First Variations of the Best Sobolev Trace Constant with Respect to the Domain

http://dx.doi.org/10.4153/CMB-2008-016-5
Canad. Math. Bull. 51(2008), 140-145

  Published:2008-03-01
 Printed: Mar 2008

In this paper we study the best constant of the Sobolev trace embedding $H^{1}(\Omega)\to L^{2}(\partial\Omega)$, where $\Omega$ is a bounded smooth domain in $\RR^N$. We find a formula for the first variation of the best constant with respect to the domain. As a consequence, we prove that the ball is a critical domain when we consider deformations that preserve volume.