THE BEST SOBOLEV TRACE CONSTANT IN PERIODIC MEDIA FOR CRITICAL AND SUBCRITICAL EXPONENTS

BONDER, JULIÁN FERNÁNDEZ ; ORIVE, RAFAEL ; ROSSI, JULIO D.

Glasgow mathematical journal, Tome 51 (2009) no. 3, pp. 619-630

Résumé

In this paper we study homogenisation problems for Sobolev trace embedding H1(Ω) ↪ Lq(∂Ω) in a bounded smooth domain. When q = 2 this leads to a Steklov-like eigenvalue problem. We deal with the best constant of the Sobolev trace embedding in rapidly oscillating periodic media, and we consider H1 and Lq spaces with weights that are periodic in space. We find that extremals for these embeddings converge to a solution of a homogenised limit problem, and the best trace constant converges to a homogenised best trace constant. Our results are in fact more general; we can also consider general operators of the form aɛ(x, ∇u) with non-linear Neumann boundary conditions. In particular, we can deal with the embedding W1,p(Ω) ↪ Lq(∂Ω).

Détail

DOI : 10.1017/S0017089509990048

Mots-clés : 35B27, 35J65, 46E35

Bibliographie

[1] 1.Adimurthi, S. L. and Yadava, S. L., Positive solution for Neumann problem with critical non linearity on boundary, Comm. Partial Diff. Eq. 16 (11) (1991), 1733–1760. Google Scholar | DOI

[2] 2.Allaire, G., Homogenization and two-scale convergence, SIAM J. Math. Anal. 23 (1992), 1482–1518. Google Scholar | DOI

[3] 3.Aubin, T., Équations différentielles non linéaires et le problîme de Yamabe concernant la courbure scalaire, J. Math. Pures Appl. 55 (1976), 269–296. Google Scholar

[4] 4.Baffico, L., Conca, C. and Rajesh, M., Homogenization of a class of nonlinear eigenvalue problems, Proc. R. Soc. Edinb. 136A (2006), 722. Google Scholar

[5] 5.Bensoussan, A., Lions, J. L. and Papanicolaou, G., Asymptotic analysis for periodic structures (North-Holland, Amsterdam, 1978). Google Scholar

[6] 6.Cherrier, P., Problèmes de Neumann non linéaires sur les variétés Riemanniennes, J. Funct. Anal. 57 (1984), 154–206. Google Scholar | DOI

[7] 7.Chiado Piat, V., Maso, G. Dal and Defranceschi, A., G-convergence of monotone operators, Ann. Inst. H. Poincaré 7 (1990), 123–160. Google Scholar | DOI

[8] 8.Cioranescu, D. and Murat, F., A strange term coming from nowhere, in Topics in the mathematical modelling of composite materials (Cherkaev, A. and Kohn, R., Editors), Progress in Nonlinear Differential Equations and Their Applications, vol. 31 (Birkhäuser, Boston, 1997), 45–93. Google Scholar | DOI

[9] 9.Maso, G. Dal, An introduction to Γ-convergence, Progress in Nonlinear Differential Equations and Their Applications, vol. 8 (Birkhäuser, Boston, 1993). Google Scholar | DOI

[10] 10.Druet, O. and Hebey, E., The AB program in geometric analysis: Sharp Sobolev inequalities and related problems, Mem. Am. Math. Soc. 160 (2002),761. Google Scholar

[11] 11.Escobar, J. F., Sharp constant in a Sobolev trace inequality, Indiana Math. J. 37 (3) (1988), 687–698. Google Scholar | DOI

[12] 12.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 (6) (2004), 795–805. Google Scholar | DOI

[13] 13.Bonder, J. Fernández, Orive, R. and Rossi, J. D., The best Sobolev trace constant in domains with holes for critical or subcritical exponents, ANZIAM J. 49 (2007), 213–230. Google Scholar

[14] 14.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 (3) (2002), 359–378. Google Scholar | DOI

[15] 15.Bonder, J. Fernández and Rossi, J. D., On the existence of extremals for the Sobolev trace embedding theorem with critical exponent, Bull. Lond. Math. Soc. 37 (1) (2005), 119–125. Google Scholar | DOI

[16] 16.Li, Y. and Zhu, M., Sharp Sobolev trace inequalities on Riemannian manifolds with boundaries. Comm. Pure Appl. Math. 50 (1997), 449–487. Google Scholar | DOI

[17] 17.Lions, J.-L., Some methods in the mathematical analysis of systems and their control (Kexue Chubanshe, Beijing, Gordon & Breach, New York, 1981). Google Scholar

[18] 18.Murat, F. and Tartar, L., H-convergence, in Topics in the Mathematical Modelling of Composite Materials (Cherkaev, A. and Kohn, R., Editors), Progress in Nonlinear Differential Equations and Their Applications, vol. 31 (Birkhäuser, Boston, 1997), 21–43. Google Scholar | DOI

[19] 19.Nguetseng, G., A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal. 20 (1989), 608–623. Google Scholar | DOI

[20] 20.Steklov, M. W., Sur les problèmes fondamentaux en physique mathématique, Ann. Sci. Ecole Norm. Sup. 19 (1902), 455–490. Google Scholar | DOI