In this talk, we propose a partially overdetermined mixed boundary value problem of Serrin type 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 the hemi-sphere orthogonally. As an application, we show a stationary point for a well-defined partially torsional rigidity under a volume constraint must be a spherical cap. An isoperimetric problem for this partially torsional rigidity are left open. This is a joint work with Jinyu Guo at Xiamen.