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September  2013, 33(9): 3861-3884. doi: 10.3934/dcds.2013.33.3861

On the non-homogeneous boundary value problem for Schrödinger equations

Corentin Audiard 1,

1. 

UPMC Univ Paris 6, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France

Received  July 2012 Revised  February 2013 Published  March 2013

We study the initial boundary value problem for the Schrödinger equation with non-homogeneous Dirichlet boundary conditions. Special care is devoted to the space where the boundary data belong. When $\Omega$ is the complement of a non-trapping obstacle, well-posedness for boundary data of optimal regularity is obtained by transposition arguments. If $\Omega^c$ is convex, a local smoothing property (similar to the one for the Cauchy problem) is proved, and used to obtain Strichartz estimates. As an application local well-posedness for a class of subcritical non-linear Schrödinger equations is derived.
Keywords: dispersive estimates., local smoothing, non homogeneous boundary value problem, Schrödinger equations, non-trapping and convex obstacles.
Mathematics Subject Classification: Primary: 35Q55; Secondary: 35A01, 35A02, 35B30, 35B4.
Citation: Corentin Audiard. On the non-homogeneous boundary value problem for Schrödinger equations. Discrete & Continuous Dynamical Systems, 2013, 33 (9) : 3861-3884. doi: 10.3934/dcds.2013.33.3861
References:
[1]

Robert A. Adams, "Sobolev Spaces," Pure and Applied Mathematics, Vol. 65, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975.  Google Scholar

[2]

Ramona Anton, Cubic nonlinear Schrödinger equation on three dimensional balls with radial data, Comm. Partial Differential Equations, 33 (2008), 1862-1889. doi: 10.1080/03605300802402591.  Google Scholar

[3]

Ramona Anton, Strichartz inequalities for Lipschitz metrics on manifolds and nonlinear Schrödinger equation on domains, Bull. Soc. Math. France, 136 (2008), 27-65.  Google Scholar

[4]

Corentin Audiard, Non-homogeneous boundary value problems for linear dispersive equations, Comm. Partial Differential Equations, 37 (2012), 1-37. doi: 10.1080/03605302.2011.587492.  Google Scholar

[5]

H. Brézis and T. Gallouet, Nonlinear Schrödinger evolution equations, Nonlinear Anal., 4 (1980), 677-681. doi: 10.1016/0362-546X(80)90068-1.  Google Scholar

[6]

Charles Bu and Walter Strauss, An inhomogeneous boundary value problem for nonlinear Schrödinger equations, J. Differential Equations, 173 (2001), 79-91. doi: 10.1006/jdeq.2000.3871.  Google Scholar

[7]

N. Burq, P. Gérard and N. Tzvetkov, On nonlinear Schrödinger equations in exterior domains, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 295-318. doi: 10.1016/S0294-1449(03)00040-4.  Google Scholar

[8]

Thierry Cazenave, "Semilinear Schrödinger Equations," Courant Lecture Notes in Mathematics, 10, New York University Courant Institute of Mathematical Sciences, New York, 2003. Google Scholar

[9]

David Gilbarg and Neil S. Trudinger, "Elliptic partial Differential Equations of Second Order," Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001.  Google Scholar

[10]

Heinrich W. Guggenheimer, "Differential Geometry," Corrected reprint of the 1963 edition, Dover Books on Advanced Mathematics, Dover Publications, Inc., New York, 1977.  Google Scholar

[11]

Justin Holmer, The initial-boundary-value problem for the 1D nonlinear Schrödinger equation on the half-line, Differential Integral Equations, 18 (2005), 647-668.  Google Scholar

[12]

Lars Hörmander, "The Analysis of Linear Partial Differential Operators. III, Pseudo-Differential Operators," Reprint of the 1994 edition, Classics in Mathematics, Springer, Berlin, 2007.  Google Scholar

[13]

Oana Ivanovici, On the Schrödinger equation outside strictly convex obstacles, Anal. PDE, 3 (2010), 261-293. doi: 10.2140/apde.2010.3.261.  Google Scholar

[14]

Oana Ivanovici and Fabrice Planchon, On the energy critical Schrödinger equation in $3D$ non-trapping domains, Ann. Inst. H. Poincaré Anal. Non Linéaire}, 27 (2010), 1153-1177. doi: 10.1016/j.anihpc.2010.04.001.  Google Scholar

[15]

Tosio Kato, On nonlinear Schrödinger equations. II. $H^s$-solutions and unconditional well-posedness, J. Anal. Math., 67 (1995), 281-306. doi: 10.1007/BF02787794.  Google Scholar

[16]

Carlos E. Kenig, Gustavo Ponce and Luis Vega, Small solutions to nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 10 (1993), 255-288.  Google Scholar

