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

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