On the harmonic characterization of the spheres: a sharp stability inequality and some of its consequences

TL;DR

This paper introduces a new parameter called Kuran gap to measure harmonic function behavior on domains, establishes a stability inequality linking it to domain geometry, and derives consequences for characterizing spheres and solving a classical surface problem.

Contribution

It defines the Kuran gap parameter, proves a sharp stability inequality relating it to domain boundary measures, and applies this to sphere characterization and classical potential theory problems.

Findings

Kuran gap bounds relate to isoperimetric differences

Stability inequality extends previous results by Preiss, Toro, Agostiniani, Magnanini

Provides new conditions for harmonic pseudospheres to be spheres

Abstract

Let $ D$ be a bounded open subset of $\mathbb R^n$ with $|\partial D| < \infty$ and let $x_0 $ be a point of $D$. We introduce a new parameter, that we call Kuran gap of $\partial D$ w.r.t. $x_0$. Roughly speaking, this parameter, denoted by $\mathcal{K}(\partial D, x_0)$, measures the gap between $u(x_0)$ and the average of $u$ on $\partial D$ for a particular family of functions $u$ harmonic in $\overline{D}$, in terms of the Poisson kernel of the biggest ball $B$ centered at $x_0$ and contained in $D$. To do that, we need the domain $D$ Lyapunov-Dini regular in at least one of the points of $\partial D$ nearest to $x_0$. Our main stability result can be described as follows: $\mathcal{K}(\partial D, x_0)$ is bounded from below by a kind of isoperimetric index, precisely the normalized difference beetween $|\partial D|$ and $|\partial B|$. This extends a stability result by Preiss and Toro, and a more recent theorem by Agostiniani and Magnanini. Moreover, from our stability inequality we obtain a new sufficient condition for a harmonic pseudosphere to be a Euclidean sphere, a result which partially improves a rigidity theorem by Lewis and Vogel. Finally, we give a new solution of the surface version of a solid ``potato'' problem by Aharonov, Schiffer and Zalcman.