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Regularity for the normalized p-Laplacian equation with an arbitrary degeneracy law

Claudemir Alcantara, Makson Santos

TL;DR

The paper addresses interior regularity of viscosity solutions to the gradient-degenerate normalized $p$-Laplacian $-\sigma(|Du|)\Delta_p^{ ext{N}}u=f$, where the degeneracy is governed by a modulus $\sigma$ whose inverse satisfies a Dini condition. It extends the regularity theory by developing a non-collapsing modulus construction and a hyperplane-approximation scheme, coupled with compactness and stability analyses for scaled, shifted-derivative equations. The main achievement is proving $u\, ext{is differentiable}, i.e.,}^1_{ ext{loc}}(B_1)$ with a gradient modulus $\gamma$, under very general degeneracy laws, and without requiring Hölder continuity of the gradient in general. This work broadens the applicability of regularity results for degenerate fully nonlinear elliptic equations and provides a robust framework for handling arbitrary degeneracy via non-collapsing sets and affine-approximation iterations.

Abstract

We examine the interior regularity of solutions to a degenerate normalized $p$-Laplace equation, where the degeneracy is governed by a modulus of continuity whose inverse satisfies a Dini continuity condition. We prove that under very general assumptions on the degeneracy law, solutions belong to the $C^1$ class. We argue by approximating the solutions by a sequence of hyperplanes, which allows us to prove the desired regularity.

Regularity for the normalized p-Laplacian equation with an arbitrary degeneracy law

TL;DR

The paper addresses interior regularity of viscosity solutions to the gradient-degenerate normalized -Laplacian , where the degeneracy is governed by a modulus whose inverse satisfies a Dini condition. It extends the regularity theory by developing a non-collapsing modulus construction and a hyperplane-approximation scheme, coupled with compactness and stability analyses for scaled, shifted-derivative equations. The main achievement is proving with a gradient modulus , under very general degeneracy laws, and without requiring Hölder continuity of the gradient in general. This work broadens the applicability of regularity results for degenerate fully nonlinear elliptic equations and provides a robust framework for handling arbitrary degeneracy via non-collapsing sets and affine-approximation iterations.

Abstract

We examine the interior regularity of solutions to a degenerate normalized -Laplace equation, where the degeneracy is governed by a modulus of continuity whose inverse satisfies a Dini continuity condition. We prove that under very general assumptions on the degeneracy law, solutions belong to the class. We argue by approximating the solutions by a sequence of hyperplanes, which allows us to prove the desired regularity.

Paper Structure

This paper contains 8 sections, 12 theorems, 149 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Definitions
  4. Assumptions and main results
  5. Auxiliary results
  6. Non-collapsing sets
  7. Compactness of the solutions
  8. Differentiability of solutions

Key Result

Theorem 2.1

Let $u \in C(B_1)$ be a viscosity solution to 1 and suppose that AAsigma-AAf are satisfied. Then $u \in C^1_{loc}(B_1)$, and there exists a modulus of continuity $\gamma: [0, +\infty) \to [0, +\infty)$, depending only on $d$, $p$ and $\|f\|_{L^\infty(B_1)}$ such that for every $x, y \in B_{1/2}$.

Theorems & Definitions (27)

  • Definition 2.1: Viscosity solution
  • Remark 2.1
  • Definition 2.2: Viscosity solution II
  • Theorem 2.1
  • Lemma 2.1
  • Remark 2.2: Scaling properties
  • Lemma 2.2
  • Definition 2.3: APPT2022, Definition 3
  • Definition 2.4: APPT2022, Definition 5
  • Proposition 2.1: APPT2022, Proposition 5
  • ...and 17 more