[17]

I. Lasiecka and R. Triggiani, Optimal regularity, exact controllability and uniform stabilization of Schrödinger equations with Dirichlet control, Differential Integral Equations, 5 (1992), 521-535.  Google Scholar

[18]

G. Lebeau, Contrôle de l'équation de Schrödinger, J. Math. Pures Appl. (9), 71 (1992), 267-291.  Google Scholar

[19]

Felipe Linares and Gustavo Ponce, "Introduction to Nonlinear Dispersive Equations," Universitext, Springer, New York, 2009.  Google Scholar

[20]

J.-L. Lions and E. Magenes, "Problèmes aux Limites Non Homogènes et Applications. Vol. 1," Travaux et Recherches Mathématiques, No. 17, Dunod, Paris, 1968.  Google Scholar

[21]

J.-L. Lions and E. Magenes, "Problèmes aux Limites Non Homogènes et Applications. Vol. 2," Travaux et Recherches Mathématiques, No. 18, Dunod, Paris, 1968.  Google Scholar

[22]

Elaine Machtyngier, Exact controllability for the Schrödinger equation, SIAM J. Control Optim., 32 (1994), 24-34. doi: 10.1137/S0363012991223145.  Google Scholar

[23]

Türker Özsarí, Varga K. Kalantarov and Irena Lasiecka, Uniform decay rates for the energy of weakly damped defocusing semilinear Schrödinger equations with inhomogeneous Dirichlet boundary control, J. Differential Equations, 251 (2011), 1841-1863. doi: 10.1016/j.jde.2011.04.003.  Google Scholar

[24]

Fabrice Planchon and Luis Vega, Bilinear virial identities and applications, Ann. Sci. Éc. Norm. Supér. (4), 42 (2009), 261-290.  Google Scholar

[25]

Lionel Rosier and Bing-Yu Zhang, Exact boundary controllability of the nonlinear Schrödinger equation, J. Differential Equations, 246 (2009), 4129-4153. doi: 10.1016/j.jde.2008.11.004.  Google Scholar

[26]

Gigliola Staffilani and Daniel Tataru, Strichartz estimates for a Schrödinger operator with nonsmooth coefficients, Comm. Partial Differential Equations, 27 (2002), 1337-1372. doi: 10.1081/PDE-120005841.  Google Scholar

[27]

D. Tataru, Boundary controllability for conservative PDEs, Appl. Math. Optim., 31 (1995), 257-295. doi: 10.1007/BF01215993.  Google Scholar

[28]

Hans Triebel, "Theory of Function Spaces," Monographs in Mathematics, 78, Birkhäuser Verlag, Basel, 1983. doi: 10.1007/978-3-0346-0416-1.  Google Scholar

[29]

Masayoshi Tsutsumi, On smooth solutions to the initial-boundary value problem for the nonlinear Schrödinger equation in two space dimensions, Nonlinear Anal., 13 (1989), 1051-1056. doi: 10.1016/0362-546X(89)90094-1.  Google Scholar

show all references

References:
[1]

Robert A. Adams, "Sobolev Spaces," Pure and Applied Mathematics, Vol. 65, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975.  Google Scholar

[2]

Ramona Anton, Cubic nonlinear Schrödinger equation on three dimensional balls with radial data, Comm. Partial Differential Equations, 33 (2008), 1862-1889. doi: 10.1080/03605300802402591.  Google Scholar

[3]

Ramona Anton, Strichartz inequalities for Lipschitz metrics on manifolds and nonlinear Schrödinger equation on domains, Bull. Soc. Math. France, 136 (2008), 27-65.  Google Scholar

[4]

Corentin Audiard, Non-homogeneous boundary value problems for linear dispersive equations, Comm. Partial Differential Equations, 37 (2012), 1-37. doi: 10.1080/03605302.2011.587492.  Google Scholar

[5]

H. Brézis and T. Gallouet, Nonlinear Schrödinger evolution equations, Nonlinear Anal., 4 (1980), 677-681. doi: 10.1016/0362-546X(80)90068-1.  Google Scholar

[6]

Charles Bu and Walter Strauss, An inhomogeneous boundary value problem for nonlinear Schrödinger equations, J. Differential Equations, 173 (2001), 79-91. doi: 10.1006/jdeq.2000.3871.  Google Scholar

[7]

N. Burq, P. Gérard and N. Tzvetkov, On nonlinear Schrödinger equations in exterior domains, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 295-318. doi: 10.1016/S0294-1449(03)00040-4.  Google Scholar

[8]

Thierry Cazenave, "Semilinear Schrödinger Equations," Courant Lecture Notes in Mathematics, 10, New York University Courant Institute of Mathematical Sciences, New York, 2003. Google Scholar

