Overdetermined problems in groups of Heisenberg type: conjectures and partial results

TL;DR

This paper explores overdetermined boundary value problems in Heisenberg-type groups, proposing conjectures, proving integral identities, and solving specific cases by leveraging invariances and symmetries in sub-Riemannian geometry.

Contribution

It formulates conjectures relating to the Koranyi-Kaplan ball characterization and provides partial solutions by establishing integral identities and exploiting group invariances.

Findings

Proved an integral identity constraining solutions to overdetermined problems.

Identified invariances that enable solving problems with partial symmetry.

Converted sub-Riemannian problems to classical $p$-Laplacian results.

Abstract

In this paper we formulate some conjectures in sub-Riemannian geometry concerning a characterisation of the Koranyi-Kaplan ball in a group of Heisenberg type through the existence of a solution to suitably overdetermined problems. We prove an integral identity that provides a rigidity constraint for one of the two problems. By exploiting some new invariances of these Lie groups, for domains having partial symmetry we solve these problems by converting them to known results for the classical $p$-Laplacian