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Second order regularity for degenerate p-Laplace type equations with log-concave weights

Carlo Alberto Antonini, Giulio Ciraolo, Francesco Pagliarin

TL;DR

This work establishes second-order regularity for weighted p-Laplace type equations with homogeneous Neumann boundary conditions in convex domains when the weight $\varrho$ is log-concave and may degenerate at the boundary. The authors develop an approximation framework for the domain, operator, and weight, and derive a generalized Reilly identity to connect divergence-form operators with the stress field $\mathcal{A}(\nabla u)=|\nabla u|^{p-2}\nabla u$, yielding sharp global estimates in bounded domains and local boundary estimates in unbounded domains. Central to the analysis are a weighted Poincaré inequality, compactness in weighted Sobolev spaces, and a robust stress-field regularization $\mathcal{A}_\varepsilon$, all used in an intricate limiting procedure to prove the main results and their sharpness. The paper also discusses extensions to anisotropic settings, and provides remarks on the Dirichlet problem, highlighting intrinsic limitations when the weight vanishes at the boundary. Overall, the work significantly extends second-order regularity theory to degenerate and log-concave weighted problems, with potential applications to nonlinear PDEs and geometric-functional inequalities.

Abstract

We consider weighted p-Laplace type equations with homogeneous Neumann boundary conditions in convex domains, where the weight is a log-concave function which may degenerate at the boundary. In the case of bounded domains, we provide sharp global second-order estimates. For unbounded domains, we prove local estimates at the boundary. The results are new even for the case p = 2.

Second order regularity for degenerate p-Laplace type equations with log-concave weights

TL;DR

This work establishes second-order regularity for weighted p-Laplace type equations with homogeneous Neumann boundary conditions in convex domains when the weight is log-concave and may degenerate at the boundary. The authors develop an approximation framework for the domain, operator, and weight, and derive a generalized Reilly identity to connect divergence-form operators with the stress field , yielding sharp global estimates in bounded domains and local boundary estimates in unbounded domains. Central to the analysis are a weighted Poincaré inequality, compactness in weighted Sobolev spaces, and a robust stress-field regularization , all used in an intricate limiting procedure to prove the main results and their sharpness. The paper also discusses extensions to anisotropic settings, and provides remarks on the Dirichlet problem, highlighting intrinsic limitations when the weight vanishes at the boundary. Overall, the work significantly extends second-order regularity theory to degenerate and log-concave weighted problems, with potential applications to nonlinear PDEs and geometric-functional inequalities.

Abstract

We consider weighted p-Laplace type equations with homogeneous Neumann boundary conditions in convex domains, where the weight is a log-concave function which may degenerate at the boundary. In the case of bounded domains, we provide sharp global second-order estimates. For unbounded domains, we prove local estimates at the boundary. The results are new even for the case p = 2.
Paper Structure (10 sections, 17 theorems, 229 equations, 2 figures)

This paper contains 10 sections, 17 theorems, 229 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Log-concave functions
  4. Approximation of log-concave weights
  5. Stress field approximation
  6. Poincaré inequality and compactness theorems
  7. Fundamental lemmas for vector fields and generalized Reilly's identity
  8. Proof of Theorem \ref{['main:thmneu']}
  9. Proof of Theorems \ref{['thm:loc1']}-\ref{['thm:loc2']}
  10. Some remarks on the Dirichlet problem

Key Result

Theorem 1.1

Let $p>1$, $\Omega\subset \mathbb R^n$ be a bounded convex domain, $\varrho: \Omega\to (0,\infty)$ be a log-concave function on $\Omega$, and suppose $f\in L^2(\Omega;\varrho)\cap L^{p'}(\Omega;\varrho)$ satifies f:compat. Assume that $u\in W^{1,p}(\Omega;\varrho)$ is a weak solution to the Neumann and there exist explicit constants $C_0=C_0(n,p)$ and $C_1=C_1(n,p,d_\Omega,\varrho)$ such that wh

Figures (2)

  • Figure 1:
  • Figure 2:

Theorems & Definitions (35)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Example 2.1
  • Example 2.2
  • Proposition 2.3
  • Theorem 2.4
  • proof : Proof of Proposition \ref{['prop:conv']}
  • Lemma 3.1
  • proof
  • ...and 25 more