Localizing estimates of the support of solutions of some nonlinear Schrödinger equations – The stationary case

Bégout, Pascal ; Díaz, Jesús Ildefonso

Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012), p. 35-58 / Harvested from Numdam

Résumé

The main goal of this paper is to study the nature of the support of the solution of suitable nonlinear Schrödinger equations, mainly the compactness of the support and its spatial localization. This question touches the very foundations underlying the derivation of the Schrödinger equation, since it is well-known a solution of a linear Schrödinger equation perturbed by a regular potential never vanishes on a set of positive measure. A fact, which reflects the impossibility of locating the particle. Here we shall prove that if the perturbation involves suitable singular nonlinear terms then the support of the solution is a compact set, and so any estimate on its spatial localization implies very rich information on places not accessible by the particle. Our results are obtained by the application of certain energy methods which connect the compactness of the support with the local vanishing of a suitable “energy function” which satisfies a nonlinear differential inequality with an exponent less than one. The results improve and extend a previous short presentation by the authors published in 2006.

Détail

Publié le : 2012-01-01

MR 2876246 | Zbl 1241.35185

DOI : https://doi.org/10.1016/j.anihpc.2011.09.001
Classification:  35B99,  35A01,  35A02,  35B65,  35J60

Bibliographie

[1] S.N. Antontsev, J.I. Díaz, S. Shmarev, Energy methods for free boundary problems, Applications to Nonlinear PDEs and Fluid Mechanics, Progress in Nonlinear Differential Equations and Their Applications vol. 48, Birkhäuser Boston Inc., Boston, MA (2002) | MR 1858749 | Zbl 0988.35002

[2] P. Bégout, J.I. Díaz, Localizing estimates of the support of solutions of some nonlinear Schrödinger equations – the evolution case, in preparation.

[3] P. Bégout, J.I. Díaz, Self-similar solutions with compactly supported profile of some nonlinear Schrödinger equations, in preparation.

[4] P. Bégout, J.I. Díaz, On a nonlinear Schrödinger equation with a localizing effect, C. R. Math. Acad. Sci. Paris 342 no. 7 (2006), 459-463 | MR 2214595 | Zbl 1094.35113

[5] P. Bégout, V. Torri, Numerical computations of the support of solutions of some localizing stationary nonlinear Schrödinger equations, in preparation.

[6] J.A. Belmonte-Beitia, Varias cuestiones sobre la ecuación de Schrödinger no lineal con coeficientes dependientes del espacio, Bol. Soc. Esp. Mat. Apl. S$ \overrightarrow{\mathrm{e}}$MA 52 (2010), 97-128

[7] H. Brezis, T. Kato, Remarks on the Schrödinger operator with singular complex potentials, J. Math. Pures Appl. (9) 58 no. 2 (1979), 137-151 | MR 539217 | Zbl 0408.35025

[8] R. Carles, C. Gallo, Finite time extinction by nonlinear damping for the Schrödinger equation, Comm. Partial Differential Equations 36 no. 6 (2011), 961-975 | MR 2765425 | Zbl 1228.35040

[9] T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics vol. 10, New York University Courant Institute of Mathematical Sciences, New York (2003) | MR 2002047 | Zbl 1055.35003

[10] T. Cazenave, An Introduction to Semilinear Elliptic Equations, Editora do Instituto de Matemática, Universidade Federal do Rio de Janeiro, Rio de Janeiro (2006)

[11] J.I. Díaz, Nonlinear Partial Differential Equations and Free Boundaries, vol. I, Elliptic Equations, Research Notes in Mathematics vol. 106, Pitman (Advanced Publishing Program), Boston, MA (1985) | MR 853732 | Zbl 0595.35100

[12] J.I. Díaz, L. Véron, Local vanishing properties of solutions of elliptic and parabolic quasilinear equations, Trans. Amer. Math. Soc. 290 no. 2 (1985), 787-814 | MR 792828 | Zbl 0579.35003

[13] I. Ekeland, R. Temam, Convex Analysis and Variational Problems, Classics in Applied Mathematics vol. 28, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1999) | MR 1727362 | Zbl 0939.49002

[14] D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin (2001) | MR 1814364 | Zbl 0691.35001

[15] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics vol. 24, Pitman (Advanced Publishing Program), Boston, MA (1985) | MR 775683 | Zbl 0695.35060

[16] L. Hörmander, Definitions of maximal differential operators, Ark. Mat. 3 (1958), 501-504 | MR 106333 | Zbl 0131.09403

[17] A. Jensen, Propagation estimates for Schrödinger-type operators, Trans. Amer. Math. Soc. 291 no. 1 (1985), 129-144 | MR 797050 | Zbl 0577.35089

[18] E. Kashdan, P. Rosenau, Compactification of nonlinear patterns and waves, Phys. Rev. Lett. 101 no. 26 (2008), 261602

[19] B.J. Lemesurier, Dissipation at singularities of the nonlinear Schrödinger equation through limits of regularisations, Phys. D 138 no. 3–4 (2000), 334-343 | MR 1744635 | Zbl 0953.35132

[20] J.-L. Lions, E. Magenes, Problèmes aux limites non homogènes, II, Ann. Inst. Fourier (Grenoble) 11 (1961), 137-178 | MR 146525 | Zbl 0101.07901

[21] J.-L. Lions, E. Magenes, Problemi ai limiti non omogenei, III, Ann. Sc. Norm. Super. Pisa (3) 15 (1961), 41-103 | Numdam | MR 146526 | Zbl 0115.31401

[22] V. Liskevich, P. Stollmann, Schrödinger operators with singular complex potentials as generators: existence and stability, Semigroup Forum 60 no. 3 (2000), 337-343 | MR 1828819 | Zbl 0961.47027

[23] P. Rosenau, S. Schochet, Compact and almost compact breathers: a bridge between an anharmonic lattice and its continuum limit, Chaos 15 no. 1 (2005), 015111 | MR 2133462 | Zbl 1080.37081

[24] W.A. Strauss, Partial Differential Equations. An Introduction, John Wiley & Sons Inc., New York (1992) | MR 1159712

[25] C. Sulem, P.-L. Sulem, The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse, Applied Mathematical Sciences vol. 139, Springer-Verlag, New York (1999) | MR 1696311 | Zbl 0928.35157