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Lorentz-Morrey regularity for nonlinear elliptic problems with irregular obstacles over Reifenberg flat domains
Schrödinger equations with rough Hamiltonians
1. | Department of Mathematics, University of Torino, via Carlo Alberto 10, 10123 Torino |
2. | Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino |
  Finally we consider nonlinear perturbations of real-analytic type and we prove local wellposedness of the corresponding initial value problem in certain modulation spaces.
References:
[1] |
A. Bényi, K. Gröchenig, K. A. Okoudjou and L. G. Rogers, Unimodular Fourier multipliers for modulation spaces, J. Funct. Anal., 246 (2007), 366-384.
doi: 10.1016/j.jfa.2006.12.019. |
[2] |
A. Bényi and K. A. Okoudjou, Local well-posedness of nonlinear dispersive equations on modulation spaces, Bull. Lond. Math. Soc., 41 (2009), 549-558.
doi: 10.1112/blms/bdp027. |
[3] |
A. Boulkhemair, Remarks on a Wiener type pseudodifferential algebra and Fourier integral operators, Math. Res. Lett., 4 (1997), 53-67.
doi: 10.4310/MRL.1997.v4.n1.a6. |
[4] |
A. Boulkhemair, Estimations $L^2$ precisées pour des integrales oscillantes. Comm. Partial Differential Equations, 22 (1997), 165-184.
doi: 10.1080/03605309708821259. |
[5] |
F. Concetti, G. Garello and J. Toft, Trace ideals for Fourier integral operators with non-smooth symbols II, Osaka J. Math., 47 (2010), 739-786. |
[6] |
E. Cordero, K. Gröchenig and F. Nicola, Approximation of Fourier integral operators by Gabor multipliers, J. Fourier Anal. Appl., 18 (2012), 661-684.
doi: 10.1007/s00041-011-9214-1. |
[7] |
E. Cordero, K. Gröchenig, F. Nicola and L. Rodino, Wiener algebras of Fourier integral operators, J. Math. Pures Appl., 99 (2013), 219-233.
doi: 10.1016/j.matpur.2012.06.012. |
[8] |
E. Cordero, K. Gröchenig, F. Nicola and L. Rodino, Generalized metaplectic operators and the Schrödinger equation with a potential in the Sjöstrand class, J. Math. Phys., 55 (2014), 081506 (17 pages). |
[9] |
E. Cordero and F. Nicola, Remarks on Fourier multipliers and applications to the wave equation, J. Math. Anal. Appl., 353 (2009), 583-591.
doi: 10.1016/j.jmaa.2008.12.027. |
[10] |
E. Cordero and F. Nicola, Metaplectic representation on Wiener amalgam spaces and applications to the Schrödinger equation, J. Funct. Anal., 254 (2008), 506-534.
doi: 10.1016/j.jfa.2007.09.015. |
[11] |
E. Cordero and F. Nicola, Boundedness of Schrödinger type propagators on modulation spaces, J. Fourier Anal. Appl., 16 (2010), 311-339.
doi: 10.1007/s00041-009-9111-z. |
[12] |
E. Cordero, F. Nicola and L. Rodino, Time-frequency analysis of Fourier integral operators, Commun. Pure Appl. Anal., 9 (2010), 1-21.
doi: 10.3934/cpaa.2010.9.1. |
[13] |
E. Cordero, F. Nicola and L. Rodino, Gabor representations of evolution operators, Trans. Amer. Math. Soc., 361 (2009), 6049-6071.
doi: 10.1090/S0002-9947-09-04848-X. |
[14] |
E. Cordero, F. Nicola and L. Rodino, Propagation of the Gabor wave front set for Schrödinger equations with non-smooth potential, Rev. Math. Phys., 27 (2015), 1550001 (33 pages).
doi: 10.1142/S0129055X15500014. |
[15] |
P. D'Ancona, V. Pierfelice and N. Visciglia, Some remarks on the Schroedinger equation with a potential in $L^r_t L^s_x$, Math. Ann., 333 (2005), 271-290.
doi: 10.1007/s00208-005-0672-0. |
[16] |
I. Daubechies, Time-frequency localization operators: A geometric phase space approach, IEEE Trans. Inf. Theory, 34 (1988), 605-612.
