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Global Lipschitz regularity in anisotropic elliptic problems with natural gradient growth

Carlo Alberto Antonini, Andrea Cianchi

TL;DR

This work establishes global Lipschitz regularity for bounded weak solutions to anisotropic elliptic equations of $p$-Laplacian type with right-hand sides that exhibit natural gradient growth. The authors develop an anisotropic Orlicz–growth framework based on a norm $H$, its associated operator $\mathcal{A}$, and carefully designed approximations to reduce gradient-growth terms, employing level-set and coarea techniques together with sharp boundary geometry via the anisotropic second fundamental form. Under minimal domain regularity (including convexity or boundary curvature in Lorentz spaces) and integrability conditions on the data $f$ (Lorentz spaces $L^{n,1}$ for $n\ge3$ or $(L^2)^{(1,1/2)}$ for $n=2$), they derive a global $L^\infty$ bound for $|\nabla u|$, hence Lipschitz continuity, with explicit dependence on the data and domain. A final step uses a Kazdan–Kramer transformation to treat genuine natural gradient growth, yielding Lipschitz estimates with an exponential factor in $\|u\|_{L^\infty}$ and extending the results to problems with gradient-dependent right-hand sides.

Abstract

We deal with homogeneous Dirichlet and Neumann boundary-value problems for anisotropic elliptic operators of p-Laplace type. They emerge as Euler-Lagrange equations of integral functionals of the Calculus of Variations built upon possibly anisotropic norms of the gradient of trial functions. We establish global Lipschitz regularity of solutions under the weakest possible assumption on right-hand side of the equation, which may also include the gradient term with natural growth exponent. The results hold in either convex domains, or domains enjoying minimal integrability assumptions on the curvature of its boundary.

Global Lipschitz regularity in anisotropic elliptic problems with natural gradient growth

TL;DR

This work establishes global Lipschitz regularity for bounded weak solutions to anisotropic elliptic equations of -Laplacian type with right-hand sides that exhibit natural gradient growth. The authors develop an anisotropic Orlicz–growth framework based on a norm , its associated operator , and carefully designed approximations to reduce gradient-growth terms, employing level-set and coarea techniques together with sharp boundary geometry via the anisotropic second fundamental form. Under minimal domain regularity (including convexity or boundary curvature in Lorentz spaces) and integrability conditions on the data (Lorentz spaces for or for ), they derive a global bound for , hence Lipschitz continuity, with explicit dependence on the data and domain. A final step uses a Kazdan–Kramer transformation to treat genuine natural gradient growth, yielding Lipschitz estimates with an exponential factor in and extending the results to problems with gradient-dependent right-hand sides.

Abstract

We deal with homogeneous Dirichlet and Neumann boundary-value problems for anisotropic elliptic operators of p-Laplace type. They emerge as Euler-Lagrange equations of integral functionals of the Calculus of Variations built upon possibly anisotropic norms of the gradient of trial functions. We establish global Lipschitz regularity of solutions under the weakest possible assumption on right-hand side of the equation, which may also include the gradient term with natural growth exponent. The results hold in either convex domains, or domains enjoying minimal integrability assumptions on the curvature of its boundary.

Paper Structure

This paper contains 13 sections, 7 theorems, 278 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Young functions
  4. The functions $H$ and $\mathcal{A}$
  5. Function spaces
  6. Classes of domains
  7. Geometric and functional inequalities
  8. Boundary value problems with Orlicz growth
  9. Identities for vector fields
  10. Proof of Theorem \ref{['thm:dirneu']}, Dirichlet problems
  11. Proof of Theorem \ref{['thm:dirneu']}, Neumann problems
  12. Convex domains: proof of Theorem \ref{['thm:conv']}
  13. Natural growth in the gradient: proof of Theorems \ref{['thm:example1']} and \ref{['thm:example']}

Key Result

Theorem 1.1

Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^n$, $n\geq 2$, such that $\partial \Omega \in W^2L^{(n-1,1)}$. Let $H$ be a norm in $\mathbb{R}^n$ such that $H^2\in C^2(\mathbb R^n\setminus\{0\})$ and fulfilling ell:H. Assume that either $n\geq 3$ and $f\in L^{n,1}(\Omega)$, or $n=2$ and where $X_n(\Omega)$ is defined as X:f. In the case when $\kappa=0$, the assumption that $u$ be boun

Figures (1)

  • Figure 1:

Theorems & Definitions (16)

  • Theorem 1.1: Global Lipschitz regularity in minimally regular domains
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4: Global Lipschitz regularity in convex domains
  • Theorem 3.1
  • proof
  • Theorem 3.2: Minimally integrable curvatures
  • Theorem 3.3: Convex domains
  • Lemma 4.1
  • proof
  • ...and 6 more