Carleman estimates for the Schrödinger operator and applications to unique continuation

Ihyeok Seo

doi:10.3934/cpaa.2012.11.1013

Article Contents

2012, Volume 11, Issue 3: 1013-1036. Doi: 10.3934/cpaa.2012.11.1013

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Carleman estimates for the Schrödinger operator and applications to unique continuation

Ihyeok Seo^1,

1.
Department of Mathematical Sciences, Seoul National University, Seoul 151-747

Received: September 30, 2010

Revised: July 31, 2011

Published: April 2012

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Abstract

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.

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Mathematics Subject Classification: Primary: 35B45, 35B60; Secondary: 35Q40, 35Q55.

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References

[1]	J. Bergh and J. Löfström, "Interpolation Spaces. An Introduction," Springer-Verlag, Berlin-New York, 1976.
[2]	J. Bourgain, On the compactness of the support of solutions of dispersive equations, Internat. Math. Res. Notices, (1997), 437-447.doi: 10.1155/S1073792897000305.
[3]	T. Carleman, Sur un problème d'unicité pour les systèmes d'équations aux derivées partielles à deux variables indépendantes, Ark. Mat., Astr. Fys., 26 (1939), 1-9.
[4]	L. Carleson, On convergence and growth of partial sums of Fourier series, Acta Math., 116 (1966), 135-157.doi: 10.1007/BF02392815.
[5]	T. Cazenave and F. B. Weissler, Rapidly decaying solutions of the nonlinear Schrödinger equation, Comm. Math. Phys., 147 (1992), 75-100.doi: 10.1007/BF02099529.
[6]	L. Escauriaza, Carleman inequalities and the heat operator, Duke Math. J., 104 (2000), 113-127.doi: 10.1215/S0012-7094-00-10415-2.
[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), 1811-1823.doi: 10.1080/03605300500530446.
[8]	L. Escauriaza and L. Vega, Carleman inequalities and the heat operator. II, Indiana Univ. Math. J., 50 (2001), 1149-1169.doi: 10.1512/iumj.2001.50.1937.
[9]	D. Foschi, Inhomogeneous Strichartz estimates, J. Hyperbolic Differ. Equ., 2 (2005), 1-24.doi: 10.1142/S0219891605000361.
[10]	J. Ginibre and G. Velo, The global Cauchy problem for the nonlinear Schrödinger equation revisited, Ann. Inst. H. Poincaré Anal. Non Linéaire, 2 (1985), 309-327.
[11]	A. D. Ionescu and C. E. Kenig, $L^p$ Carleman inequalities and uniqueness of solutions of nonlinear Schrödinger equations, Acta Math., 193 (2004), 193-239.doi: 10.1007/BF02392564.
[12]	D. Jerison and C. E. Kenig, Unique continuation and absence of positive eigenvalues of Schrödinger operators, Ann. of Math., 121 (1985), 463-488.doi: 10.2307/1971205.
[13]	T. Kato, An $L^{q,r}$-theory for nonlinear Schrödinger equations}, Spectral and scattering theory and applications, Adv. Stud. Pure Math., vol. 23, Math. Soc. Japan, Toyko, 1994, pp. 223-238.
[14]	M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.doi: 10.1353/ajm.1998.0039.
[15]	C. E. Kenig and N. Nadirashvili, A counterexample in unique continuation, Math. Res. Lett., 7 (2000), 625-630.
[16]	C. E. Kenig, G. Ponce and L. Vega, On unique continuation for nonlinear Schrödinger equations, Comm. Pure Appl. Math., 56 (2003), 1247-1262.doi: 10.1002/cpa.10094.
[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), 329-347.doi: 10.1215/S0012-7094-87-05518-9.
[18]	C. E. Kenig and C. D. Sogge, A note on unique continuation for Schrödinger's operator, Proc. Amer. Math. Soc., 103 (1988), 543-546.doi: 10.2307/2047176.
[19]	H. Koch and D. Tataru, Sharp counterexamples in unique continuation for second order elliptic equations, J. Reine Angew. Math., 542 (2002), 133-146.doi: 10.1515/crll.2002.003.
[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), 505-515.doi: 10.1002/cpa.3160070304.
[21]	E. M. Stein, "Singular Integrals and Differentiability Properties of Functions," Princeton University Press, Princeton, N.J., 1970.
[22]	E. M. Stein, Oscillatory integrals in Fourier analysis, in "Beijing Lectures in Harmonic Analysis" (ed. E. M. Stein), Princeton University Press, (1986), 307-355.
[23]	E. M. Stein, "Harmonic Analysis. Real-variable Methods, Orthogonality and Oscillatory Integrals," Princeton University. Press, Princeton, N.J., 1993.
[24]	R. S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J., 44 (1977), 705-714.doi: 10.1215/S0012-7094-77-04430-1.
[25]	P. A. Tomas, A restriction theorem for the Fourier transform, Bull. Amer. Math. Soc., 81 (1975), 477-478.doi: 10.1090/S0002-9904-1975-13790-6.
[26]	H. Triebel, "Interpolation Theory, Function Spaces, Differential operator," North-Holland, New York, 1978.
[27]	M. C. Vilela, Inhomogeneous Strichartz estimates for the Schrödinger equation, Trans. Amer. Math. Soc., 359 (2007), 2123-2136.doi: 10.1090/S0002-9947-06-04099-2.
[28]	T. H. Wolff, Note on counterexamples in strong unique continuation problems, Proc. Amer. Math. Soc., 114 (1992), 351-356.doi: 10.1090/S0002-9939-1992-1014648-2.
[29]	B. -Y. Zhang, Unique continuation properties of the nonlinear Schrödinger equation, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 191-205.doi: 10.1017/S0308210500023581.