doi: 10.1109/18.9761. |
[17] |
H. G. Feichtinger, Modulation spaces on locally compact abelian groups, in Wavelets and their Applications (eds. M. Krishna, R. Radhaand and S. Thangavelu), Chennai, India, Allied Publishers, New Delhi, 2003, 99-140; Updated version of a technical report, University of Vienna, 1983. |
[18] |
K. Gröchenig, Foundations of Time-Frequency Analysis, Applied and Numerical Harmonic Analysis, Birkhäuser Boston, Inc., Boston, MA, 2001.
doi: 10.1007/978-1-4612-0003-1. |
[19] |
K. Gröchenig, Time-frequency analysis of Sjöstrand's class, Rev. Mat. Iberoam., 22 (2006), 703-724.
doi: 10.4171/RMI/471. |
[20] |
K. Gröchenig and Z. Rzeszotnik, Banach algebras of pseudodifferential operators and their almost diagonalization, Ann. Inst. Fourier, 58 (2008), 2279-2314. |
[21] |
F. Herau, Melin-Hörmander inequality in a Wiener type pseudo-differential algebra, Ark. för Mat., 39 (2001), 311-338.
doi: 10.1007/BF02384559. |
[22] |
L. Hörmander, The Analysis of Linear Partial Differential Operators, III, Springer-Verlag, 1985. |
[23] |
K. Kato, M. Kobayashi and S. Ito, Representation of Schrödinger operator of a free particle via short time Fourier transform and its applications, Tohoku Math. J., 64 (2012), 223-231.
doi: 10.2748/tmj/1341249372. |
[24] |
K. Kato, M. Kobayashi and S. Ito, Remark on wave front sets of solutions to Schrödinger equation of a free particle and a harmonic oscillator, SUT J. Math., 47 (2011), 175-183. |
[25] |
K. Kato, M. Kobayashi and S. Ito, Remarks on Wiener Amalgam space type estimates for Schrödinger equation, RIMS Kôkyûroku Bessatsu, B33, Res. Inst. Math. Sci. (RIMS), Kyoto, (2012), 41-48. |
[26] |
K. Kato, M. Kobayashi and S. Ito, Estimates on modulation spaces for Schrödinger evolution operators with quadratic and sub-quadratic potentials, J. Funct. Anal., 266 (2014), 733-753.
doi: 10.1016/j.jfa.2013.08.017. |
[27] |
H. Koch and D. Tataru, Dispersive estimates for principally normal pseudodifferential operators, Comm. Pure Appl. Math., 58 (2005), 217-284.
doi: 10.1002/cpa.20067. |
[28] |
N. Lerner, Metrics on the Phase Space and Non-Selfadjoint Pseudo-Differential Operators, Pseudo-Differential Operators. Theory and Applications, 3, Birkhäuser Verlag, Basel, 2010.
doi: 10.1007/978-3-7643-8510-1. |
[29] |
N. Lerner and Y. Morimoto, On the Fefferman-Phong inequality and a Wiener-type algebra of pseudodifferential operators, Publications of the Research Institute for Mathematical Sciences (Kyoto University), 43 (2007), 329-371. |
[30] |
A. Miyachi, F. Nicola, S. Rivetti, A. Tabacco and N. Tomita, Estimates for unimodular Fourier multipliers on modulation spaces, Proc. Amer. Math. Soc., 137 (2009), 3869-3883.
doi: 10.1090/S0002-9939-09-09968-7. |
[31] |
F. Nicola, Phase space analysis of semilinear parabolic equations, J. Funct. Anal., 267 (2014), 727-743.
doi: 10.1016/j.jfa.2014.05.007. |
[32] |
R. Rochberg and K. Tachizawa, Pseudodifferential operators, Gabor frames, and local trigonometric bases, in Gabor Analysis and Algorithms, Appl. Numer. Harmon. Anal., Birkhäuser Boston, Boston, MA, 1998, 171-192. |
[33] |
M. Ruzhansky, M. Sugimoto and B. Wang, Modulation spaces and nonlinear evolution equations, in Evolution Equations of Hyperbolic and Schrödinger Type, Progress in Mathematics, 301, Birkhäuser, 2012, 267-283.
