  In this paper we study the problem of minimizing the Sobolev trace Rayleigh
quotient $\|u\|_{W^{1,p}(\Omega)}^p / \|u\|_{L^q(\partial\Omega)}^p$ among
functions that vanish in a set contained on the boundary $\partial\Omega$ 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.
