American Institute of Mathematical Sciences

Advanced
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 - A, 2013, 33 (9) : 3861-3884. doi: 10.3934/dcds.2013.33.3861
References:
[1]

Robert A. Adams, "Sobolev Spaces,", Pure and Applied Mathematics, (1975).   Google Scholar

[2]

Ramona Anton, Cubic nonlinear Schrödinger equation on three dimensional balls with radial data,, Comm. Partial Differential Equations, 33 (2008), 1862.  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.   Google Scholar

[4]

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

[5]

H. Brézis and T. Gallouet, Nonlinear Schrödinger evolution equations,, Nonlinear Anal., 4 (1980), 677.  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.  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.  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, (2003).   Google Scholar

[9]

David Gilbarg and Neil S. Trudinger, "Elliptic partial Differential Equations of Second Order,", Reprint of the 1998 edition, (1998).   Google Scholar

[10]

Heinrich W. Guggenheimer, "Differential Geometry,", Corrected reprint of the 1963 edition, (1963).   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.   Google Scholar

[12]

Lars Hörmander, "The Analysis of Linear Partial Differential Operators. III, Pseudo-Differential Operators,", Reprint of the 1994 edition, (1994).   Google Scholar

[13]

Oana Ivanovici, On the Schrödinger equation outside strictly convex obstacles,, Anal. PDE, 3 (2010), 261.  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.  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.  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.   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.   Google Scholar

[18]

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

[19]

Felipe Linares and Gustavo Ponce, "Introduction to Nonlinear Dispersive Equations,", Universitext, (2009).   Google Scholar

[20]

J.-L. Lions and E. Magenes, "Problèmes aux Limites Non Homogènes et Applications. Vol. 1,", Travaux et Recherches Mathématiques, (1968).   Google Scholar

[21]

J.-L. Lions and E. Magenes, "Problèmes aux Limites Non Homogènes et Applications. Vol. 2,", Travaux et Recherches Mathématiques, (1968).   Google Scholar

[22]

Elaine Machtyngier, Exact controllability for the Schrödinger equation,, SIAM J. Control Optim., 32 (1994), 24.  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.  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.   Google Scholar

[25]

Lionel Rosier and Bing-Yu Zhang, Exact boundary controllability of the nonlinear Schrödinger equation,, J. Differential Equations, 246 (2009), 4129.  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.  doi: 10.1081/PDE-120005841.  Google Scholar

[27]

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

[28]

Hans Triebel, "Theory of Function Spaces," Monographs in Mathematics, 78,, Birkhäuser Verlag, (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.  doi: 10.1016/0362-546X(89)90094-1.  Google Scholar

show all references

References:
[1]

Robert A. Adams, "Sobolev Spaces,", Pure and Applied Mathematics, (1975).   Google Scholar

[2]

Ramona Anton, Cubic nonlinear Schrödinger equation on three dimensional balls with radial data,, Comm. Partial Differential Equations, 33 (2008), 1862.  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.   Google Scholar

[4]

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

[5]

H. Brézis and T. Gallouet, Nonlinear Schrödinger evolution equations,, Nonlinear Anal., 4 (1980), 677.  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.  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.  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, (2003).   Google Scholar

[9]

David Gilbarg and Neil S. Trudinger, "Elliptic partial Differential Equations of Second Order,", Reprint of the 1998 edition, (1998).   Google Scholar

[10]

Heinrich W. Guggenheimer, "Differential Geometry,", Corrected reprint of the 1963 edition, (1963).   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.   Google Scholar

[12]

Lars Hörmander, "The Analysis of Linear Partial Differential Operators. III, Pseudo-Differential Operators,", Reprint of the 1994 edition, (1994).   Google Scholar

[13]

Oana Ivanovici, On the Schrödinger equation outside strictly convex obstacles,, Anal. PDE, 3 (2010), 261.  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.  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.  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.   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.   Google Scholar

[18]

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

[19]

Felipe Linares and Gustavo Ponce, "Introduction to Nonlinear Dispersive Equations,", Universitext, (2009).   Google Scholar

[20]

J.-L. Lions and E. Magenes, "Problèmes aux Limites Non Homogènes et Applications. Vol. 1,", Travaux et Recherches Mathématiques, (1968).   Google Scholar

[21]

J.-L. Lions and E. Magenes, "Problèmes aux Limites Non Homogènes et Applications. Vol. 2,", Travaux et Recherches Mathématiques, (1968).   Google Scholar

[22]

Elaine Machtyngier, Exact controllability for the Schrödinger equation,, SIAM J. Control Optim., 32 (1994), 24.  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.  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.   Google Scholar

[25]

