Brownian motion with killing and reflection and the "hot--spots" problem

We investigate the “hot–spots” property for the survival time probability of Brownian motion with killing and reflection in planar convex domains whose boundary consists of two curves, one of which is an arc of a circle, intersecting at acute angles. This leads to the “hot–spots” property for the mixed Dirichlet–Neumann eigenvalue problem in the domain with Neumann conditions on one of the curves and Dirichlet conditions on the other