Optimal potentials for Schrödinger operators

[Potentiels optimaux pour les opérateurs de Schrödinger]

Buttazzo, Giuseppe ; Gerolin, Augusto ; Ruffini, Berardo ; Velichkov, Bozhidar

Journal de l'École polytechnique - Mathématiques, Tome 1 (2014), p. 71-100 / Harvested from Numdam

Résumé

Nous considérons l’opérateur de Schrödinger $-\Delta +V(x)$ sur $H^1_0(\Omega )$, où $\Omega $ est un domaine fixé de $\mathbb{R}^d$. Nous étudions certains problèmes d’optimisation pour lesquels un potentiel optimal $V\ge 0$ doit être déterminé dans une certaine classe admissible et pour certains critères d’optimisation tels que l’énergie ou les valeurs propres de Dirichlet.

Abstract

We consider the Schrödinger operator $-\Delta +V(x)$ on $H^1_0(\Omega )$, where $\Omega $ is a given domain of $\mathbb{R}^d$. Our goal is to study some optimization problems where an optimal potential $V\ge 0$ has to be determined in some suitable admissible classes and for some suitable optimization criteria, like the energy or the Dirichlet eigenvalues.

Détail

Publié le : 2014-01-01
DOI : https://doi.org/10.5802/jep.4
Classification:  49J45,  35J10,  49R05,  35P15,  35J05
Mots clés: Opérateurs de Schrödinger, potentiels optimaux, optimisation spectrale, capacité

Bibliographie

[1] Alt, H. W.; Caffarelli, L. A. Existence and regularity for a minimum problem with free boundary, J. Reine Angew. Math., Tome 325 (1981), pp. 105-144 | MR 618549 | Zbl 0449.35105

[2] Ashbaugh, M. S.; Harrell, E. M. Ii Maximal and minimal eigenvalues and their associated nonlinear equations, J. Math. Phys., Tome 28 (1987) no. 8, pp. 1770-1786 | Article | MR 899179 | Zbl 0628.47032

[3] Briançon, T.; Lamboley, J. Regularity of the optimal shape for the first eigenvalue of the Laplacian with volume and inclusion constraints, Ann. Inst. H. Poincaré Anal. Non Linéaire, Tome 26 (2009) no. 4, pp. 1149-1163 | Article | Numdam | MR 2542718 | Zbl 1194.49059

[4] Bucur, D.; Buttazzo, G. Variational methods in shape optimization problems, Birkhäuser Boston, Inc., Boston, MA, Progress in Nonlinear Differential Equations and their Applications, Tome 65 (2005), pp. viii+216 | MR 2150214 | Zbl 1117.49001

[5] Bucur, D.; Buttazzo, G. On the characterization of the compact embedding of Sobolev spaces, Calc. Var. Partial Differential Equations, Tome 44 (2012) no. 3-4, pp. 455-475 | Article | MR 2915329 | Zbl 1241.49023

[6] Bucur, D.; Buttazzo, G.; Velichkov, B. Spectral optimization problems for potentials and measures (Preprint available at http://cvgmt.sns.it)

[7] Buttazzo, G. Semicontinuity, relaxation and integral representation in the calculus of variations, Longman Scientific & Technical, Harlow, Pitman Research Notes in Mathematics Series, Tome 207 (1989), pp. iv+222 | MR 1020296 | Zbl 0669.49005

[8] Buttazzo, G. Spectral optimization problems, Rev. Mat. Univ. Complut. Madrid, Tome 24 (2011) no. 2, pp. 277-322 | Article | MR 2806346 | Zbl 1226.49038

[9] Buttazzo, G.; Dal Maso, G. Shape optimization for Dirichlet problems: relaxed formulation and optimality conditions, Appl. Math. Optim., Tome 23 (1991) no. 1, pp. 17-49 | Article | MR 1076053 | Zbl 0762.49017

[10] Buttazzo, G.; Dal Maso, G. An existence result for a class of shape optimization problems, Arch. Rational Mech. Anal., Tome 122 (1993) no. 2, pp. 183-195 | Article | MR 1217590 | Zbl 0811.49028

[11] Buttazzo, G.; Varchon, N.; Zoubairi, H. Optimal measures for elliptic problems, Ann. Mat. Pura Appl. (4), Tome 185 (2006) no. 2, pp. 207-221 | Article | MR 2214133 | Zbl 1232.49049

[12] Carlen, E. A.; Frank, R. L.; Lieb, E. H. Stability estimates for the lowest eigenvalue of a Schrödinger operator, Geom. Funct. Anal., Tome 24 (2014) no. 1, pp. 63-84 | Article | MR 3177378

[13] Dal Maso, G. An introduction to $\Gamma $-convergence, Birkhäuser Boston, Inc., Boston, MA, Progress in Nonlinear Differential Equations and their Applications, Tome 8 (1993), pp. xiv+340 | Article | MR 1201152 | Zbl 0816.49001

[14] Dal Maso, G.; Mosco, U. Wiener’s criterion and $\Gamma $-convergence, Appl. Math. Optim., Tome 15 (1987) no. 1, pp. 15-63 | Article | MR 866165 | Zbl 0644.35033

[15] Egnell, H. Extremal properties of the first eigenvalue of a class of elliptic eigenvalue problems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), Tome 14 (1987) no. 1, pp. 1-48 | Numdam | MR 937535 | Zbl 0649.35072

[16] Essén, M. On estimating eigenvalues of a second order linear differential operator, General inequalities, 5 (Oberwolfach, 1986), Birkhäuser, Basel (Internat. Schriftenreihe Numer. Math.) Tome 80 (1987), pp. 347-366 | MR 1018159 | Zbl 0625.34019

[17] Evans, L. C. Partial differential equations, American Mathematical Society, Providence, RI, Graduate Texts in Math., Tome 19 (2010), pp. xxii+749 | MR 2597943 | Zbl 1194.35001

[18] Evans, L. C.; Gariepy, R. F. Measure theory and fine properties of functions, CRC Press, Boca Raton, FL, Studies in Advanced Mathematics (1992), pp. viii+268 | MR 1158660 | Zbl 0804.28001

[19] Gilbarg, D.; Trudinger, N. S. Elliptic partial differential equations of second order, Springer-Verlag, Berlin, Classics in Mathematics (2001), pp. xiv+517 | MR 1814364 | Zbl 1042.35002

[20] Harrell, E. M. Ii Hamiltonian operators with maximal eigenvalues, J. Math. Phys., Tome 25 (1984) no. 1, pp. 48-51 (Erratum: J. Math. Phys. 27 (1986) no. 1, p. 419) | Article | MR 728885 | Zbl 0555.35098

[21] Henrot, A. Minimization problems for eigenvalues of the Laplacian, J. Evol. Equ., Tome 3 (2003) no. 3, pp. 443-461 | Article | MR 2019029 | Zbl 1049.49029

[22] Henrot, A. Extremum problems for eigenvalues of elliptic operators, Birkhäuser Verlag, Basel, Frontiers in Mathematics (2006), pp. x+202 | MR 2251558 | Zbl 1109.35081

[23] Henrot, A.; Pierre, M. Variation et optimisation de formes. Une analyse géométrique, Springer, Berlin, Mathématiques & Applications (Berlin), Tome 48 (2005), pp. xii+334 | MR 2512810 | Zbl 1098.49001

[24] Talenti, G. Estimates for eigenvalues of Sturm-Liouville problems, General inequalities, 4 (Oberwolfach, 1983), Birkhäuser, Basel (Internat. Schriftenreihe Numer. Math.) Tome 71 (1984), pp. 341-350 | MR 821811 | Zbl 0591.34019