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

Abstract

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.

13 pages

Cited by in corpus (1)

• Optimal boundary holes for the Sobolev trace constant