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A quantitative symmetry result for $p$-Laplace equations with discontinuous nonlinearities

Giulio Ciraolo, Xiaoliang Li

Abstract

In this paper, we study positive solutions $u$ of the homogeneous Dirichlet problem for the $p$-Laplace equation $-Δ_p \,u=f(u)$ in a bounded domain $Ω\subset\mathbb{R}^N$, where $N\ge 2$, $1<p<+\infty$ and $f$ is a discontinuous function. We address the quantitative stability of a Gidas-Ni-Nirenberg type symmetry result for $u$, which was established by Lions and Serra when $Ω$ is a ball. By exploiting a quantitative version of the Pólya-Szegö principle, we prove that the deviation of $u$ from its Schwarz symmetrization can be estimated in terms of the isoperimetric deficit of $Ω$.

A quantitative symmetry result for $p$-Laplace equations with discontinuous nonlinearities

Abstract

In this paper, we study positive solutions of the homogeneous Dirichlet problem for the -Laplace equation in a bounded domain , where , and is a discontinuous function. We address the quantitative stability of a Gidas-Ni-Nirenberg type symmetry result for , which was established by Lions and Serra when is a ball. By exploiting a quantitative version of the Pólya-Szegö principle, we prove that the deviation of from its Schwarz symmetrization can be estimated in terms of the isoperimetric deficit of .

Paper Structure

This paper contains 9 sections, 9 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Strategy of the proof and organization of the paper
  3. Preliminaries
  4. The isoperimetric deficit and a Poincaré trace inequality
  5. Rearrangements
  6. Quantitative Pólya--Szegö principle
  7. Estimate on the quantity $\mathfrak{D}$ via the isoperimetric deficit
  8. A comparison result of Talenti type
  9. Proof of Theorem \ref{['thm:main']}

Key Result

Theorem 1.1

Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^N$ with $N\ge 2$, and let $f$ be a nonnegative Borel function on $[0,+\infty)$ that is locally bounded. Let $u\in C^1(\overline{\Omega})$ be a weak solution to in-eq:pro. Assume $m_f>0$ and that one of the following two conditions is fulfille or for some nonincreasing function $\phi\ge 0$ and number $s>0$. Then there exist a constant $\rho

Theorems & Definitions (22)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • ...and 12 more