doi: 10.1007/978-3-0348-0454-7_14. |
[34] |
H. F. Smith, A parametrix construction for wave equations with $C^{1,1}$ coefficients, Ann. Inst. Fourier (Grenoble), 48 (1998), 797-835. |
[35] |
G. Staffilani and D. Tataru, Strichartz estimates for the Schrödinger operator with nonsmooth coefficients, Comm. Partial Differential Equations, 27 (2002), 1337-1372.
doi: 10.1081/PDE-120005841. |
[36] |
J. Sjöstrand, An algebra of pseudodifferential operators, Math. Res. Lett., 1 (1994), 185-192.
doi: 10.4310/MRL.1994.v1.n2.a6. |
[37] |
J. Sjöstrand, Wiener type algebras of pseudodifferential operators, in Séminaire sur les Équations aux Dérivées Partielles, 1994-1995, Exp. No. IV, École Polytech., Palaiseau, 1995, 21pp. |
[38] |
D. Tataru, Phase space transforms and microlocal analysis, in Phase Space Analysis of Partial Differential Equations, Vol. II, Pubbl. Cent. Ric. Mat. Ennio Giorgi, Scuola Norm. Sup., Pisa, 2004, 505-524. |
[39] |
D. Tataru, Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients. III, J. Amer. Math. Soc., 15 (2002), 419-442.
doi: 10.1090/S0894-0347-01-00375-7. |
[40] |
B. Wang, Globally well and ill posedness for non-elliptic derivative Schrödinger equations with small rough data, J. Funct. Anal., 265 (2013), 3009-3052.
doi: 10.1016/j.jfa.2013.08.009. |
[41] |
B. Wang and C. Huang, Frequency-uniform decomposition method for the generalized BO, KdV and NLS equations, J. Differential Equations, 239 (2007), 213-250.
doi: 10.1016/j.jde.2007.04.009. |
[42] |
B. Wang and H. Hudzik, The global Cauchy problem for the NLS and NLKG with small rough data, J. Differential Equations, 232 (2007), 36-73.
doi: 10.1016/j.jde.2006.09.004. |
[43] |
B. Wang, Z. Huo, C. Hao and Z. Guo, Harmonic Analysis Method for Nonlinear Evolution Equations. I, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2011.
doi: 10.1142/9789814360746. |
[44] |
B. Wang, Z. Lifeng and G. Boling, Isometric decomposition operators, function spaces $E^\lambda_p,q$ and applications to nonlinear evolution equations, J. Funct. Anal., 233 (2006), 1-39.
doi: 10.1016/j.jfa.2005.06.018. |
show all references
References:
[1] |
A. Bényi, K. Gröchenig, K. A. Okoudjou and L. G. Rogers, Unimodular Fourier multipliers for modulation spaces, J. Funct. Anal., 246 (2007), 366-384.
doi: 10.1016/j.jfa.2006.12.019. |
[2] |
A. Bényi and K. A. Okoudjou, Local well-posedness of nonlinear dispersive equations on modulation spaces, Bull. Lond. Math. Soc., 41 (2009), 549-558.
doi: 10.1112/blms/bdp027. |
[3] |
A. Boulkhemair, Remarks on a Wiener type pseudodifferential algebra and Fourier integral operators, Math. Res. Lett., 4 (1997), 53-67.
doi: 10.4310/MRL.1997.v4.n1.a6. |
[4] |
A. Boulkhemair, Estimations $L^2$ precisées pour des integrales oscillantes. Comm. Partial Differential Equations, 22 (1997), 165-184.
doi: 10.1080/03605309708821259. |
[5] |
F. Concetti, G. Garello and J. Toft, Trace ideals for Fourier integral operators with non-smooth symbols II, Osaka J. Math., 47 (2010), 739-786. |
[6] |
E. Cordero, K. Gröchenig and F. Nicola, Approximation of Fourier integral operators by Gabor multipliers, J. Fourier Anal. Appl., 18 (2012), 661-684.
doi: 10.1007/s00041-011-9214-1. |
[7] |
E. Cordero, K. Gröchenig, F. Nicola and L. Rodino, Wiener algebras of Fourier integral operators, J. Math. Pures Appl., 99 (2013), 219-233.
doi: 10.1016/j.matpur.2012.06.012. |
[8] |
E. Cordero, K. Gröchenig, F. Nicola and L. Rodino, Generalized metaplectic operators and the Schrödinger equation with a potential in the Sjöstrand class, J. Math. Phys., 55 (2014), 081506 (17 pages). |
[9] |
E. Cordero and F. Nicola, Remarks on Fourier multipliers and applications to the wave equation, J. Math. Anal. Appl., 353 (2009), 583-591.
