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ABP estimate and Harnack inequality for a class of degenerate fully nonlinear pseudo-$p$-Laplacian equations

Sun-Sig Byun, Hongsoo Kim

TL;DR

The paper addresses the regularity theory for a class of degenerate fully nonlinear nondivergence equations of pseudo-$p$-Laplacian type by proving Aleksandrov-Bakelman-Pucci estimates and a Harnack inequality for viscosity solutions. The authors adapt the sliding paraboloid method to anisotropic, coordinatewise degenerate operators, using an anisotropic paraboloid $\varphi$ and a barrier function tailored to the degeneracy, together with the area formula to convert gradient information into measure estimates. Key contributions include ABP-type bounds that control the solution via $\|f^-\|_{L^n(\Gamma^+(u))}$ or $\|f^+\|_{L^n(\Gamma^+( -u))}$ and a Harnack inequality with $L^n$ right-hand side data, which yield Hölder regularity. The results advance understanding of regularity in anisotropic diffusion models and have potential applications to variational problems with directional growth and layered media.

Abstract

We prove Aleksandrov-Bakelman-Pucci estimates and Harnack inequalities for viscosity solutions of a class of degenerate fully nonlinear pseudo-$p$-Laplacian equations in nondivergence form. Our main approach is an adaptation of the sliding paraboloid method with anisotropic functions tailored to the coordinatewise degeneracy.

ABP estimate and Harnack inequality for a class of degenerate fully nonlinear pseudo-$p$-Laplacian equations

TL;DR

The paper addresses the regularity theory for a class of degenerate fully nonlinear nondivergence equations of pseudo--Laplacian type by proving Aleksandrov-Bakelman-Pucci estimates and a Harnack inequality for viscosity solutions. The authors adapt the sliding paraboloid method to anisotropic, coordinatewise degenerate operators, using an anisotropic paraboloid and a barrier function tailored to the degeneracy, together with the area formula to convert gradient information into measure estimates. Key contributions include ABP-type bounds that control the solution via or and a Harnack inequality with right-hand side data, which yield Hölder regularity. The results advance understanding of regularity in anisotropic diffusion models and have potential applications to variational problems with directional growth and layered media.

Abstract

We prove Aleksandrov-Bakelman-Pucci estimates and Harnack inequalities for viscosity solutions of a class of degenerate fully nonlinear pseudo--Laplacian equations in nondivergence form. Our main approach is an adaptation of the sliding paraboloid method with anisotropic functions tailored to the coordinatewise degeneracy.

Paper Structure

This paper contains 5 sections, 9 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Notations and Preliminaries
  3. Proof of ABP estimates
  4. Basic measure estimate
  5. barrier function and doubling property

Key Result

Theorem 1.1

Let $p \ge 0$. Let $u \in C(\Omega)$ satisfy Then there exists a constant $C = C(n,\lambda,\Lambda,p,d) > 0$ such that where $d=\operatorname{diam}(\Omega)$ and $\Gamma^+(u)$ is the upper contact set defined in upp. Analogously, if $u \in C(\Omega)$ satisfies Then there exists a constant $C=C(n,\lambda,\Lambda,p,d)>0$ such that

Theorems & Definitions (18)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.1
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.1
  • Remark 2.1
  • Lemma 2.1: Caffarelli95
  • proof : Proof of Theorem \ref{['Main1']}
  • Lemma 4.1
  • ...and 8 more