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The unique continuation property for second order evolution PDEs. (English) Zbl 1480.35080

Summary: We present a simple and self-contained approach to establish the unique continuation property for some classical evolution equations of second order in a cylindrical domain. We namely discuss this property for wave, parabolic and Schödinger operators with time-independent principal part. Our method is builds on two-parameter Carleman inequalities combined with unique continuation across a pseudo-convex hypersurface with respect to the space variable. The most results we demonstrate in this work are more or less classical. Some of them are not stated exactly as in their original form.

MSC:

35B60 Continuation and prolongation of solutions to PDEs
35A23 Inequalities applied to PDEs involving derivatives, differential and integral operators, or integrals
35J15 Second-order elliptic equations
35K10 Second-order parabolic equations
35L10 Second-order hyperbolic equations
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[1] Alinhac, S., Non-unicité du problème de Cauchy, Ann. Math. (2), 117, 77-108 (1983) · Zbl 0516.35018
[2] Alinhac, S.; Baouendi, MS, A non uniqueness result for operators of principal type, Math. Z., 220, 561-568 (1995) · Zbl 0851.35003
[3] Baudouin, L., Puel, J.-P.: Détermination du potentiel dans l’équation de Schrödinger à partir de mesures sur une partie du bord. C. R. Math. Acad. Sci. Paris 334(11), 967-972 (2002) · Zbl 1027.35156
[4] Baudouin, L.; Puel, J-P, Uniqueness and stability in an inverse problem for the Schrödinger equation, Inverse Probl., 18, 6, 1537-1554 (2002) · Zbl 1023.35091
[5] Baudouin, L., Puel, J.-P.: Corrigendum: “Uniqueness and stability in an inverse problem for the Schrödinger equation” [Inverse Problems 18 (6) (2002), 1537-1554]. Inverse Probl. 23(3), 1327-1328 (2007)
[6] Bellassoued, M., Choulli, M.: Global logarithmic stability of the Cauchy problem for anisotropic wave equations. arXiv:1902.05878 · Zbl 1163.35040
[7] Bellassoued, M., Yamamoto, M.: Carleman Estimates and Applications to Inverse Problems for Hyperbolic Systems, Springer Monographs in Mathematics, pp. xii+260. Springer, Tokyo (2017) · Zbl 1412.35002
[8] Bourgeois, L., Quantification of the unique continuation property for the heat equation, Math. Control Relat. Fields, 7, 3, 347-367 (2017) · Zbl 1366.35234
[9] Bourgeois, L., About stability and regularization of ill-posed elliptic Cauchy problems: the case of \(C^{1,1}\)-domains, M2AN Math, Model. Numer. Anal., 44, 4, 715-735 (2010) · Zbl 1194.35497
[10] Choulli, M., New global logarithmic stability result for the Cauchy problem for elliptic equations, Bull. Aust. Math. Soc., 101, 1, 141-145 (2020) · Zbl 1433.35040
[11] Choulli, M.: Applications of Elliptic Carleman Inequalities to Cauchy and Inverse Problems, SpringerBriefs in Mathematics, BCAM SpringerBriefs, pp. ix+81. Springer, Bilbao (2016) · Zbl 1351.35260
[12] Choulli, M.: Une Introduction Aux Problèmes Inverses Elliptiques et Paraboliques, Mathématiques & Applications, vol. 65, pp. xxii+249. Springer, Berlin (2009) · Zbl 1192.35187
[13] Choulli, M., Yamamoto, M.: Logarithmic global stability of parabolic Cauchy problems. J. Inverse Ill Posed Probl. (to appear) · Zbl 1231.35298
[14] Duyckaerts, T., Zhang, X., Zuazua, E.: On the optimality of the observability inequalities for parabolic and hyperbolic systems with potentials. Annales de l’Institut Henri Poincaré (C), Analyse Non Linéaire 25, 1-41 (2008) · Zbl 1248.93031
[15] Fernàndez-Cara, E.; Guerrero, S., Global Carleman inequalities for parabolic systems and applications to controllability, SIAM J. Control. Optim., 45, 4, 1399-1446 (2006) · Zbl 1121.35017
[16] Fu, X., Lü, Q., Zhang, X.: Carleman Estimates for Second Order Partial Differential Operators and Applications. A Unified Approach, SpringerBriefs in Mathematics. BCAM SpringerBriefs, pp. xi+127. Springer (2019)
[17] Fursikov, A.V., Imanuvilov, O.Yu.: Controllability of Evolution Equations, Lecture Notes Series. Seoul National Univ. (1996) · Zbl 0862.49004
[18] Hörmander, L., Linear Partial Differential Operators, Fourth Printing, 285 (1976), Berlin: Springer, Berlin · Zbl 0321.35001