[9]

David Gilbarg and Neil S. Trudinger, "Elliptic partial Differential Equations of Second Order," Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001.  Google Scholar

[10]

Heinrich W. Guggenheimer, "Differential Geometry," Corrected reprint of the 1963 edition, Dover Books on Advanced Mathematics, Dover Publications, Inc., New York, 1977.  Google Scholar

[11]

Justin Holmer, The initial-boundary-value problem for the 1D nonlinear Schrödinger equation on the half-line, Differential Integral Equations, 18 (2005), 647-668.  Google Scholar

[12]

Lars Hörmander, "The Analysis of Linear Partial Differential Operators. III, Pseudo-Differential Operators," Reprint of the 1994 edition, Classics in Mathematics, Springer, Berlin, 2007.  Google Scholar

[13]

Oana Ivanovici, On the Schrödinger equation outside strictly convex obstacles, Anal. PDE, 3 (2010), 261-293. doi: 10.2140/apde.2010.3.261.  Google Scholar

[14]

Oana Ivanovici and Fabrice Planchon, On the energy critical Schrödinger equation in $3D$ non-trapping domains, Ann. Inst. H. Poincaré Anal. Non Linéaire}, 27 (2010), 1153-1177. doi: 10.1016/j.anihpc.2010.04.001.  Google Scholar

[15]

Tosio Kato, On nonlinear Schrödinger equations. II. $H^s$-solutions and unconditional well-posedness, J. Anal. Math., 67 (1995), 281-306. doi: 10.1007/BF02787794.  Google Scholar

[16]

Carlos E. Kenig, Gustavo Ponce and Luis Vega, Small solutions to nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 10 (1993), 255-288.  Google Scholar

[17]

I. Lasiecka and R. Triggiani, Optimal regularity, exact controllability and uniform stabilization of Schrödinger equations with Dirichlet control, Differential Integral Equations, 5 (1992), 521-535.  Google Scholar

[18]

G. Lebeau, Contrôle de l'équation de Schrödinger, J. Math. Pures Appl. (9), 71 (1992), 267-291.  Google Scholar

[19]

Felipe Linares and Gustavo Ponce, "Introduction to Nonlinear Dispersive Equations," Universitext, Springer, New York, 2009.  Google Scholar

[20]

J.-L. Lions and E. Magenes, "Problèmes aux Limites Non Homogènes et Applications. Vol. 1," Travaux et Recherches Mathématiques, No. 17, Dunod, Paris, 1968.  Google Scholar

[21]

J.-L. Lions and E. Magenes, "Problèmes aux Limites Non Homogènes et Applications. Vol. 2," Travaux et Recherches Mathématiques, No. 18, Dunod, Paris, 1968.  Google Scholar

[22]

Elaine Machtyngier, Exact controllability for the Schrödinger equation, SIAM J. Control Optim., 32 (1994), 24-34. doi: 10.1137/S0363012991223145.  Google Scholar

[23]

Türker Özsarí, Varga K. Kalantarov and Irena Lasiecka, Uniform decay rates for the energy of weakly damped defocusing semilinear Schrödinger equations with inhomogeneous Dirichlet boundary control, J. Differential Equations, 251 (2011), 1841-1863. doi: 10.1016/j.jde.2011.04.003.  Google Scholar

[24]

Fabrice Planchon and Luis Vega, Bilinear virial identities and applications, Ann. Sci. Éc. Norm. Supér. (4), 42 (2009), 261-290.  Google Scholar

[25]

Lionel Rosier and Bing-Yu Zhang, Exact boundary controllability of the nonlinear Schrödinger equation, J. Differential Equations, 246 (2009), 4129-4153. doi: 10.1016/j.jde.2008.11.004.  Google Scholar

[26]

Gigliola Staffilani and Daniel Tataru, Strichartz estimates for a Schrödinger operator with nonsmooth coefficients, Comm. Partial Differential Equations, 27 (2002), 1337-1372. doi: 10.1081/PDE-120005841.  Google Scholar

[27]

D. Tataru, Boundary controllability for conservative PDEs, Appl. Math. Optim., 31 (1995), 257-295. doi: 10.1007/BF01215993.  Google Scholar

[28]

Hans Triebel, "Theory of Function Spaces," Monographs in Mathematics, 78, Birkhäuser Verlag, Basel, 1983. doi: 10.1007/978-3-0346-0416-1.  Google Scholar

[29]

Masayoshi Tsutsumi, On smooth solutions to the initial-boundary value problem for the nonlinear Schrödinger equation in two space dimensions, Nonlinear Anal., 13 (1989), 1051-1056. doi: 10.1016/0362-546X(89)90094-1.  Google Scholar

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