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A rigidity theorem for Kolmogorov-type operators

Alessia E. Kogoj, E. Lanconelli

Abstract

Let $D\subseteq \mathbb{R}^n$, $n\geq 3$, be a bounded open set and let $x_0\in D$. Assume that the Newtonian potential of $D$ is proportional outside $D$ to the Newtonian potential of a mass concentrated at $\{x_0\}.$ Then $D$ is a Euclidean ball centered at $x_0$. This Theorem, proved by Aharonov, Shiffer and Zalcman in 1981, was extended to the caloric setting by Suzuki and Watson in 2001. In this note, we show that Suzuki--Watson Theorem is a particular case of a more general rigidity result related to a class of Kolmogorov-type PDEs.

A rigidity theorem for Kolmogorov-type operators

Abstract

Let , , be a bounded open set and let . Assume that the Newtonian potential of is proportional outside to the Newtonian potential of a mass concentrated at Then is a Euclidean ball centered at . This Theorem, proved by Aharonov, Shiffer and Zalcman in 1981, was extended to the caloric setting by Suzuki and Watson in 2001. In this note, we show that Suzuki--Watson Theorem is a particular case of a more general rigidity result related to a class of Kolmogorov-type PDEs.

Paper Structure

This paper contains 14 sections, 2 theorems, 117 equations.

Table of Contents

  1. Introduction
  2. Rigidity results in harmonic and caloric settings
  3. Our operators
  4. Mean Value formula
  5. Main Theorem
  6. Basic notions and results from Potential Analysis for $\mathcal{L}$ and $\mathcal{L}^\ast$
  7. Harmonic and subharmonic functions for $\mathcal{L}$ and $\mathcal{L}^\ast$
  8. Maximum principle for $\mathcal{L}^\ast$-subharmonic functions
  9. Propagation of maxima along drift-trajectories
  10. $\Gamma$-potentials
  11. An inequality for the $\Gamma$-potentials of the $\mathcal{L}$-balls
  12. A convolution continuity result
  13. Proof of Theorem \ref{['main']}
  14. Proof of Corollary \ref{['cor']}

Key Result

Theorem 1.1

Let $z_0\in{\mathbb{R}^{{n + {1}}}}$ and let $D$ be a bounded open subset of ${\mathbb{R}^{{n + {1}}}}$ such that, for a suitable $r>0$, If, moreover, then $D={\Omega_r(z_0)}.$

Theorems & Definitions (3)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 1.3