[19] Hörmander, L.: The Analysis of Linear Partial Differential Operators IV. Fourier Integral Operators, Reprint of the 1994 Edition, Classics in Mathematics, pp. viii+352. Springer, Berlin (2009) · Zbl 1178.35003
[20] Huang, X.: Carleman Estimate for a General Second-order Hyperbolic Equation, Inverse Problems and Related Topics, Springer Proceedings in Mathematics and Statistics, pp. 149-165. Springer, Singapore (2020) · Zbl 1443.35189
[21] Isakov, V.: Inverse Problems for Partial Differential Equations, Applied Mathematical Sciences, vol. 127, 3rd edn, pp. xv+406. Springer, Cham (2017) · Zbl 1366.65087
[22] John, F.: Partial Differential Equations, Applied Mathematical Sciences, vol. 1, 4th edn, pp. x+249. Springer, New York (1986)
[23] Lasiecka, I.; Triggiani, R.; Zhang, X., Global uniqueness, observability and stabilization of nonconservative Schrödinger equations via pointwise Carleman estimates. Part: \(I H^1(\Omega )\)-estimates, J. Inverse Ill Posed Probl., 12, 43-123 (2004) · Zbl 1057.35042
[24] Laurent, C.; Léautaud, M., Quantitative unique continuation for operators with partially analytic coefficients. Application to approximate control for waves, J. Eur. Math. Soc., 21, 4, 957-1069 (2019) · Zbl 1428.35066
[25] Le Rousseau, J., Lebeau, G.: On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations. ESAIM Control Optim. Calc. Var. 18(3), 712-747 (2012) · Zbl 1262.35206
[26] Mercado, A.; Osses, A.; Rosier, L., Carleman inequalities and inverse problems for the Schrödinger equation, C. R. Math. Acad. Sci. Paris, 346, 1-2, 53-58 (2008) · Zbl 1135.35098
[27] Mercado, A.; Osses, A.; Rosier, L., Inverse problems for the Schrödinger equation via Carleman inequalities with degenerate weights, Inverse Probl., 24, 1, 015017 (2008) · Zbl 1153.35407
[28] Nirenberg, L., Uniqueness in Cauchy problems for differential equations with constant leading coefficients, Commun. Pure Appl. Math., 10, 89-105 (1957) · Zbl 0077.09402
[29] Ouhabaz, E. M.: Analysis of Heat Equations on Domains, London Mathematical Society Monographs Series, vol. 31, pp. xiv+284. Princeton University Press, Princeton (2005) · Zbl 1082.35003
[30] Renardy, M., Rogers, R.C.: An Introduction to Partial Differential Equations, Texts in Applied Mathematics, vol. 13, pp. xiv+428. Springer, New York (1993) · Zbl 0917.35001
[31] Robbiano, L.: Théorème d’unicité adapté au contrôle des solutions des problèmes hyperboliques. Commun. Partial Differ. Equ. 16(4-5), 789-800 (1991) · Zbl 0735.35086
[32] Saut, J-C; Scheurer, B., Un théorème de prolongement unique pour des opérateurs elliptiques dont les coefficients ne sont pas localement bornés, C. R. Acad. Sci. Paris Sér. A-B, 290, 13, A595-A598 (1980) · Zbl 0429.35020
[33] Saut, J.-C., Scheurer, B.: Sur l’unicité du problème de Cauchy et le prolongement unique pour des équations elliptiques à coefficients non localement bornés. J. Differ. Equ. 43(1), 28-43 (1982) · Zbl 0431.35017
[34] Saut, J-C; Scheurer, B., Remarques sur un théorème de prolongement unique de Mizohata, C. R. Acad. Sci. Paris Sér. I Math., 296, 6, 307-310 (1983) · Zbl 0555.35055
[35] Saut, J-C; Scheurer, B., Unique continuation for some evolution equations, J. Differ. Equ., 66, 1, 118-139 (1987) · Zbl 0631.35044
[36] Shao, A., On Carleman and observability estimates for wave equations on time-dependent domains, Proc. Lond. Math. Soc., 119, 998-1064 (2019) · Zbl 1437.35441
[37] Yao, P-F, On the observability inequalities for exact controllability of wave equations with variable coefficients, SIAM J. Control. Optim., 37, 5, 1568-1599 (1999) · Zbl 0951.35069
[38] Zhang, X., Explicit observability estimate for the wave equation with potential and its application, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 456, 1101-1115 (2000) · Zbl 0976.93038
[39] Zhang, X., Exact controllability of the semilinear plate equations, Asymptot. Anal., 27, 95-125 (2001) · Zbl 1007.35008
[40] Zuily, C.: Uniqueness and Nonuniqueness in the Cauchy Problem, Progress in Mathematics, vol. 33. Birkhüser Boston, Inc., Boston, pp. xi+168 (1983) · Zbl 0521.35003
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