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Rigidity of balls in the solid mean value property for polyharmonic functions

Nicola Abatangelo

Abstract

We show that balls are the only open bounded domains for which the mean value formula for polyharmonic functions holds. We do so by adapting an argument of Ü. Kuran for harmonic functions. Also, we provide a quantitative version of the same result.

Rigidity of balls in the solid mean value property for polyharmonic functions

Abstract

We show that balls are the only open bounded domains for which the mean value formula for polyharmonic functions holds. We do so by adapting an argument of Ü. Kuran for harmonic functions. Also, we provide a quantitative version of the same result.
Paper Structure (4 sections, 8 theorems, 62 equations)

This paper contains 4 sections, 8 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Rigidity of balls: proof of Theorem rigid-poly
  3. Stability via the Kuran gap: proof of Theorem stab-poly
  4. The coefficient matrix

Key Result

Lemma 1.1

Let $D\subseteq\mathbb{R}^n$ be open and $m\in\mathbb{N}$ with $m\geq 2$. A function $u\in L^1(D)\cap C^0(D)$ is $m$-polyharmonic in $D$ if and only if for any $x_0\in D$, $r>0$ such that $B_r(x_0)\subseteq D$, and $0<\alpha_1<\cdots<\alpha_m\leq 1$ it holds where $c_1,\ldots,c_m>0$ are constants depending on $m$ and $\alpha_1,\ldots,\alpha_m$ satisfying

Theorems & Definitions (15)

  • Lemma 1.1
  • Lemma 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Lemma 2.1
  • proof
  • proof : Proof of Theorem \ref{['thm:rigid-poly']}
  • Lemma 3.1
  • proof
  • ...and 5 more