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The best Sobolev trace constant in domains with holes for critical or subcritical exponents

Published online by Cambridge University Press:  17 February 2009

J. Fernandezbonder ,
R. Orive  and
J. D. Rossi
J. Fernandezbonder
Affiliation:
Departamento de Matematica FCEYN Universidad de Buenos Aires Pabellon I Ciudad Universitaria (1428), Buenos Aires Argentina; email: jfbonder@dm.uba.ar.
R. Orive
Affiliation:
Departamento de Matematicas Universidad Autonoma de Madrid Crta Colmenar Viejo km 15 28049Madrid Spain; email: rafael.orive@uam.es.
J. D. Rossi
Affiliation:
Instituto de Matematicas y Fi′sica Fundamental Consejo Superior de Investigaciones Cientfficas Serrano 123 Madrid Spain on leave from Departamento de Matematica FCEyN UBA (1428), Buenos Aires Argentina email: jrossi@dm.uba.ar.
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Abstract

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In this paper we study the best constant in the Sobolev trace embedding H1 (Ω) →Lq(∂Ω) in a bounded smooth domain for 1 < q < 2+ = 2(N - 1)/(N - 2), that is, critical or subcritical q. First, we consider a domain with periodically distributed holes inside which we impose that the involved functions vanish. There exists a critical size of the holes for which the limit problem has an extra term. For sizes larger than critical the best trace constant diverges to infinity and for sizes smaller than critical it converges to the best constant in the domain without holes. Also, we study the problem with the holes located on the boundary of the domain. In this case another critical exists and its extra term appears on the boundary.

Keywords

homogenizationnonlinear boundary conditionsSobolev trace embedding.
Type
Articles
Information
The ANZIAM Journal , Volume 49 , Issue 2 , October 2007 , pp. 213 - 230
Copyright
Copyright © Australian Mathematical Society 2007

References

[1]Adimurthi, and Yadava, S. L., “Positive solution for Neumann problem with critical non linearity on boundary”, Comm Partial Differential Equations 16 (1991) 1733–1760.CrossRefGoogle Scholar
[2]Aubin, T “Equations differentielles non lineaires et le probleme de Yamabe concernant la courbure scalaire”, J Math Pures et Appl. 55 (1976) 269–296.Google Scholar
[3]Azorero, Garcia JPeral, I and Rossi, J. D., “A convex-concave problem with a nonlinear boundary condition”, J Differential Equations 198 (2004) 91–128.CrossRefGoogle Scholar
[4]Biezuner, R. J., “Best constants in Sobolev trace inequalities”, Nonlinear Analysis 54 (2003) 575–589.CrossRefGoogle Scholar
[5]Bonder, Fernandez JLamidozo, E. and Rossi, J. D., “Symmetry properties for the extremals of the Sobolev trace embedding”, Ann Inst H Poincare Anal Non Lineaire 21 (2004) 795–805.CrossRefGoogle Scholar
[6]Bonder, Fernandez J and Rossi, J. D., “On the existence of extremals for the Sobolev trace embedding theorem with critical exponent”, Bull London Math Soc. 37 (2005) 119–125.CrossRefGoogle Scholar
[7]Bonder, Fernandez JRossi, J. D. and Wolanski, N “Regularity of the free boundary in an optimization problem related to the best Sobolev trace constant”, SIAM J Control Optim. 44 (2005) 1614–1635.CrossRefGoogle Scholar
[8]Bonder, Fernandez JRossi, J. D. and Wolanski, N “Behavior of the best Sobolev trace constant and extremals in domains with holes”, Bull Sci Math 130 (2006) 565–579.CrossRefGoogle Scholar
[9]Cherrier, P “Problemes de Neumann non lineaires sur les varietes Riemanniennes”, J Fund Anal. 57 (1984) 154–206.CrossRefGoogle Scholar
[10]Cioranescu, D and Murat, F “Un terme etrange venu d′ailleurs”, in Nonlinear partial differential equations and their applications. College de France Seminar, Vol. II (Paris, 1979/1980), Volume 60 of Res. Notes in Math., English translation: A Strange Term Coming from Nowhere, in Topics in the Mathematical Modelling of Composite Materials, Cherkaev, A et al. eds, Progress in Nonlinear Differential Equations and Their Applications 31, Birkhauser, Boston (1997) 45–93, (Pitman, Boston, Mass., 1982) 98–138, 389–390.Google Scholar
[11]Cioranescu, D and Paulin, Saint Jean J, “Homogenization in open sets with holes”, J Math Anal Appl. 71 (1979) 590–607.CrossRefGoogle Scholar
[12]Cioranescu, Doi′na and Murat, Francois, “Un terme etrange venu d′ailleurs. II”, in Nonlinear partial differential equations and their applications. College de France Seminar, Vol. Ill (Paris, 1980/1981), Volume 70 of Res. Notes in Math., (Pitman, Boston, Mass., 1982) 154–178,425–426.Google Scholar
[13]Conca, C and Donato, P “Non homogeneous Neumann problems in domains with small holes”, RAIRO Model Math Anal Numer. 22 (1988) 561–607.Google Scholar
[14]Damlamian, A and T.-T. Li, “Boundary homogenization for elliptic problems”, J Math Pures Appl. 66(1987) 351–361.Google Scholar
[15]Druet, O and Hebey, E “The AB program in geometric analysis: sharp Sobolev inequalities and related problems”, Mem Amer Math Soc 160 (761) (2002).Google Scholar
[16]Escobar, J. F., “Sharp constant in a Sobolev trace inequality”, Indiana Math J 37 (1988) 687–698.CrossRefGoogle Scholar
[17]Li, Y and Zhu, M “Sharp Sobolev trace inequalities on Riemannian manifolds with boundaries”, Comm Pure Appl Math. 50 (1997) 449–487.Google Scholar
[18]Lobo, M and Perez, E “On vibrations of a body with many concentrated masses near the boundary”, Math Models Methods Appl Sci. 3 (1993) 249–273.Google Scholar
[19]Lobo, M and Perez, E “Vibrations of a membrane with many concentrated masses near the boundary”, Math Models Methods Appl Sci. 5 (1995) 565–585.Google Scholar
[20]Murat, F “The Neumann sieve”, in Nonlinear variationalproblems (Isola d′Elba, 1983), Volume 127 of Res. Notes in Math, (Pitman, Boston, MA, 1985) 24–32.Google Scholar
[21]Rauch, J and Taylor, M “Potential and scattering theory on wildly perturbed domains”, J Fund Anal. 18 (1975) 27–59.CrossRefGoogle Scholar
[22]Steklov, M. W., “Sur les problemes fondamentaux en physique mathematique”, Ann Sci Ecole Norm Sup. 19 (1902) 455–490.Google Scholar