Symmetry problems for gauge balls in the Heisenberg group

Abstract

In this note we focus on possible characterizations of gauge-symmetric functions in the Heisenberg group. We discuss a family of inverse problems in potential theory relating solid and surface weighted mean-value formulas, and we show a partial solution to such problems. To this aim, we review a uniqueness result for gauge balls obtained with V. Martino in [23] by means of overdetermined problems of Serrin-type. The class of competitor sets we consider enjoys partial symmetries of toric and cylindrical type.