ANZIAM J. 49 (2007), no. 2, pp. 213–230.

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

J. Fernández Bonder J. D. Rossi
Departamento de Matemática
FCEyN, Universidad de Buenos Aires
Pabellon I
Ciudad Universitaria (1428)
Buenos Aires
Argentina
jfbonder@dm.uba.ar
Instituto de Matemáticas y Física Fundamental
Consejo Superior de Investigaciones Científicas
Serrano 123
Madrid
Spain

on leave from

Departamento de Matemática
FCEyN UBA (1428)
Buenos Aires
Argentina
jrossi@dm.uba.ar
R. Orive
Departamento de Matemáticas
Universidad Autonoma de Madrid Crta.
Colmenar Viejo km. 15
28049 Madrid
Spain
rafael.orive@uam.es
Received 8 November, 2006

Abstract

In this paper we study the best constant in the Sobolev trace embedding H^1(\Omega ) \hookrightarrow L^q (\partial \Omega ) in a bounded smooth domain for 1 < q\le 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.

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2000 Mathematics Subject Classification: primary 35B27, 35J65; secondary 46E35
(Metadata: XML, RSS, BibTeX) MathSciNet: MR2376???
indicates author for correspondence

References

  1. Adimurthi and S. L. Yadava, “Positive solution for Neumann problem with critical non linearity on boundary”, Comm. Partial Differential Equations 16 (1991) 1733–1760. MR1135918
  2. T. Aubin, “Équations différentielles non linéaires et le problème de Yamabe concernant la courbure scalaire”, J. Math. Pures et Appl. 55 (1976) 269–296. MR431287
  3. J. Garcia Azorero, I. Peral and J. D. Rossi, “A convex-concave problem with a nonlinear boundary condition”, J. Differential Equations 198 (2004) 91–128. MR2037751
  4. R. J. Biezuner, “Best constants in Sobolev trace inequalities”, Nonlinear Analysis 54 (2003) 575–589. MR1978428
  5. J. Fernández Bonder, E. Lami Dozo and J. D. Rossi, “Symmetry properties for the extremals of the Sobolev trace embedding”, Ann. Inst. H. Poincaré. Anal. Non Linéaire 21 (2004) 795–805. MR2097031
  6. J. Fernández Bonder and J. D. Rossi, “On the existence of extremals for the Sobolev trace embedding theorem with critical exponent”, Bull. London Math. Soc. 37 (2005) 119–125. MR2105826
  7. J. Fernández Bonder, J. D. Rossi and N. Wolanski, “Regularity of the free boundary in an optimization problem related to the best Sobolev trace constant”, SIAM J. Control Optim. 44 (2005) 1614–1635. MR2193498
  8. J. Fernández Bonder, J. D. Rossi and N. Wolanski, “Behavior of the best Sobolev trace constant and extremals in domains with holes”, Bull. Sci. Math. 130 (2006) 565–579. MR2261964
  9. P. Cherrier, “Problèmes de Neumann non linéaires sur les variétés Riemanniennes”, J. Funct. Anal. 57 (1984) 154–206. MR749522
  10. D. Cioranescu and F. Murat, “Un terme étrange venu d'ailleurs”, in Nonlinear partial differential equations and their applications. Collège 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, A. Cherkaev et al. eds, Progress in Nonlinear Differential Equations and Their Applications 31, Birkhäuser, Boston (1997) 45–93, (Pitman, Boston, Mass., 1982) 98–138, 389–390. MR0652509
  11. D. Cioranescu and J. Saint Jean Paulin, “Homogenization in open sets with holes”, J. Math. Anal. Appl. 71 (1979) 590–607. MR548785
  12. Doïna Cioranescu and François Murat, “Un terme étrange venu d'ailleurs. II”, in Nonlinear partial differential equations and their applications. Collège de France Seminar, Vol. III (Paris, 1980/1981), Volume 70 of Res. Notes in Math., (Pitman, Boston, Mass., 1982) 154–178, 425–426. MR0670272
  13. C. Conca and P. Donato, “Non homogeneous Neumann problems in domains with small holes”, RAIRO Modél. Math Anal. Numér. 22 (1988) 561–607. MR974289
  14. A. Damlamian and T.-T. Li, “Boundary homogenization for elliptic problems”, J. Math. Pures Appl. 66 (1987) 351–361. MR928268
  15. O. Druet and E. Hebey, “The AB program in geometric analysis: sharp Sobolev inequalities and related problems”, Mem. Amer. Math. Soc. 160 (761) (2002). MR1938183
  16. J. F. Escobar, “Sharp constant in a Sobolev trace inequality”, Indiana Math. J. 37 (1988) 687–698. MR962929
  17. Y. Li and M. Zhu, “Sharp Sobolev trace inequalities on Riemannian manifolds with boundaries”, Comm. Pure Appl. Math. 50 (1997) 449–487. MR1443055
  18. M. Lobo and E. Pérez, “On vibrations of a body with many concentrated masses near the boundary”, Math. Models Methods Appl. Sci. 3 (1993) 249–273. MR1212942
  19. M. Lobo and E. Pérez, “Vibrations of a membrane with many concentrated masses near the boundary”, Math. Models Methods Appl. Sci. 5 (1995) 565–585. MR1347148
  20. F. Murat, “The Neumann sieve”, in Nonlinear variational problems (Isola d'Elba, 1983), Volume 127 of Res. Notes in Math., (Pitman, Boston, MA, 1985) 24–32. MR807534
  21. J. Rauch and M. Taylor, “Potential and scattering theory on wildly perturbed domains”, J. Funct. Anal. 18 (1975) 27–59. MR377303
  22. M. W. Steklov, “Sur les problèmes fondamentaux en physique mathématique”, Ann. Sci. Ecole Norm. Sup. 19 (1902) 455–490.
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