May  2012, 11(3): 1013-1036. doi: 10.3934/cpaa.2012.11.1013

Carleman estimates for the Schrödinger operator and applications to unique continuation

1. 

Department of Mathematical Sciences, Seoul National University, Seoul 151-747, South Korea

Received  October 2010 Revised  August 2011 Published  December 2012

We extend previously known Carleman estimates [18, 16, 11] for the (time-dependent) Schrödinger operator $i\partial_t+\Delta$ to a wider range for which inhomogeneous Strichartz estimates ([9, 27]) are known to hold. Then we apply them to obtain new results on unique continuation for the Schrödinger equation which include more general classes of potentials. Also, we obtain a unique continuation result for nonlinear Schrödinger equations.
Citation: Ihyeok Seo. Carleman estimates for the Schrödinger operator and applications to unique continuation. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1013-1036. doi: 10.3934/cpaa.2012.11.1013
References:
[1]

J. Bergh and J. Löfström, "Interpolation Spaces. An Introduction,", Springer-Verlag, (1976). Google Scholar

[2]

J. Bourgain, On the compactness of the support of solutions of dispersive equations,, Internat. Math. Res. Notices, (1997), 437. doi: 10.1155/S1073792897000305. Google Scholar

[3]

T. Carleman, Sur un problème d'unicité pour les systèmes d'équations aux derivées partielles à deux variables indépendantes,, Ark. Mat., 26 (1939), 1. Google Scholar

[4]

L. Carleson, On convergence and growth of partial sums of Fourier series,, Acta Math., 116 (1966), 135. doi: 10.1007/BF02392815. Google Scholar

[5]

T. Cazenave and F. B. Weissler, Rapidly decaying solutions of the nonlinear Schrödinger equation,, Comm. Math. Phys., 147 (1992), 75. doi: 10.1007/BF02099529. Google Scholar

[6]

L. Escauriaza, Carleman inequalities and the heat operator,, Duke Math. J., 104 (2000), 113. doi: 10.1215/S0012-7094-00-10415-2. Google Scholar

[7]

L. Escauriaza, C. E. Kenig, G. Ponce and L. Vega, On uniqueness properties of solutions of Schrödinger equations,, Comm. Partial Differential Equations, 31 (2006), 1811. doi: 10.1080/03605300500530446. Google Scholar

[8]

L. Escauriaza and L. Vega, Carleman inequalities and the heat operator. II,, Indiana Univ. Math. J., 50 (2001), 1149. doi: 10.1512/iumj.2001.50.1937. Google Scholar

[9]

D. Foschi, Inhomogeneous Strichartz estimates,, J. Hyperbolic Differ. Equ., 2 (2005), 1. doi: 10.1142/S0219891605000361. Google Scholar

[10]

J. Ginibre and G. Velo, The global Cauchy problem for the nonlinear Schrödinger equation revisited,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 2 (1985), 309. Google Scholar

[11]

A. D. Ionescu and C. E. Kenig, $L^p$ Carleman inequalities and uniqueness of solutions of nonlinear Schrödinger equations,, Acta Math., 193 (2004), 193. doi: 10.1007/BF02392564. Google Scholar

[12]

D. Jerison and C. E. Kenig, Unique continuation and absence of positive eigenvalues of Schrödinger operators,, Ann. of Math., 121 (1985), 463. doi: 10.2307/1971205. Google Scholar

[13]

T. Kato, An $L^{q,r}$-theory for nonlinear Schrödinger equations},, Spectral and scattering theory and applications, (1994), 223. Google Scholar

[14]

M. Keel and T. Tao, Endpoint Strichartz estimates,, Amer. J. Math., 120 (1998), 955. doi: 10.1353/ajm.1998.0039. Google Scholar

[15]

C. E. Kenig and N. Nadirashvili, A counterexample in unique continuation,, Math. Res. Lett., 7 (2000), 625. Google Scholar

[16]

C. E. Kenig, G. Ponce and L. Vega, On unique continuation for nonlinear Schrödinger equations,, Comm. Pure Appl. Math., 56 (2003), 1247. doi: 10.1002/cpa.10094. Google Scholar

