A partially overdetermined problem in a half ball

TL;DR

This paper characterizes domains in a half ball where a partially overdetermined boundary value problem admits solutions, showing they must be spherical caps intersecting orthogonally, with implications for torsional rigidity.

Contribution

It proves a uniqueness result for solutions to a partially overdetermined problem, identifying spherical caps as the only possible domains under certain conditions.

Findings

Domains admitting solutions are spherical caps intersecting orthogonally.

Stationary points of partially torsional rigidity are spherical caps.

Provides a geometric characterization of solutions in a half ball.

Abstract

In this paper, we study a partially overdetermined mixed boundary value problem in a half ball. We prove that a domain in which this partially overdetermined problem admits a solution if and only if the domain is a spherical cap intersecting $\ss^{n-1}$ orthogonally. As an application, we show a stationary point for a partially torsional rigidity under a volume constraint must be a spherical cap.