[论文解读] Local and global $C^{1,β}$-regularity for uniformly elliptic quasilinear equations of $p$-Laplace and Orlicz-Laplace type

Carlo Alberto Antonini|arXiv (Cornell University)|Jan 12, 2026

Nonlinear Partial Differential Equations被引用 0

一句话总结

论文证明在一致椭圆拟线性方程组（具有 Orlicz 增长）的解的 interior 与边界梯度 Hölder 连续性，适用于 Dirichlet 或 Neumann 条件下的 p-Laplac e 与 Orlicz-Laplace 类型。

ABSTRACT

We establish gradient Hölder continuity for solutions to quasilinear, uniformly elliptic equations, including $p$-Laplace and Orlicz-Laplace type operators. We revisit and improve upon the results existing in the literature, proving gradient regularity both in the interior and up to the boundary, under Dirichlet or Neumann boundary conditions.

研究动机与目标

• Motivate and establish gradient Hölder continuity for solutions to quasilinear elliptic equations with Orlicz-type growth.
• Generalize interior regularity results to include boundary (Dirichlet and Neumann) problems.
• Provide global $C^{1,β}$ estimates under Dirichlet or Neumann boundary conditions on suitable domains.
• Develop a unified framework that handles both isotropic Orlicz-Laplace and anisotropic variants with nonstandard growth.
• Deliver quantitative estimates that depend on data such as $f$, boundary regularity, and domain geometry.

提出的方法

• Study the autonomous homogeneous problem to obtain $C^{1,α}$ regularity via Bernstein-type bounds and a fundamental alternative argument.
• Use a perturbation (local to global) scheme to transfer interior regularity to nonhomogeneous and nonautonomous settings.
• Derive $L^1$-excess decay estimates for gradients and employ comparison/barrier arguments in half-balls to handle boundary regimes.
• Reduce boundary problems to half-ball domains through flattening and obtain boundary $C^{1,β}$ estimates via a tailored Bernstein argument for the normal derivative.
• Handle anisotropic and nonautonomous variants by verifying structural conditions that ensure uniform ellipticity and controlled growth (e.g., Orlicz-type growth, Hölder continuity in $(x,u)$).
• Provide global estimates by combining interior regularity with boundary control and domain regularity assumptions (Lipschitz and $C^{1,α}$).

实验结果

研究问题

• RQ1 Under what structural and regularity assumptions on the operator and data can one obtain gradient Hölder continuity up to the boundary for Dirichlet problems?
• RQ2 How does the gradient regularity interiorly extend to global (boundary) settings for Neumann and conormal problems within Orlicz-growth quasilinear systems?
• RQ3 What are the precise quantitative dependencies of the $C^{1,β}$ estimates on data such as the right-hand side $f$, domain geometry, and boundary data?
• RQ4 Can the interior gradient estimates be extended to anisotropic or nonautonomous operators with Orlicz-type growth?

主要发现

• Local $C^{1,β}$ regularity: solutions are locally in $C^{1,β}$ with a beta determined by dimension, ellipticity, growth indices, and data.
• Boundary regularity: Dirichlet and Neumann problems admit global $C^{1,β}$ regularity on appropriate domains (Lipschitz or $C^{1,α}$ boundaries) with explicit dependence on boundary data and domain geometry.
• Global estimates: The paper provides quantitative bounds for the $C^{1,β}$ norm in terms of data including $f$, boundary data, and the integrals of $u$ and $B(|Du|)$.
• Key technical tools: a Bernstein method, a fundamental alternative (with new De Giorgi-type inequalities), $L^1$-excess decay for gradients, and perturbation arguments to transfer regularity from homogeneous to inhomogeneous problems.
• The results apply to a broad class of operators including Orlicz-Laplace and anisotropic variants, and remain valid under Dirichlet or Neumann boundary conditions.
• The approach yields an interior gradient Hölder estimate (Theorem 1.1) and global gradient Hölder estimates for Dirichlet (Theorem 1.2) and Neumann problems (Theorem 1.3) with corollaries for the global Dirichlet and Neumann settings (Corollaries 1.4–1.5).