no. 3
On nonlinear Schrödinger equations in exterior domains
Annales de l'I.H.P. Analyse non linéaire, Tome 21 (2004) no. 3, pp. 295-318 
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     author = {Burq, N and G\'erard, P and Tzvetkov, N},
     title = {On nonlinear {Schr\"odinger} equations in exterior domains},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {295--318},
     year = {2004},
     publisher = {Elsevier},
     volume = {21},
     number = {3},
     doi = {10.1016/j.anihpc.2003.03.002},
     mrnumber = {2068304},
     zbl = {1061.35126},
     language = {en},
     url = {https://www.numdam.org/articles/10.1016/j.anihpc.2003.03.002/}
}
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Burq, N; Gérard, P; Tzvetkov, N. On nonlinear Schrödinger equations in exterior domains. Annales de l'I.H.P. Analyse non linéaire, Tome 21 (2004) no. 3, pp. 295-318. doi: 10.1016/j.anihpc.2003.03.002

[1] Ben Artzi M, Klainerman S, Decay and regularity for the Schrödinger equation, J. Anal. Math. 58 (1992) 25-37. | Zbl | MR

[2] Bergh J, Löfstrom J, Interpolation Spaces, Springer-Verlag, 1976. | Zbl

[3] Bourgain J, Global Solutions of Nonlinear Schrödinger Equations, Colloq. Publications, American Math. Soc, 1999. | Zbl | MR

[4] Bourgain J, Problems in Hamiltonian PDE's. GAFA 2000 (Tel Aviv, 1999), Geom. Funct. Anal. (Special Volume, Part I) (2000) 32-56. | Zbl | MR

[5] Brézis H, Gallouet T, Nonlinear Schrödinger evolution equations, Nonlinear Anal. 4 (1980) 677-681. | Zbl | MR

[6] Burq N, Contrôle de l'équation des ondes dans des ouverts peu réguliers, Asymptotic Anal. 14 (1997) 157-191. | Zbl | MR

[7] Burq N, Décroissance de l'énergie locale de l'équation des ondes pour le problème extérieur et absence de résonance au voisinage du réel, Acta Math. 180 (1998) 1-29. | Zbl | MR

[8] Burq N, Semi-classical estimates for the resolvent in nontrapping geometries, Internat. Math. Res. Notices (2002) 221-241. | Zbl | MR

[9] N. Burq, P. Gérard, N. Tzvetkov, Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds, Amer. J. Math., in press. | Zbl | MR

[10] Burq N, Gérard P, Tzvetkov N, The Cauchy problem for the nonlinear Schrödinger equation on a compact manifold, Proceedings of the Öresund Symposium on Partial Differential Equations, Lund, May 23-25, 2002, J. Nonlinear Math. Phys. 10 (2003) 12-27, Supplement 1. | MR

[11] Burq N, Gérard P, Tzvetkov N, Two singular dynamics of the nonlinear Schrödinger equation on a plane domain, Geom. Funct. Anal. 13 (2003) 1-19. | Zbl | MR

[12] Burq N, Gérard P, Tzvetkov N, An example of singular dynamics for the nonlinear Schrödinger equation on bounded domains, in: Colombini F, Nishitani T (Eds.), Proceedings of the Conference on Hyperbolic PDEs and Related Topics, Cortona, 2002, submitted for publication. | Zbl | MR

[13] Cazenave T, An introduction to nonlinear Schrödinger equations, second ed., Textos de Métodos Matemáticos 26 (1993).

[14] Constantin P, Saut J.C, Local smoothing properties of dispersive equations, J. Amer. Math. Soc. 1 (1988) 413-439. | Zbl | MR

[15] Constantin P, Saut J.C, Local smoothing properties of Schrödinger equations, Indiana Univ. Math. J. 38 (1989) 791-810. | Zbl | MR

[16] Christ M, Kiselev A, Maximal functions associated to filtrations, J. Funct. Anal. 179 (2001) 409-425. | Zbl | MR

[17] Doi S.I, Smoothing effects of Schrödinger evolution groups on Riemannian manifolds, Duke Math. J. 82 (1996) 679-706. | Zbl | MR

[18] Kato T, On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor. 46 (1987) 113-129. | Zbl | MR | Numdam | EuDML

[19] Kavian O, A remark on the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations, Trans. Amer. Math. Soc. 299 (1987) 193-203. | Zbl | MR

[20] S. Keraani, Étude de quelques régimes asymptotiques de l'équation de Schrödinger, PhD Thesis, Université de Paris-Sud, December 2000.

[21] Lax P, Phillips R, Scattering Theory, Pure Appl. Math., vol. 26, Academic Press, 1989. | Zbl | MR

[22] Lions J.-L, Quelques méthodes de résolution des équations aux dérivées partielles non linéaires, Dunod, Paris, 1969. | Zbl

[23] Melrose R.B, Sjöstrand J, Singularities of boundary value problems I, Comm. Pure Appl. Math. 31 (1978) 593-617. | Zbl | MR

[24] Melrose R.B, Sjöstrand J, Singularities of boundary value problems II, Comm. Pure Appl. Math. 35 (1982) 129-168. | Zbl | MR

[25] Ogawa T, Ozawa T, Trudinger type inequalities and uniqueness of weak solutions for the nonlinear Schrödinger equations, J. Math. Anal. Appl. 155 (1991) 531-540. | Zbl | MR

[26] Segal I, Nonlinear semi-groups, Ann. Math. 78 (1963) 339-364. | Zbl | MR

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

[28] Sjölin P, Regularity of solutions to Schrödinger equations, Duke Math. J. 55 (1987) 699-715. | Zbl | MR

[29] Smith H, Sogge C, On the critical semilinear wave equation outside convex obstacles, J. Amer. Math. Soc. 8 (1995) 879-916. | Zbl | MR

[30] Staffilani G, Tataru D, Strichartz estimates for a Schrödinger operator with nonsmooth coefficients, Comm. Partial Differential Equations 27 (2002) 1337-1372. | Zbl | MR

[31] Taylor M, Partial Differential Equations, Applied Mathematical Sciences, vols. 115, 116, 117, Springer-Verlag, New York, 1996. | Zbl | MR

[32] Tsutsumi M, On smooth solutions to the initial-boundary value problem for the nonlinear Schrödinger equation in two space dimensions, Nonlinear Anal. 13 (1989) 1051-1056. | Zbl | MR

[33] Tsutsumi M, On global solutions to the initial-boundary value problem for the nonlinear Schrödinger equation in exterior domains, Comm. Partial Differential Equations 16 (1991) 885-907. | Zbl | MR

[34] Tsutsumi Y, Global solutions of the nonlinear Schrödinger equation in exterior domains, Comm. Partial Differential Equations 8 (1984) 1337-1374. | Zbl | MR

[35] Vainberg B.R, Asymptotic Methods in Equations of Mathematical Physics, Gordon and Breach, New York, 1988. | Zbl | MR

[36] Vasy A, Zworski M, Semiclassical estimates in asymptotically euclidean scattering, Comm. Math. Phys. 212 (2000) 205-217. | Zbl | MR

[37] Vega L, Schrödinger equations: pointwise convergence to the initial data, Proc. Amer. Math. Soc. 102 (1988) 874-878. | Zbl | MR

[38] Vladimirov M.V, On the solvability of mixed problem for a nonlinear equation of Schrödinger type, Soviet Math. Dokl. 29 (1984) 281-284. | Zbl

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