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

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.