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Abstract:

In this paper we find estimates for the optimal constant in the critical Sobolev trace inequality λ1(Ω) ∥u∥L1(∂Ω) ≤ ∥u∥ W1,1(Ω) that are independent of Ω. These estimates generalize those of [J. Fernandez Bonder, N. Saintier, Estimates for the Sobolev trace constant with critical exponent and applications, Ann. Mat. Pura. Aplicata (in press)] concerning the p-Laplacian to the case p = 1. We apply our results to prove the existence of an extremal for this embedding.We then study an optimal design problem related to λ1, and eventually compute the shape derivative of the functional Ω → λ1(Ω) © 2007 Elsevier Ltd. All rights reserved.

Registro:

Documento: Artículo
Título:Estimates of the best Sobolev constant of the embedding of BV (Ω) into L1(∂Ω) and related shape optimization problems
Autor:Saintier, N.
Filiación:Departamento de Matemática, FCEyN UBA, (1428), Buenos Aires, Argentina
Palabras clave:1-Laplacian; Critical exponents; Functions of bounded variations; Optimal design problems; Shape analysis; Sobolev trace embedding; Optimal systems; Optimization; 1-Laplacian; Critical exponent; Functions of bounded variations; Optimal design; Shape analysis; Sobolev; Shape optimization
Año:2008
Volumen:69
Número:8
Página de inicio:2479
Página de fin:2491
DOI: http://dx.doi.org/10.1016/j.na.2007.08.026
Título revista:Nonlinear Analysis, Theory, Methods and Applications
Título revista abreviado:Nonlinear Anal Theory Methods Appl
ISSN:0362546X
CODEN:NOAND
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_0362546X_v69_n8_p2479_Saintier

Referencias:

  • Ambrosio, L., Fusco, N., Pallara, D., Functions of bounded variations and free discontinuity problems (2000) Oxford Mathematical Monographs, , The Clarendon Press, Oxford University Press, New York
  • Andreu, F., Mazon, J.M., Rossi, J.D., The best constant for the Sobolev embedding form W1,1(Ω) into L1(∂Ω) (2004) Nonlinear Anal., 59, pp. 1125-1145
  • Aubin, T., Equations différentielles non-linéaires et le problème de Yamabé concernant la courbure scalaire (1976) J. Math. Pures Appl., 55, pp. 269-296
  • 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
  • Demengel, F., On some nonlinear equation involving the 1-laplacian and trace map inequalities (2002) Nonlinear Anal., 48, pp. 1151-1163
  • Demengel, F., On some nonlinear partial differential equations involving the 1-Laplacian and critical Sobolev exponent (1999) ESAIM, 4, pp. 667-686
  • Druet, O., Hebey, E., The AB program in geometric analysis: Sharp Sobolev inequalities and related problems (2002) Mem. Amer. Math. Soc., 160
  • Evans, L.C., Gariepy, R.F., Measure theory and fine properties of functions (1992) Studies in Advanced Math., , CRC Press, Ann Harbor
  • Fernandez Bonder, J., Saintier, N., Estimates for the Sobolev trace constant with critical exponent and applications Ann. Mat. Pura. Aplicata, , in press
  • Fernández Bonder, J., Rossi, J.D., Wolanski, N., On 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 (2006) SIAM J. Control Optim., 44 (5), pp. 1614-1635
  • Giusti, E., Minimal surfaces and functions of bounded variation (1984) Monographs in Mathematics, , Birkhäuser
  • Hebey, E., Saintier, N., Stability and perturbations of the domain for the first eigenvalue of the 1-laplacian (2006) Arch. Math., 86 (4), pp. 340-351
  • Henrot, A., Pierre, M., Variation et optimisation de formes - une analyse géomé trique (2005) Mathématiques et Applications, 48. , Springer, Berlin, New York
  • Ionescu, I.R., Lachand-Robert, T., Generalized Cheeger sets related to landslides (2005) Calc. Var. Partial Differential Equations, 23, pp. 227-249
  • Lions, P.L., The concentration-compactness principle in the calculus of variations - The limit case part. 2 (1985) Rev. Mat. Iberoamericana, 1 (2), pp. 45-121
  • Luckhaus, S., Modica, L., The Gibbs-Thompson relation within the gradient theory of phase transitions ARMA, 107 (1), pp. 71-83
  • Maz'ya, V.G., Sobolev Spaces (1985) Springer Series in Soviet Mathematics, , Berlin, New York
  • Motron, M., Around the best constants for the Sobolev trace map from W 1,1(Ω) into L1(∂Ω) (2002) Asymptot. Anal., 29, pp. 69-90
  • Reshetnyak, Yu.G., Weak convergence of completely additive vector functions on a set (1968) Siberian Math. J., 9, pp. 1039-1045
  • (1968) Sibirskii Mathematicheskii Zhurnal, 9, pp. 1386-1394. , Translated from
  • Saintier, N., Shape Derivative of the First Eigenvalue of the 1-Laplacian, , submitted for publication
  • Ziemer, W.P., Weakly differentiable functions. Sobolev spaces and functions of bounded variations (1989) Graduate Texts in Mathematics, 120. , Springer-Verlag

Citas:

---------- APA ----------
(2008) . Estimates of the best Sobolev constant of the embedding of BV (Ω) into L1(∂Ω) and related shape optimization problems. Nonlinear Analysis, Theory, Methods and Applications, 69(8), 2479-2491.
http://dx.doi.org/10.1016/j.na.2007.08.026
---------- CHICAGO ----------
Saintier, N. "Estimates of the best Sobolev constant of the embedding of BV (Ω) into L1(∂Ω) and related shape optimization problems" . Nonlinear Analysis, Theory, Methods and Applications 69, no. 8 (2008) : 2479-2491.
http://dx.doi.org/10.1016/j.na.2007.08.026
---------- MLA ----------
Saintier, N. "Estimates of the best Sobolev constant of the embedding of BV (Ω) into L1(∂Ω) and related shape optimization problems" . Nonlinear Analysis, Theory, Methods and Applications, vol. 69, no. 8, 2008, pp. 2479-2491.
http://dx.doi.org/10.1016/j.na.2007.08.026
---------- VANCOUVER ----------
Saintier, N. Estimates of the best Sobolev constant of the embedding of BV (Ω) into L1(∂Ω) and related shape optimization problems. Nonlinear Anal Theory Methods Appl. 2008;69(8):2479-2491.
http://dx.doi.org/10.1016/j.na.2007.08.026