Optimization problem for extremals of the trace inequality in domains with holes

We study the Sobolev trace constant for functions defined in a bounded domain $\O$ that vanish in the subset $A.$ We find a formula for the first variation of the Sobolev trace with respect to hole. As a consequence of this formula, we prove that when $\O$ is a centered ball, the symmetric hole is critical when we consider deformation that preserve volume but is not optimal for some case.