@article{Cupini_Lanconelli_2024, title={On the surface average for harmonic functions: a stability inequality}, volume={14}, url={https://mathematicalanalysis.unibo.it/article/view/18860}, DOI={10.6092/issn.2240-2829/18860}, abstractNote={<div>In this article we present some of the main aspects and the most recent results related to the following question: If the surface mean integral of every harmonic function on the boundary of an open set D is "almost’’ equal to the value of these functions at x<sub>0</sub> in D, then is D "almost’’ a ball with center x<sub>0</sub>? This is the stability counterpart of the rigidity question (the statement above, without the two "almost’’) for which several positive answers are known in literature. A positive answer to the stability problem has been given in a paper by Preiss and Toro, by assuming a condition that turns out to be sufficient for ∂D to be geometrically close to a sphere. This condition, however, is not necessary, even for small Lipschitz perturbations of smooth domains, as shown in our recent paper, in which a stability inequality is obtained by assuming only a local regularity property of the boundary of D in at least one of its points closest to x<sub>0</sub>.</div>}, number={2}, journal={Bruno Pini Mathematical Analysis Seminar}, author={Cupini, Giovanni and Lanconelli, Ermanno}, year={2024}, month={Jan.}, pages={129–138} }