Lionel Rosier and Bing-Yu Zhang, Exact boundary controllability of the nonlinear Schrödinger equation,, J. Differential Equations, 246 (2009), 4129.  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.  doi: 10.1081/PDE-120005841.  Google Scholar

[27]

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

[28]

Hans Triebel, "Theory of Function Spaces," Monographs in Mathematics, 78,, Birkhäuser Verlag, (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.  doi: 10.1016/0362-546X(89)90094-1.  Google Scholar

[1]

Shenghao Li, Min Chen, Bing-Yu Zhang. A non-homogeneous boundary value problem of the sixth order Boussinesq equation in a quarter plane. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2505-2525. doi: 10.3934/dcds.2018104
[2]

Angelo Favini, Rabah Labbas, Stéphane Maingot, Maëlis Meisner. Boundary value problem for elliptic differential equations in non-commutative cases. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 4967-4990. doi: 10.3934/dcds.2013.33.4967
[3]

Hideo Takaoka. Energy transfer model and large periodic boundary value problem for the quintic nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2020, 40 (11) : 6351-6378. doi: 10.3934/dcds.2020283
[4]

Michael Goldberg. Strichartz estimates for Schrödinger operators with a non-smooth magnetic potential. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 109-118. doi: 10.3934/dcds.2011.31.109
[5]

Niels Jacob, Feng-Yu Wang. Higher order eigenvalues for non-local Schrödinger operators. Communications on Pure & Applied Analysis, 2018, 17 (1) : 191-208. doi: 10.3934/cpaa.2018012
[6]

Nurullah Yilmaz, Ahmet Sahiner. On a new smoothing technique for non-smooth, non-convex optimization. Numerical Algebra, Control & Optimization, 2020, 10 (3) : 317-330. doi: 10.3934/naco.2020004
[7]

M. Burak Erdoǧan, William R. Green. Dispersive estimates for matrix Schrödinger operators in dimension two. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4473-4495. doi: 10.3934/dcds.2013.33.4473
[8]

Hongjie Dong, Doyoon Kim. Schauder estimates for a class of non-local elliptic equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2319-2347. doi: 10.3934/dcds.2013.33.2319
[9]

Nassif Ghoussoub. Superposition of selfdual functionals in non-homogeneous boundary value problems and differential systems. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 187-220. doi: 10.3934/dcds.2008.21.187
[10]

Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825
[11]

Fabio Camilli, Paola Loreti, Naoki Yamada. Systems of convex Hamilton-Jacobi equations with implicit obstacles and the obstacle problem. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1291-1302. doi: 10.3934/cpaa.2009.8.1291
[12]

Giovanna Cerami, Riccardo Molle. On some Schrödinger equations with non regular potential at infinity. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 827-844. doi: 10.3934/dcds.2010.28.827
[13]

Serena Dipierro, Enrico Valdinoci. (Non)local and (non)linear free boundary problems. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 465-476. doi: 10.3934/dcdss.2018025
[14]

Eugenio Montefusco, Benedetta Pellacci, Marco Squassina. Energy convexity estimates for non-degenerate ground states of nonlinear 1D Schrödinger systems. Communications on Pure & Applied Analysis, 2010, 9 (4) : 867-884. doi: 10.3934/cpaa.2010.9.867
[15]

Türker Özsarı, Nermin Yolcu. The initial-boundary value problem for the biharmonic Schrödinger equation on the half-line. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3285-3316. doi: 10.3934/cpaa.2019148
[16]

Shouchuan Hu, Haiyan Wang. Convex solutions of boundary value problem arising from Monge-Ampère equations. Discrete & Continuous Dynamical Systems - A, 2006, 16 (3) : 705-720. doi: 10.3934/dcds.2006.16.705
[17]

Giuseppe Maria Coclite, Lorenzo di Ruvo. A note on the non-homogeneous initial boundary problem for an Ostrovsky-Hunter type equation. Discrete & Continuous Dynamical Systems - S, 2020, 13 (12) : 3357-3389. doi: 10.3934/dcdss.2020236
[18]

C. M. Elliott, B. Gawron, S. Maier-Paape, E. S. Van Vleck. Discrete dynamics for convex and non-convex smoothing functionals in PDE based image restoration. Communications on Pure & Applied Analysis, 2006, 5 (1) : 181-200. doi: 10.3934/cpaa.2006.5.181
[19]

Nakao Hayashi, Pavel I. Naumkin, Patrick-Nicolas Pipolo. Smoothing effects for some derivative nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 1999, 5 (3) : 685-695. doi: 10.3934/dcds.1999.5.685
[20]

Franck Boyer, Pierre Fabrie. Outflow boundary conditions for the incompressible non-homogeneous Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2007, 7 (2) : 219-250. doi: 10.3934/dcdsb.2007.7.219

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (37)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences