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

Carlo Alberto Antonini

TL;DR

The paper develops a comprehensive gradient regularity theory for scalar solutions to uniformly elliptic quasilinear equations with Orlicz-type growth, including the $p$-Laplace and anisotropic Orlicz–Laplace models. The authors integrate an interior analysis based on autonomous homogeneous problems (using Bernstein-type bounds and a fundamental alternative) with a perturbation argument to transfer regularity to nonhomogeneous, nonautonomous settings. They extend the gradient Hölder continuity up to the boundary for both Dirichlet and Neumann problems, employing boundary-flattening maps and precise domain-regularity handling to obtain global $C^{1,\beta}$ estimates with quantitative dependence on data. The framework accommodates general Orlicz growth, including nearly linear cases, and encompasses isotropic as well as anisotropic operators, yielding new boundary regularity results and an $L^1$-excess decay estimate for the homogeneous Neumann problem.

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.

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

TL;DR

The paper develops a comprehensive gradient regularity theory for scalar solutions to uniformly elliptic quasilinear equations with Orlicz-type growth, including the -Laplace and anisotropic Orlicz–Laplace models. The authors integrate an interior analysis based on autonomous homogeneous problems (using Bernstein-type bounds and a fundamental alternative) with a perturbation argument to transfer regularity to nonhomogeneous, nonautonomous settings. They extend the gradient Hölder continuity up to the boundary for both Dirichlet and Neumann problems, employing boundary-flattening maps and precise domain-regularity handling to obtain global estimates with quantitative dependence on data. The framework accommodates general Orlicz growth, including nearly linear cases, and encompasses isotropic as well as anisotropic operators, yielding new boundary regularity results and an -excess decay estimate for the homogeneous Neumann problem.

Abstract

We establish gradient Hölder continuity for solutions to quasilinear, uniformly elliptic equations, including -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.
Paper Structure (20 sections, 61 theorems, 899 equations)

This paper contains 20 sections, 61 theorems, 899 equations.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. Young functions
  4. Orlicz Lebesgue and Sobolev spaces
  5. The stress field A
  6. Classes of domains and boundary flattening
  7. Auxiliary results and lemmas
  8. Boundedness of solutions
  9. Interior gradient regularity: homogeneous problems
  10. Boundary regularity for equations in trace form
  11. Boundary gradient regularity: homogeneous Dirichlet problems
  12. Boundary gradient regularity: homogeneous Neumann problems
  13. Some properties of B
  14. Bounds for C on the flat boundary.
  15. Gradient boundedness
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $u\in W^{1,B}_{loc}(\Omega)$ be a local weak solution to eq1 under the assumptions iasa, reg:A-A:hold, ab=1 and Then there exists $\beta\in (0,1)$ depending on $n,\lambda,\Lambda,i_a,s_a,\alpha,d$ such that and for every $\Omega"\Subset \Omega'\Subset \Omega$, there holds

Theorems & Definitions (128)

  • Theorem 1.1: Local $C^{1,\beta}$ regularity
  • Theorem 1.2: $C^{1,\beta}$ regularity, Dirichlet problems
  • Theorem 1.3: $C^{1,\beta}$ regularity, Neumann problems
  • Corollary 1.4: Global $C^{1,\beta}$ regularity, Dirichlet problems
  • Corollary 1.5: Global $C^{1,\beta}$ regularity, Neumann problems
  • Lemma 2.1
  • proof
  • Remark 2.2: Uniformity in $\varepsilon$
  • Remark 2.3
  • Lemma 2.4
  • ...and 118 more