On the Harmonic Characterization Of The Spheres: A Sharp Stability Inequality

Let $ D$ be a bounded open subset of $\mathbb{R}^n$ with finite $(n-1)$-dimensional Hausdorff measure $|\partial D|$ and let $x_0 $ be a point of $D$. We introduce a new harmonic invariant, that we call Kuran gap of $\partial D$ w.r.t. $x_0$. To define this new invariant, denoted $\mathcal{K}(\partial D, x_0)$, we use a family of harmonic functions introduced by \"Ulk\"u Kuran in \cite{kuran}. Our main stability result can be described as follows: if $\partial D$ is sufficiently regular just in one of the points of $\partial D$ nearest to $x_0$, then $\mathcal{K}(\partial D, x_0)$ is bounded from below by a kind of isoperimetric index, precisely the normalized difference between $|\partial D|$ and $|\partial B|$, being $B$ the biggest ball contained in $D$ and centered at $x_0$. This partially extends and improves a stability result by Preiss and Toro. By our stability result, we also obtain new rigidity results: (i) a characterization of the Euclidean spheres in terms of single-layer potentials, improving previous theorems by Fichera and by Shahgholian; (ii) a sufficient condition for a harmonic pseudosphere to be a Euclidean sphere, partially extending and improving rigidity results by Lewis and Vogel.