In this note I give a short overview about convexity properties of solutions to elliptic equations in convex domains and convex rings and show a result about the optimal concavity of the Newtonian potential of a bounded convex domain in ℝn , n ≥ 3, namely: if the Newtonian potential of a bounded domain is ”sufficiently concave”, then the domain is necessarily a ball. This result can be considered an unconventional overdetermined problem.This paper is based on a talk given by the author in Bologna at the ”Bruno Pini Mathematical Analysis Seminar”, which in turn was based on the paper P. Salani, A characterization of balls through optimal concavity for potential functions, Proc. AMS 143 (1) (2015), 173-183.
@article{7795,
title = {Optimal Concavity for Newtonian Potentials},
journal = {Bruno Pini Mathematical Analysis Seminar},
year = {2018},
doi = {10.6092/issn.2240-2829/7795},
language = {EN},
url = {http://dml.mathdoc.fr/item/7795}
}
Salani, Paolo. Optimal Concavity for Newtonian Potentials. Bruno Pini Mathematical Analysis Seminar, (2018), . doi : 10.6092/issn.2240-2829/7795. http://gdmltest.u-ga.fr/item/7795/