Optimal boundary holes for the Sobolev trace constant

In this paper we study the problem of minimizing the Sobolev trace Rayleigh quotient $\mathrm{\parallel }u{\mathrm{\parallel }}_{W^{1,p}(\mathrm{\Omega })}^p\mathrm{/}\mathrm{\parallel }u{\mathrm{\parallel }}_{L^q(\mathrm{\partial }\mathrm{\Omega })}^p$ among functions that vanish in a set contained on the boundary $\mathrm{\partial }\mathrm{\Omega }$ of given boundary measure. We prove existence of extremals for this problem, and analyze some particular cases where information about the location of the optimal boundary set can be given. Moreover, we further study the shape derivative of the Sobolev trace constant under regular perturbations of the boundary set.