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

Let D be an open subset of Rn with finite measure, and let x∈ D. We introduce the p-Gauss gap of D w.r.t. x to measure how far are the averages over D of the harmonic functions u∈ Lp(D) from u(x). We estimate from below this gap in terms of the ball gap of D w.r.t. x, i.e., the normalized Lebesgue measure of D B, being B the biggest ball centered at x contained in D. From these stability estimates of the mean value formula for harmonic functions in Lp-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 W1,p′, where p′ is the conjugate exponent of p.