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Harnack inequalities for quasilinear anisotropic elliptic equations with a first order term

Domenico Vuono

TL;DR

This work extends Harnack theory to anisotropic, quasilinear elliptic equations with a first-order term governed by a $Finsler$ norm, $-\

Abstract

We consider weak solutions of the equation $$-Δ_p^H u+a(x,u)H^q(\nabla u)=f(x,u) \quad \text{in } Ω,$$ where $H$ is in some cases called Finsler norm, $Ω$ is a domain of $\mathbb R^N$, $p>1$, $q\ge \max\{p-1,1\}$, and $a(\cdot,u)$, $f(\cdot,u)$ are functions satisfying suitable assumptions. We exploit the Moser iteration technique to prove a Harnack type comparison inequality for solutions of the equation and a Harnack type inequality for solutions of the linearized operator. As a consequence, we deduce a Strong Comparison Principle for solutions of the equation and a strong Maximum Principle for solutions of the linearized operator.

Harnack inequalities for quasilinear anisotropic elliptic equations with a first order term

TL;DR

This work extends Harnack theory to anisotropic, quasilinear elliptic equations with a first-order term governed by a norm, $-\

Abstract

We consider weak solutions of the equation where is in some cases called Finsler norm, is a domain of , , , and , are functions satisfying suitable assumptions. We exploit the Moser iteration technique to prove a Harnack type comparison inequality for solutions of the equation and a Harnack type inequality for solutions of the linearized operator. As a consequence, we deduce a Strong Comparison Principle for solutions of the equation and a strong Maximum Principle for solutions of the linearized operator.
Paper Structure (5 sections, 11 theorems, 124 equations)

This paper contains 5 sections, 11 theorems, 124 equations.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. Finsler norms
  4. The Harnack comparison inequality
  5. The weak Harnack inequality for the linearized operator

Key Result

Theorem 1.1

Suppose that assumptions $(h_H)$ and $(h_p)$ hold. Let $p>(2N+2)/(N+2)$ and let $u,v\,\in\,C^{1,\alpha}_{loc}(\Omega)$ solutions to eq:Euler-Lagrange in $\Omega$, with $q\geq\max\,\{p-1,1\}.$ Suppose that $\overline{B(x_0,6\delta)}\subset\Omega'\subset\subset \Omega$ for some $\delta>0$ and that Then there exists $C=C(p,q,\delta,L, \|v\|_{L^{\infty}(\Omega')}, \|\nabla u\|_{L^{\infty}(\Omega')},

Theorems & Definitions (24)

  • Theorem 1.1
  • Theorem 1.2: Strong Comparison Principle
  • Definition 1.3: CES1CDSDS1
  • Definition 1.4
  • Remark 1.5
  • Theorem 1.6
  • Theorem 1.7: Strong Maximum Principle
  • Theorem 2.1
  • Theorem 2.2
  • Remark 2.3
  • ...and 14 more