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 $Ø$ 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 $Ø$ is a centered ball, the symmetric hole is critical when we consider deformation that preserve volume but is not optimal for some case.

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