Geodesic
First Variations of the Best Sobolev Trace Constant with Respect to the Domain
Canadian mathematical bulletin, Tome 51 (2008) no. 1, pp. 140-145

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DOI

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.
DOI : 10.4153/CMB-2008-016-5
Mots-clés : 35J65, 35B33, nonlinear boundary conditions, Sobolev trace embedding 
Rossi, Julio D. First Variations of the Best Sobolev Trace Constant with Respect to the Domain. Canadian mathematical bulletin, Tome 51 (2008) no. 1, pp. 140-145. doi: 10.4153/CMB-2008-016-5
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     title = {First {Variations} of the {Best} {Sobolev} {Trace} {Constant} with {Respect} to the {Domain}},
     journal = {Canadian mathematical bulletin},
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     year = {2008},
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     doi = {10.4153/CMB-2008-016-5},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2008-016-5/}
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