[17]

C. E. Kenig, A. Ruiz and C. D. Sogge, Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators,, Duke Math. J., 55 (1987), 329. doi: 10.1215/S0012-7094-87-05518-9. Google Scholar

[18]

C. E. Kenig and C. D. Sogge, A note on unique continuation for Schrödinger's operator,, Proc. Amer. Math. Soc., 103 (1988), 543. doi: 10.2307/2047176. Google Scholar

[19]

H. Koch and D. Tataru, Sharp counterexamples in unique continuation for second order elliptic equations,, J. Reine Angew. Math., 542 (2002), 133. doi: 10.1515/crll.2002.003. Google Scholar

[20]

C. Müller, On the behavior of the solutions of the differential equation $\Delta U=F(x,U)$ in the neighborhood of a point,, Comm. Pure Appl. Math., 7 (1954), 505. doi: 10.1002/cpa.3160070304. Google Scholar

[21]

E. M. Stein, "Singular Integrals and Differentiability Properties of Functions,", Princeton University Press, (1970). Google Scholar

[22]

E. M. Stein, Oscillatory integrals in Fourier analysis,, in, (1986), 307. Google Scholar

[23]

E. M. Stein, "Harmonic Analysis. Real-variable Methods, Orthogonality and Oscillatory Integrals,", Princeton University. Press, (1993). Google Scholar

[24]

R. S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations,, Duke Math. J., 44 (1977), 705. doi: 10.1215/S0012-7094-77-04430-1. Google Scholar

[25]

P. A. Tomas, A restriction theorem for the Fourier transform,, Bull. Amer. Math. Soc., 81 (1975), 477. doi: 10.1090/S0002-9904-1975-13790-6. Google Scholar

[26]

H. Triebel, "Interpolation Theory, Function Spaces, Differential operator,", North-Holland, (1978). Google Scholar

[27]

M. C. Vilela, Inhomogeneous Strichartz estimates for the Schrödinger equation,, Trans. Amer. Math. Soc., 359 (2007), 2123. doi: 10.1090/S0002-9947-06-04099-2. Google Scholar

[28]

T. H. Wolff, Note on counterexamples in strong unique continuation problems,, Proc. Amer. Math. Soc., 114 (1992), 351. doi: 10.1090/S0002-9939-1992-1014648-2. Google Scholar

[29]

B. -Y. Zhang, Unique continuation properties of the nonlinear Schrödinger equation,, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 191. doi: 10.1017/S0308210500023581. Google Scholar

show all references

References:
[1]

J. Bergh and J. Löfström, "Interpolation Spaces. An Introduction,", Springer-Verlag, (1976). Google Scholar

[2]

J. Bourgain, On the compactness of the support of solutions of dispersive equations,, Internat. Math. Res. Notices, (1997), 437. doi: 10.1155/S1073792897000305. Google Scholar

[3]

T. Carleman, Sur un problème d'unicité pour les systèmes d'équations aux derivées partielles à deux variables indépendantes,, Ark. Mat., 26 (1939), 1. Google Scholar

[4]

L. Carleson, On convergence and growth of partial sums of Fourier series,, Acta Math., 116 (1966), 135. doi: 10.1007/BF02392815. Google Scholar

[5]

T. Cazenave and F. B. Weissler, Rapidly decaying solutions of the nonlinear Schrödinger equation,, Comm. Math. Phys., 147 (1992), 75. doi: 10.1007/BF02099529. Google Scholar

[6]

L. Escauriaza, Carleman inequalities and the heat operator,, Duke Math. J., 104 (2000), 113. doi: 10.1215/S0012-7094-00-10415-2. Google Scholar

[7]

L. Escauriaza, C. E. Kenig, G. Ponce and L. Vega, On uniqueness properties of solutions of Schrödinger equations,, Comm. Partial Differential Equations, 31 (2006), 1811. doi: 10.1080/03605300500530446. Google Scholar

[8]

L. Escauriaza and L. Vega, Carleman inequalities and the heat operator. II,, Indiana Univ. Math. J., 50 (2001), 1149. doi: 10.1512/iumj.2001.50.1937. Google Scholar

