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

Rossi, Julio D.

Canadian mathematical bulletin, Tome 51 (2008) no. 1, pp. 140-145

Résumé

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.

Détail

DOI : 10.4153/CMB-2008-016-5

Mots-clés : 35J65, 35B33, nonlinear boundary conditions, Sobolev trace embedding

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