Navegar

Colección

Datos Estadísticas

Artículo

La versión final de este artículo es de uso interno de la institución.
Consulte el artículo en la página del editor
Consulte la política de Acceso Abierto del editor

Abstract:

Let Ω ⊂ ℝN be a bounded, smooth domain. We deal with the best constant of the Sobolev trace embedding W1,p(Ω) Rightwards arrow with hook sign Lq (∂Ω) for functions that vanish in a subset A ⊂ Ω, which we call the hole, i.e. we deal with the minimization problem SA = inf||u||W 1,p(ω)p/||u||Lq(∂ω) for functions that verify u|A = 0. It is known that there exists an optimal hole that minimizes the best constant SA among subsets of Ω of the prescribed volume. In this paper, we look for optimal holes and extremals in thin domains. We find a limit problem (when the thickness of the domain goes to zero), that is a standard Neumann eigenvalue problem with weights and prove that when the domain is contracted to a segment, it is better to concentrate the hole on one side of the domain. © 2008 World Scientific Publishing Company.

Registro:

Documento: Artículo
Título:An optimization problem related to the best Sobolev trace constant in thin domains
Autor:Bonder, J.F.; Rossi, J.D.; SchÖnlieb, C.-B.
Filiación:Departamento de Matemática, FCEyN UBA (1428), Buenos Aires, Argentina
IMDEA Matematicas, C-IX Campus UAM, Madrid, Spain
DAMTP, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, United Kingdom
Faculty of Mathematics, University of Vienna, Nordbergstr. 15, 1090, Vienna, Austria
Palabras clave:Calculus of variations; Optimal design; Sobolev trace embedding
Año:2008
Volumen:10
Número:5
Página de inicio:633
Página de fin:650
DOI: http://dx.doi.org/10.1142/S0219199708002922
Título revista:Communications in Contemporary Mathematics
Título revista abreviado:Commun. Contemp. Math.
ISSN:02191997
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_02191997_v10_n5_p633_Bonder

Referencias:

  • Aubin, T., Équations différentielles non linéaires et le problème de Yamabe concernant la courbure scalaire (1976) J. Math. Pures Appl, 55, pp. 269-296
  • Biezuner, R.J., Best constants in Sobolev trace inequalities (2003) Nonlinear Anal, 54, pp. 575-589
  • Binding, P., Boulton, L., Cepicka, J., Drabek, P., Girg, P., Basic properties of eigen-functions of the p-Laplacian (2006) Proc. Amer. Math. Soc, 134 (12), pp. 3487-3494
  • Cherkaev, A., Cherkaeva, E., Optimal design for uncertain loading condition (1999) Ser. Adv. Math. Appl. Sci, 50, pp. 193-213. , Homogenization, World Sci. Publishing, River Edge, NJ
  • Druet, O., Hebey, E., The AB program in geometric analysis: Sharp Sobolev inequalities and related problems (2002) Mem. Amer. Math. Soc, 160 (761). , viii, 98pp
  • del Pino, M., Flores, C., Asymptotic behavior of best constants and extremals for trace embeddings in expanding domains (2001) Comm. Partial Differential Equations, 26 (11-12), pp. 2189-2210
  • Escobar, J.F., Sharp constant in a Sobolev trace inequality (1988) Indiana Math. J, 37 (3), pp. 687-698
  • Fernández Bonder, J., Lami Dozo, E., Rossi, J.D., Symmetry properties for the extremals of the Sobolev trace embedding (2004) Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (6), pp. 795-805
  • Fernandez Bonder, J., Ferreira, R., Rossi, J.D., Uniform bounds for the best Sobolev trace constant (2003) Adv. Nonlinear Stud, 3 (2), pp. 181-192
  • Fernández Bonder, J., Groisman, P., Rossi, J.D., Optimization of the first Steklov eigenvalue in domains with holes: A shape derivative approach (2007) Ann. Mat. Pura Appl, 186 (2), pp. 341-358
  • Fernandez Bonder, J., Martinez, S., Rossi, J.D., The behavior of the best Sobolev trace constant and extremals in thin domains (2004) J. Differential Equations, 198 (1), pp. 129-148
  • Fernández Bonder, J., Rossi, J.D., Asymptotic behavior of the best Sobolev trace constant in expanding and contracting domains (2002) Comm. Pure Appl. Anal, 1 (3), pp. 359-378
  • Fernández Bonder, J., Rossi, J.D., Wolanski, N., Behavior of the best Sobolev trace constant and extremals in domains with holes (2006) Bull. Sci. Math, 130, pp. 565-579
  • Fernández Bonder, J., Rossi, J.D., Wolanski, N., Regularity of the free boundary in an optimization problem related to the best Sobolev trace constant (2005) SIAM J. Control Optim, 44 (5), pp. 1614-1635
  • Henrot, A., Minimization problems for eigenvalues of the Laplacian (2003) J. Evol. Equ, 3 (3), pp. 443-461
  • Lami Dozo, E., Torne, O., Symmetry and symmetry breaking for minimizers in the trace inequality (2005) Commun. Contemp. Math, 7 (6), pp. 727-756
  • Li, Y., Zhu, M., Sharp Sobolev trace inequalities on Riemannian manifolds with boundaries (1997) Comm. Pure Appl. Math, 50, pp. 449-487
  • Martinez, S., Rossi, J.D., Isolation and simplicity for the first eigenvalue of the p-Laplacian with a nonlinear boundary condition (2002) Abstr. Appl. Anal, 7 (5), pp. 287-293

Citas:

---------- APA ----------
Bonder, J.F., Rossi, J.D. & SchÖnlieb, C.-B. (2008) . An optimization problem related to the best Sobolev trace constant in thin domains. Communications in Contemporary Mathematics, 10(5), 633-650.
http://dx.doi.org/10.1142/S0219199708002922
---------- CHICAGO ----------
Bonder, J.F., Rossi, J.D., SchÖnlieb, C.-B. "An optimization problem related to the best Sobolev trace constant in thin domains" . Communications in Contemporary Mathematics 10, no. 5 (2008) : 633-650.
http://dx.doi.org/10.1142/S0219199708002922
---------- MLA ----------
Bonder, J.F., Rossi, J.D., SchÖnlieb, C.-B. "An optimization problem related to the best Sobolev trace constant in thin domains" . Communications in Contemporary Mathematics, vol. 10, no. 5, 2008, pp. 633-650.
http://dx.doi.org/10.1142/S0219199708002922
---------- VANCOUVER ----------
Bonder, J.F., Rossi, J.D., SchÖnlieb, C.-B. An optimization problem related to the best Sobolev trace constant in thin domains. Commun. Contemp. Math. 2008;10(5):633-650.
http://dx.doi.org/10.1142/S0219199708002922