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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: jrossi@dm.uba.ar
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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

[1] Adimurthi, and Yadava, S. L., Positive solution for Neumann problem with critical non linearity on boundary. Comm. Partial Differential Equations, 16(1991), no. 11, 1733–1760.Google Scholar
[2] Aubin, T., Équations différentielles non linéaires et le problème de Yamabe concernant la courbure scalaire. J. Math. Pures Appl. 55(1976), no. 3, 269–296.Google Scholar
[3] Biezuner, R. J., Best constants in Sobolev trace inequalities. Nonlinear Anal. 54(2003), no. 3, 575–589.Google Scholar
[4] Druet, O. and Hebey, E., The AB program in geometric analysis: sharp Sobolev inequalities and related problems. Mem. Amer. Math. Soc. 160(2002), no. 761.Google Scholar
[5] del Pino, M. and Flores, C., Asymptotic behavior of best constants and extremals for trace embeddings in expanding domains. Comm. Partial Differential Equations 26(2001), no. 11-12, 2189–2210.Google Scholar
[6] Escobar, J. F., Sharp constant in a Sobolev trace inequality. Indiana Math. J. 37(1988), no. 3, 687–698.Google Scholar
[7] Bonder, J. Fernández, Dozo, E. Lami and Rossi, J. D., Symmetry properties for the extremals of the Sobolev trace embedding. Ann. Inst. H. Poincaré. Anal. Non Linéaire 21(2004), no. 6, 795–805.Google Scholar
[8] Bonder, J. Fernández and Rossi, J. D., Asymptotic behavior of the best Sobolev trace constant in expanding and contracting domains. Comm. Pure Appl. Anal. 1(2002), no. 3, 359–378.Google Scholar
[9] Bonder, J. Fernández and Rossi, J. D., On the existence of extremals for the Sobolev trace embedding theorem with critical exponent. Bull. London Math. Soc. 37(2005), no. 1, 119–125.Google Scholar
[10] García-Meli án, J. and de Lis, J. Sabina, On the perturbation of eigenvalues for the p-Laplacian. C. R. Acad. Sci. Paris Sér I Math 332(2001), no. 10, 893–898.Google Scholar
[11] Henrot, A.,Minimization problems for eigenvalues of the Laplacian. J. Evol. Equ. 3(2003), no. 3, 443–461.Google Scholar
[12] Henrot, A. and Pierre, M., Variation et optimization de formes : une analyse geometrique. Mathématiques et Applications 48, Springer, New York, 2005.Google Scholar
[13] Kato, T., Perturbation theory for linear operators. Springer-Verlag, Berlin, 1995.Google Scholar
[14] Dozo, E. Lami and Torne, O., Symmetry and symmetry breaking for minimizers in the trace inequality. Commun. Contemp. Math. 7(2005), no. 6, 727–746.Google Scholar
[15] Li, Y. and Zhu, M., Sharp Sobolev trace inequalities on Riemannian manifolds with boundaries. Comm. Pure Appl. Math. 50(1997), no. 5, 449–487.Google Scholar
[16] Martinez, S. and Rossi, J. D., Isolation and simplicity for the first eigenvalue of the p-laplacian with a nonlinear boundary condition. Abstr. Appl. Anal. 7(2002), no. 5, 287–293.Google Scholar