Stability of the mean value formula for harmonic functions in Lebesgue spaces

Abstract

Let D be an open subset of ${\mathbb {R}}^n$ with finite measure, and let $x_0 \in D$. We introduce the p-Gauss gap of D w.r.t. $x_0$ to measure how far are the averages over D of the harmonic functions $u \in L^p(D)$ from $u(x_0)$. We estimate from below this gap in terms of the ball gap of D w.r.t. $x_0$, i.e., the normalized Lebesgue measure of $D \setminus B$, being B the biggest ball centered at $x_0$ contained in D. From these stability estimates of the mean value formula for harmonic functions in $L^p$-spaces, we straightforwardly obtain rigidity properties of the Euclidean balls. We also prove a continuity result of the p-Gauss gap in the Sobolev space $W^{1,p'}$, where $p'$ is the conjugate exponent of p.