounded generalized solutions of elliptic equations in divergence form whose nonlinearity is governed by a Musielak–Orlicz function G(x, t). The authors treat both Dirichlet and Neumann boundary conditions within a unified framework, thereby extending a large body of earlier work that dealt separately with variable‑exponent, Orlicz, (p,q)‑growth, and double‑phase models.

Setting and assumptions.
Let Ω⊂ℝⁿ (n≥2) be a bounded Lipschitz domain satisfying a quantitative interior thickness condition (A). The growth function G(x, t)=∫₀^{|t|}g(x,s)ds is assumed to be a generalized N‑function satisfying four structural hypotheses:

• (G₀) 1<g⁻≤t g(x,t) G(x,t)≤g⁺<n for all x, t>0.
• (G₁) Uniform bounds on G(x,1): F⁻¹≤G(x,1)≤F.
• (G₂) Log‑type continuity of the inverse G⁻¹ on balls of size ≤1.
• (G₃) A logarithmic modulus of continuity for r(x,t)=t g(x,t) G(x,t) on small balls, i.e. η(R)≤L₀|ln(2R)|.

These conditions are more flexible than the usual log‑Hölder continuity required in variable‑exponent spaces and encompass double‑phase and multi‑phase growth as special cases.

The differential operator is \