[9]

D. Foschi, Inhomogeneous Strichartz estimates,, J. Hyperbolic Differ. Equ., 2 (2005), 1. doi: 10.1142/S0219891605000361. Google Scholar

[10]

J. Ginibre and G. Velo, The global Cauchy problem for the nonlinear Schrödinger equation revisited,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 2 (1985), 309. Google Scholar

[11]

A. D. Ionescu and C. E. Kenig, $L^p$ Carleman inequalities and uniqueness of solutions of nonlinear Schrödinger equations,, Acta Math., 193 (2004), 193. doi: 10.1007/BF02392564. Google Scholar

[12]

D. Jerison and C. E. Kenig, Unique continuation and absence of positive eigenvalues of Schrödinger operators,, Ann. of Math., 121 (1985), 463. doi: 10.2307/1971205. Google Scholar

[13]

T. Kato, An $L^{q,r}$-theory for nonlinear Schrödinger equations},, Spectral and scattering theory and applications, (1994), 223. Google Scholar

[14]

M. Keel and T. Tao, Endpoint Strichartz estimates,, Amer. J. Math., 120 (1998), 955. doi: 10.1353/ajm.1998.0039. Google Scholar

[15]

C. E. Kenig and N. Nadirashvili, A counterexample in unique continuation,, Math. Res. Lett., 7 (2000), 625. Google Scholar

[16]

C. E. Kenig, G. Ponce and L. Vega, On unique continuation for nonlinear Schrödinger equations,, Comm. Pure Appl. Math., 56 (2003), 1247. doi: 10.1002/cpa.10094. Google Scholar

[17]

C. E. Kenig, A. Ruiz and C. D. Sogge, Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators,, Duke Math. J., 55 (1987), 329. doi: 10.1215/S0012-7094-87-05518-9. Google Scholar

[18]

C. E. Kenig and C. D. Sogge, A note on unique continuation for Schrödinger's operator,, Proc. Amer. Math. Soc., 103 (1988), 543. doi: 10.2307/2047176. Google Scholar

[19]

H. Koch and D. Tataru, Sharp counterexamples in unique continuation for second order elliptic equations,, J. Reine Angew. Math., 542 (2002), 133. doi: 10.1515/crll.2002.003. Google Scholar

[20]

C. Müller, On the behavior of the solutions of the differential equation $\Delta U=F(x,U)$ in the neighborhood of a point,, Comm. Pure Appl. Math., 7 (1954), 505. doi: 10.1002/cpa.3160070304. Google Scholar

[21]

E. M. Stein, "Singular Integrals and Differentiability Properties of Functions,", Princeton University Press, (1970). Google Scholar

[22]

E. M. Stein, Oscillatory integrals in Fourier analysis,, in, (1986), 307. Google Scholar

[23]

E. M. Stein, "Harmonic Analysis. Real-variable Methods, Orthogonality and Oscillatory Integrals,", Princeton University. Press, (1993). Google Scholar

[24]

R. S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations,, Duke Math. J., 44 (1977), 705. doi: 10.1215/S0012-7094-77-04430-1. Google Scholar

[25]

P. A. Tomas, A restriction theorem for the Fourier transform,, Bull. Amer. Math. Soc., 81 (1975), 477. doi: 10.1090/S0002-9904-1975-13790-6. Google Scholar

[26]

H. Triebel, "Interpolation Theory, Function Spaces, Differential operator,", North-Holland, (1978). Google Scholar

[27]

M. C. Vilela, Inhomogeneous Strichartz estimates for the Schrödinger equation,, Trans. Amer. Math. Soc., 359 (2007), 2123. doi: 10.1090/S0002-9947-06-04099-2. Google Scholar

[28]

T. H. Wolff, Note on counterexamples in strong unique continuation problems,, Proc. Amer. Math. Soc., 114 (1992), 351. doi: 10.1090/S0002-9939-1992-1014648-2. Google Scholar

[29]

B. -Y. Zhang, Unique continuation properties of the nonlinear Schrödinger equation,, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 191. doi: 10.1017/S0308210500023581. Google Scholar

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