Estimates of the best Sobolev constant of the embedding of \(\mathrm{BV}(\Omega)\) into \(L^1(\partial \Omega)\) and related shape optimization problems. (English) Zbl 1151.49036
Summary: We find estimates for the optimal constant in the critical Sobolev trace inequality \(\lambda _{1}(\Omega )\| u\| _{L^1(\partial \Omega)}\leq \| u\| _{W^{1,1}}(\Omega)\) that are independent of \(\Omega \). These estimates generalize those of J. Fernández Bonder and N. Saintier, Ann. Mat. Pura Appl. (4) 187, No. 4, 683–704 (2008; Zbl 1223.35131)] 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 \(\lambda _{1}\), and eventually compute the shape derivative of the functional \(\Omega \rightarrow \lambda _{1}(\Omega )\).
We apply our results to prove the existence of an extremal for this embedding. We then study an optimal design problem related to \(\lambda _{1}\), and eventually compute the shape derivative of the functional \(\Omega \rightarrow \lambda _{1}(\Omega )\).
MSC:
| 49Q10 | Optimization of shapes other than minimal surfaces |
| 35P15 | Estimates of eigenvalues in context of PDEs |
| 49Q20 | Variational problems in a geometric measure-theoretic setting |
Keywords:
Sobolev trace embedding; optimal design problems; critical exponents; shape analysis; functions of bounded variations; 1-LaplacianCitations:
Zbl 1223.35131References:
| [1] | Ambrosio, L.; Fusco, N.; Pallara, D., Functions of bounded variations and free discontinuity problems, (Oxford Mathematical Monographs (2000), The Clarendon Press, Oxford University Press: The Clarendon Press, Oxford University Press New York) · Zbl 0957.49001 |
| [2] | Andreu, F.; Mazon, J. M.; Rossi, J. D., The best constant for the Sobolev embedding form \(W^{1, 1}(\Omega)\) into \(L^1(\partial \Omega)\), Nonlinear Anal., 59, 1125-1145 (2004) · Zbl 1108.35067 |
| [3] | Aubin, T., Equations différentielles non-linéaires et le problème de Yamabé concernant la courbure scalaire, J. Math. Pures Appl., 55, 269-296 (1976) · Zbl 0336.53033 |
| [4] | Cherkaev, A.; Cherkaeva, E., Optimal design for uncertain loading condition, (Homogenization. Homogenization, Ser. Adv. Math. Appl. Sci., vol. 50 (1999), World Sci. Publishing: World Sci. Publishing River Edge, NJ), 193-213 · Zbl 1055.74549 |
| [5] | Demengel, F., On some nonlinear equation involving the 1-laplacian and trace map inequalities, Nonlinear Anal., 48, 1151-1163 (2002) · Zbl 1016.49001 |
| [6] | Demengel, F., On some nonlinear partial differential equations involving the 1-Laplacian and critical Sobolev exponent, ESAIM, 4, 667-686 (1999) · Zbl 0939.35070 |
| [7] | Druet, O.; Hebey, E., The \(A B\) program in geometric analysis: Sharp Sobolev inequalities and related problems, Mem. Amer. Math. Soc., 160 (2002) · Zbl 1023.58009 |
| [8] | Evans, L. C.; Gariepy, R. F., Measure theory and fine properties of functions, (Studies in Advanced Math. (1992), CRC Press: CRC Press Ann Harbor) · Zbl 0626.49007 |
| [9] | J. Fernandez Bonder, N. Saintier, Estimates for the Sobolev trace constant with critical exponent and applications, Ann. Mat. Pura. Aplicata (in press); J. Fernandez Bonder, N. Saintier, Estimates for the Sobolev trace constant with critical exponent and applications, Ann. Mat. Pura. Aplicata (in press) · Zbl 1223.35131 |
| [10] | Fernández Bonder, J.; Rossi, J. D.; Wolanski, N., On the best Sobolev trace constant and extremals in domains with holes, Bull. Sci. Math., 130, 565-579 (2006) · Zbl 1115.35046 |
| [11] | Fernández Bonder, J.; Rossi, J. D.; Wolanski, N., Regularity of the free boundary in an optimization problem related to the best Sobolev trace constant, SIAM J. Control Optim., 44, 5, 1614-1635 (2006) · Zbl 1134.35036 |
| [12] | Giusti, E., Minimal surfaces and functions of bounded variation, (Monographs in Mathematics (1984), Birkhäuser) · Zbl 0545.49018 |
| [13] | Hebey, E.; Saintier, N., Stability and perturbations of the domain for the first eigenvalue of the 1-laplacian, Arch. Math., 86, 4, 340-351 (2006) · Zbl 1099.58012 |
| [14] | Henrot, A.; Pierre, M., Variation et optimisation de formes - une analyse géométrique, (Mathématiques et applications, vol. 48 (2005), Springer: Springer Berlin, New York) · Zbl 1098.49001 |
| [15] | Ionescu, I. R.; Lachand-Robert, T., Generalized Cheeger sets related to landslides, Calc. Var. Partial Differential Equations, 23, 227-249 (2005) · Zbl 1062.49036 |
| [16] | Lions, P. L., The concentration-compactness principle in the calculus of variations—the limit case part. 2, Rev. Mat. Iberoamericana, 1, 2, 45-121 (1985) · Zbl 0704.49006 |
| [17] | S. Luckhaus, L. Modica, The Gibbs-Thompson relation within the gradient theory of phase transitions, ARMA 107 (1) 71-83; S. Luckhaus, L. Modica, The Gibbs-Thompson relation within the gradient theory of phase transitions, ARMA 107 (1) 71-83 · Zbl 0681.49012 |
| [18] | V.G. Maz’ya, Sobolev Spaces, in: Springer Series in Soviet Mathematics, Berlin, New York, 1985; V.G. Maz’ya, Sobolev Spaces, in: Springer Series in Soviet Mathematics, Berlin, New York, 1985 |
| [19] | Motron, M., Around the best constants for the Sobolev trace map from \(W^{1, 1}(\Omega)\) into \(L^1(\partial \Omega)\), Asymptot. Anal., 29, 69-90 (2002) · Zbl 1147.46305 |
| [20] | Reshetnyak, Yu. G., Weak convergence of completely additive vector functions on a set, Siberian Math. J., 9, 1039-1045 (1968), Translated from Sibirskii Mathematicheskii Zhurnal 9 (1968) 1386-1394 · Zbl 0176.44402 |
| [21] | N. Saintier, Shape derivative of the first eigenvalue of the 1-Laplacian (submitted for publication); N. Saintier, Shape derivative of the first eigenvalue of the 1-Laplacian (submitted for publication) |
| [22] | Ziemer, W. P., Weakly differentiable functions. Sobolev spaces and functions of bounded variations, (Graduate Texts in Mathematics, vol. 120 (1989), Springer-Verlag) · Zbl 0165.06802 |
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