Nous considérons le problème de Cauchy pour l'équation de Schrödinger non linéaire sur un domaine du plan avec des conditions aux limites de Dirichlet. Nous prouvons que le problème est bien posé et qu'il existe une solution globale pour une non linéarité polynomiale défocalisante. La preuve repose sur une inégalité de Strichartz généralisée sur des variétés munies d'une métrique de Lipschitz.
We prove wellposedness of the Cauchy problem for the nonlinear Schrödinger equation for any defocusing power nonlinearity on a domain of the plane with Dirichlet boundary conditions. The main argument is based on a generalized Strichartz inequality on manifolds with Lipschitz metric.
@article{BSMF_2008__136_1_27_0,
author = {Anton, Ramona},
title = {Strichartz inequalities for Lipschitz metrics on manifolds and nonlinear Schr\"odinger equation on domains},
journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
volume = {136},
year = {2008},
pages = {27-65},
doi = {10.24033/bsmf.2548},
mrnumber = {2415335},
zbl = {1157.35100},
language = {en},
url = {http://dml.mathdoc.fr/item/BSMF_2008__136_1_27_0}
}
Anton, Ramona. Strichartz inequalities for Lipschitz metrics on manifolds and nonlinear Schrödinger equation on domains. Bulletin de la Société Mathématique de France, Tome 136 (2008) pp. 27-65. doi : 10.24033/bsmf.2548. http://gdmltest.u-ga.fr/item/BSMF_2008__136_1_27_0/
[1] & - Opérateurs pseudo-différentiels et théorème de Nash-Moser, Savoirs Actuels, InterÉditions, 1991. | Zbl 0791.47044
[2] & - « Équations d'ondes quasilinéaires et estimations de Strichartz », Amer. J. Math. 121 (1999), p. 1337-1377. | MR 1719798 | Zbl 0952.35073
[3] - « Dispersion and Strichartz inequalities for Schrödinger equations with singular coefficients », SIAM J. Math. Anal. 35 (2003), p. 868-883. | MR 2049025 | Zbl 1058.35048
[4] - Global solutions of nonlinear Schrödinger equations, American Mathematical Society Colloquium Publications, vol. 46, American Mathematical Society, 1999. | MR 1691575 | Zbl 0933.35178
[5] & - « Nonlinear Schrödinger evolution equations », Nonlinear Anal. 4 (1980), p. 677-681. | MR 582536 | Zbl 0451.35023
[6] , & - « Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds », Amer. J. Math. 126 (2004), p. 569-605. | MR 2058384 | Zbl 1067.58027
[7] & - « Smoothing and dispersive estimates for 1D Schrödinger equations with BV coefficients and applications », J. Funct. Anal. 236 (2006), p. 265-298. | MR 2227135 | Zbl pre05037260
[8] & - « Concentration and lack of observability of waves in highly heterogeneous media », Arch. Ration. Mech. Anal. 164 (2002), p. 39-72. | MR 1921162 | Zbl 1016.35003
[9] - Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics, vol. 10, New York University Courant Institute of Mathematical Sciences, 2003. | Zbl 1055.35003
[10] & - « The Cauchy problem for the critical nonlinear Schrödinger equation in », Nonlinear Anal. 14 (1990), p. 807-836. | MR 1055532 | Zbl 0706.35127
[11] & - Elliptic partial differential equations of second order, Classics in Mathematics, Springer, 2001, Reprint of the 1998 edition. | MR 1814364 | Zbl 1042.35002
[12] & - « The global Cauchy problem for the nonlinear Schrödinger equation revisited », Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (1985), p. 309-327. | Numdam | MR 801582 | Zbl 0586.35042
[13] -, « Scattering theory in the energy space for a class of nonlinear Schrödinger equations », J. Math. Pures Appl. (9) 64 (1985), p. 363-401. | MR 839728 | Zbl 0535.35069
[14] - The analysis of linear partial differential operators. I, Classics in Mathematics, Springer, 2003. | MR 1996773 | Zbl 1028.35001
[15] - « On nonlinear Schrödinger equations », Ann. Inst. H. Poincaré Phys. Théor. 46 (1987), p. 113-129. | Numdam | MR 877998 | Zbl 0632.35038
[16] & - « Endpoint Strichartz estimates », Amer. J. Math. 120 (1998), p. 955-980. | MR 1646048 | Zbl 0922.35028
[17] & - « Trudinger type inequalities and uniqueness of weak solutions for the nonlinear Schrödinger mixed problem », J. Math. Anal. Appl. 155 (1991), p. 531-540. | MR 1097298 | Zbl 0733.35095
[18] & - Methods of modern mathematical physics. I, second éd., Academic Press Inc. [Harcourt Brace Jovanovich Publishers], 1980, Functional analysis. | MR 751959 | Zbl 0459.46001
[19] - « Dispersion and Strichartz inequalities for the one-dimensional Schrödinger equation with variable coefficients », Int. Math. Res. Not. (2005), p. 687-700. | MR 2146323 | Zbl 1160.35509
[20] & - « bounds for spectral clusters », in Phase Space Analysis of PDEs, Pisa, 2004. | MR 2208882 | Zbl 1122.35064
[21] - A comprehensive introduction to differential geometry. Vol. One, Published by M. Spivak, Brandeis Univ., Waltham, Mass., 1970. | MR 267467 | Zbl 0306.53003
[22] & - « Strichartz estimates for a Schrödinger operator with nonsmooth coefficients », Comm. Partial Differential Equations 27 (2002), p. 1337-1372. | MR 1924470 | Zbl 1010.35015
[23] - « Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations », Duke Math. J. 44 (1977), p. 705-714. | MR 512086 | Zbl 0372.35001
[24] & - The nonlinear Schrödinger equation, Applied Mathematical Sciences, vol. 139, Springer, 1999, Self-focusing and wave collapse. | MR 1696311 | Zbl 0928.35157
[25] - « Strichartz estimates for operators with nonsmooth coefficients and the nonlinear wave equation », Amer. J. Math. 122 (2000), p. 349-376. | MR 1749052 | Zbl 0959.35125
[26] - « On smooth solutions to the initial-boundary value problem for the nonlinear Schrödinger equation in two space dimensions », Nonlinear Anal. 13 (1989), p. 1051-1056. | MR 1013309 | Zbl 0693.35133
[27] - « On the solvability of mixed problem for a nonlinear equation of Schrödinger type », Sov. Math. Dokl. 29 (1984), p. 281-284. | Zbl 0585.35019
[28] - « Existence of solutions for Schrödinger evolution equations », Comm. Math. Phys. 110 (1987), p. 415-426. | MR 891945 | Zbl 0638.35036