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

Published:2008-03-01
Printed: Mar 2008
• Julio D. Rossi
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## Abstract

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.
 Keywords: nonlinear boundary conditions, Sobolev trace embedding
 MSC Classifications: 35J65 - Nonlinear boundary value problems for linear elliptic equations 35B33 - Critical exponents

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