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

Published online by Cambridge University Press:  20 November 2018

Julio D. Rossi
Julio D. Rossi*
Affiliation:
Departamento de Matemática, FCEyN UBA, (1428) Buenos Aires, Argentina e-mail: [email protected]
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Abstract

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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 ${{\mathbb{R}}^{N}}.$ 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

35J6535B33nonlinear boundary conditionsSobolev trace embedding
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 51 , Issue 1 , 01 March 2008 , pp. 140 - 145
Copyright
Copyright © Canadian Mathematical Society 2008

References

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