doi: 10.1016/j.jmaa.2008.12.027. |
[10] |
E. Cordero and F. Nicola, Metaplectic representation on Wiener amalgam spaces and applications to the Schrödinger equation, J. Funct. Anal., 254 (2008), 506-534.
doi: 10.1016/j.jfa.2007.09.015. |
[11] |
E. Cordero and F. Nicola, Boundedness of Schrödinger type propagators on modulation spaces, J. Fourier Anal. Appl., 16 (2010), 311-339.
doi: 10.1007/s00041-009-9111-z. |
[12] |
E. Cordero, F. Nicola and L. Rodino, Time-frequency analysis of Fourier integral operators, Commun. Pure Appl. Anal., 9 (2010), 1-21.
doi: 10.3934/cpaa.2010.9.1. |
[13] |
E. Cordero, F. Nicola and L. Rodino, Gabor representations of evolution operators, Trans. Amer. Math. Soc., 361 (2009), 6049-6071.
doi: 10.1090/S0002-9947-09-04848-X. |
[14] |
E. Cordero, F. Nicola and L. Rodino, Propagation of the Gabor wave front set for Schrödinger equations with non-smooth potential, Rev. Math. Phys., 27 (2015), 1550001 (33 pages).
doi: 10.1142/S0129055X15500014. |
[15] |
P. D'Ancona, V. Pierfelice and N. Visciglia, Some remarks on the Schroedinger equation with a potential in $L^r_t L^s_x$, Math. Ann., 333 (2005), 271-290.
doi: 10.1007/s00208-005-0672-0. |
[16] |
I. Daubechies, Time-frequency localization operators: A geometric phase space approach, IEEE Trans. Inf. Theory, 34 (1988), 605-612.
doi: 10.1109/18.9761. |
[17] |
H. G. Feichtinger, Modulation spaces on locally compact abelian groups, in Wavelets and their Applications (eds. M. Krishna, R. Radhaand and S. Thangavelu), Chennai, India, Allied Publishers, New Delhi, 2003, 99-140; Updated version of a technical report, University of Vienna, 1983. |
[18] |
K. Gröchenig, Foundations of Time-Frequency Analysis, Applied and Numerical Harmonic Analysis, Birkhäuser Boston, Inc., Boston, MA, 2001.
doi: 10.1007/978-1-4612-0003-1. |
[19] |
K. Gröchenig, Time-frequency analysis of Sjöstrand's class, Rev. Mat. Iberoam., 22 (2006), 703-724.
doi: 10.4171/RMI/471. |
[20] |
K. Gröchenig and Z. Rzeszotnik, Banach algebras of pseudodifferential operators and their almost diagonalization, Ann. Inst. Fourier, 58 (2008), 2279-2314. |
[21] |
F. Herau, Melin-Hörmander inequality in a Wiener type pseudo-differential algebra, Ark. för Mat., 39 (2001), 311-338.
doi: 10.1007/BF02384559. |
[22] |
L. Hörmander, The Analysis of Linear Partial Differential Operators, III, Springer-Verlag, 1985. |
[23] |
K. Kato, M. Kobayashi and S. Ito, Representation of Schrödinger operator of a free particle via short time Fourier transform and its applications, Tohoku Math. J., 64 (2012), 223-231.
doi: 10.2748/tmj/1341249372. |
[24] |
K. Kato, M. Kobayashi and S. Ito, Remark on wave front sets of solutions to Schrödinger equation of a free particle and a harmonic oscillator, SUT J. Math., 47 (2011), 175-183. |
[25] |
K. Kato, M. Kobayashi and S. Ito, Remarks on Wiener Amalgam space type estimates for Schrödinger equation, RIMS Kôkyûroku Bessatsu, B33, Res. Inst. Math. Sci. (RIMS), Kyoto, (2012), 41-48. |
[26] |
K. Kato, M. Kobayashi and S. Ito, Estimates on modulation spaces for Schrödinger evolution operators with quadratic and sub-quadratic potentials, J. Funct. Anal., 266 (2014), 733-753.
doi: 10.1016/j.jfa.2013.08.017. |
[27] |
H. Koch and D. Tataru, Dispersive estimates for principally normal pseudodifferential operators, Comm. Pure Appl. Math., 58 (2005), 217-284.
