On the energy critical Schrödinger equation in 3D non-trapping domains
Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 5, pp. 1153-1177.

We prove that the quintic Schrödinger equation with Dirichlet boundary conditions is locally well posed for H 0 1 (Ω) data on any smooth, non-trapping domain Ω 3 . The key ingredient is a smoothing effect in L x 5 (L t 2 ) for the linear equation. We also derive scattering results for the whole range of defocusing sub quintic Schrödinger equations outside a star-shaped domain.

@article{AIHPC_2010__27_5_1153_0,
     author = {Ivanovici, Oana and Planchon, Fabrice},
     title = {On the energy critical {Schr\"odinger} equation in {3\protect\emph{D}} non-trapping domains},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1153--1177},
     publisher = {Elsevier},
     volume = {27},
     number = {5},
     year = {2010},
     doi = {10.1016/j.anihpc.2010.04.001},
     mrnumber = {2683754},
     zbl = {1200.35066},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2010.04.001/}
}
TY  - JOUR
AU  - Ivanovici, Oana
AU  - Planchon, Fabrice
TI  - On the energy critical Schrödinger equation in 3D non-trapping domains
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2010
SP  - 1153
EP  - 1177
VL  - 27
IS  - 5
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2010.04.001/
DO  - 10.1016/j.anihpc.2010.04.001
LA  - en
ID  - AIHPC_2010__27_5_1153_0
ER  - 
%0 Journal Article
%A Ivanovici, Oana
%A Planchon, Fabrice
%T On the energy critical Schrödinger equation in 3D non-trapping domains
%J Annales de l'I.H.P. Analyse non linéaire
%D 2010
%P 1153-1177
%V 27
%N 5
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2010.04.001/
%R 10.1016/j.anihpc.2010.04.001
%G en
%F AIHPC_2010__27_5_1153_0
Ivanovici, Oana; Planchon, Fabrice. On the energy critical Schrödinger equation in 3D non-trapping domains. Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 5, pp. 1153-1177. doi : 10.1016/j.anihpc.2010.04.001. http://www.numdam.org/articles/10.1016/j.anihpc.2010.04.001/

[1] Ramona Anton, Global existence for defocusing cubic NLS and Gross–Pitaevskii equations in three dimensional exterior domains, J. Math. Pures Appl. (9) 89 no. 4 (2008), 335-354 | MR | Zbl

[2] N. Burq, P. Gérard, N. Tzvetkov, On nonlinear Schrödinger equations in exterior domains, Ann. Inst. H. Poincaré Anal. Non Linéaire 21 no. 3 (2004), 295-318 | EuDML | Numdam | MR | Zbl

[3] Nicolas Burq, Estimations de Strichartz pour des perturbations à longue portée de l'opérateur de Schrödinger, Séminaire: Équations aux Dérivées Partielles, 2001–2002, Sémin. Équ. Dériv. Partielles, École Polytech., Palaiseau (2002) | EuDML

[4] Nicolas Burq, Gilles Lebeau, Fabrice Planchon, Global existence for energy critical waves in 3-D domains, J. Amer. Math. Soc. 21 no. 3 (2008), 831-845 | MR | Zbl

[5] Nicolas Burq, Fabrice Planchon, Smoothing and dispersive estimates for 1D Schrödinger equations with BV coefficients and applications, J. Funct. Anal. 236 no. 1 (2006), 265-298 | MR | Zbl

[6] Nicolas Burq, Fabrice Planchon, Global existence for energy critical waves in 3-D domains: Neumann boundary conditions, Amer. J. Math. 131 no. 6 (2009), 1715-1742 | MR | Zbl

[7] Thierry Cazenave, Fred B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in H s , Nonlinear Anal. 14 no. 10 (1990), 807-836 | MR | Zbl

[8] Michael Christ, Alexander Kiselev, Maximal functions associated to filtrations, J. Funct. Anal. 179 no. 2 (2001), 409-425 | Zbl

[9] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao, Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in 3 , Ann. of Math. (2) 167 no. 3 (2008), 767-865 | Zbl

[10] J. Ginibre, G. Velo, The global Cauchy problem for the nonlinear Schrödinger equation revisited, Ann. Inst. H. Poincaré Anal. Non Linéaire 2 no. 4 (1985), 309-327 | EuDML | Numdam | Zbl

[11] Oana Ivanovici, Precise smoothing effect in the exterior of balls, Asymptot. Anal. 53 no. 4 (2007), 189-208 | Zbl

[12] Oana Ivanovici, Counter example to Strichartz estimates for the wave equation in domains, Math. Ann. 347 (2010), 627-673, http://dx.doi.org/10.1007/s00208-009-0454-1 | Zbl

[13] Oana Ivanovici, On the Schrodinger equation outside strictly convex obstacles, arXiv:0809.1060 [math.AP] (2008) | Zbl

[14] Oana Ivanovici, Fabrice Planchon, Square function and heat flow estimates on domains, arXiv:0812.2733 [math.AP] (2008)

[15] Fabrice Planchon, Dispersive estimates and the 2D cubic NLS equation, J. Anal. Math. 86 (2002), 319-334 | Zbl

[16] Fabrice Planchon, Luis Vega, Bilinear virial identities and applications, Ann. Sci. École. Norm. Sup. 42 (2009), 261-290 | EuDML | Numdam | Zbl

[17] Hart F. Smith, Christopher D. Sogge, On the critical semilinear wave equation outside convex obstacles, J. Amer. Math. Soc. 8 no. 4 (1995), 879-916 | Zbl

[18] Hart F. Smith, Christopher D. Sogge, On the L p norm of spectral clusters for compact manifolds with boundary, Acta Math. 198 no. 1 (2007), 107-153 | Zbl

[19] Gigliola Staffilani, Daniel Tataru, Strichartz estimates for a Schrödinger operator with nonsmooth coefficients, Comm. Partial Differential Equations 27 no. 7–8 (2002), 1337-1372 | Zbl

Cité par Sources :