A partially overdetermined problem in a half ball

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