doi: 10.1002/cpa.20067. |
[28] |
N. Lerner, Metrics on the Phase Space and Non-Selfadjoint Pseudo-Differential Operators, Pseudo-Differential Operators. Theory and Applications, 3, Birkhäuser Verlag, Basel, 2010.
doi: 10.1007/978-3-7643-8510-1. |
[29] |
N. Lerner and Y. Morimoto, On the Fefferman-Phong inequality and a Wiener-type algebra of pseudodifferential operators, Publications of the Research Institute for Mathematical Sciences (Kyoto University), 43 (2007), 329-371. |
[30] |
A. Miyachi, F. Nicola, S. Rivetti, A. Tabacco and N. Tomita, Estimates for unimodular Fourier multipliers on modulation spaces, Proc. Amer. Math. Soc., 137 (2009), 3869-3883.
doi: 10.1090/S0002-9939-09-09968-7. |
[31] |
F. Nicola, Phase space analysis of semilinear parabolic equations, J. Funct. Anal., 267 (2014), 727-743.
doi: 10.1016/j.jfa.2014.05.007. |
[32] |
R. Rochberg and K. Tachizawa, Pseudodifferential operators, Gabor frames, and local trigonometric bases, in Gabor Analysis and Algorithms, Appl. Numer. Harmon. Anal., Birkhäuser Boston, Boston, MA, 1998, 171-192. |
[33] |
M. Ruzhansky, M. Sugimoto and B. Wang, Modulation spaces and nonlinear evolution equations, in Evolution Equations of Hyperbolic and Schrödinger Type, Progress in Mathematics, 301, Birkhäuser, 2012, 267-283.
doi: 10.1007/978-3-0348-0454-7_14. |
[34] |
H. F. Smith, A parametrix construction for wave equations with $C^{1,1}$ coefficients, Ann. Inst. Fourier (Grenoble), 48 (1998), 797-835. |
[35] |
G. Staffilani and D. Tataru, Strichartz estimates for the Schrödinger operator with nonsmooth coefficients, Comm. Partial Differential Equations, 27 (2002), 1337-1372.
doi: 10.1081/PDE-120005841. |
[36] |
J. Sjöstrand, An algebra of pseudodifferential operators, Math. Res. Lett., 1 (1994), 185-192.
doi: 10.4310/MRL.1994.v1.n2.a6. |
[37] |
J. Sjöstrand, Wiener type algebras of pseudodifferential operators, in Séminaire sur les Équations aux Dérivées Partielles, 1994-1995, Exp. No. IV, École Polytech., Palaiseau, 1995, 21pp. |
[38] |
D. Tataru, Phase space transforms and microlocal analysis, in Phase Space Analysis of Partial Differential Equations, Vol. II, Pubbl. Cent. Ric. Mat. Ennio Giorgi, Scuola Norm. Sup., Pisa, 2004, 505-524. |
[39] |
D. Tataru, Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients. III, J. Amer. Math. Soc., 15 (2002), 419-442.
doi: 10.1090/S0894-0347-01-00375-7. |
[40] |
B. Wang, Globally well and ill posedness for non-elliptic derivative Schrödinger equations with small rough data, J. Funct. Anal., 265 (2013), 3009-3052.
doi: 10.1016/j.jfa.2013.08.009. |
[41] |
B. Wang and C. Huang, Frequency-uniform decomposition method for the generalized BO, KdV and NLS equations, J. Differential Equations, 239 (2007), 213-250.
doi: 10.1016/j.jde.2007.04.009. |
[42] |
B. Wang and H. Hudzik, The global Cauchy problem for the NLS and NLKG with small rough data, J. Differential Equations, 232 (2007), 36-73.
doi: 10.1016/j.jde.2006.09.004. |
[43] |
B. Wang, Z. Huo, C. Hao and Z. Guo, Harmonic Analysis Method for Nonlinear Evolution Equations. I, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2011.
doi: 10.1142/9789814360746. |
[44] |
B. Wang, Z. Lifeng and G. Boling, Isometric decomposition operators, function spaces $E^\lambda_p,q$ and applications to nonlinear evolution equations, J. Funct. Anal., 233 (2006), 1-39.
doi: 10.1016/j.jfa.2005.06.018. |
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