Analysis & PDE
- Anal. PDE
- Volume 6, Number 7 (2013), 1601-1648.
Carleman estimates for anisotropic elliptic operators with jumps at an interface
Jérôme Le Rousseau and Nicolas Lerner
Full-text: Open access
Abstract
We consider a second-order self-adjoint elliptic operator with an anisotropic diffusion matrix having a jump across a smooth hypersurface. We prove the existence of a weight function such that a Carleman estimate holds true. We also prove that the conditions imposed on the weight function are sharp.
Article information
Source
Anal. PDE, Volume 6, Number 7 (2013), 1601-1648.
Dates
Received: 31 January 2012
Revised: 8 March 2013
Accepted: 13 April 2013
First available in Project Euclid: 20 December 2017
Permanent link to this document
https://projecteuclid.org/euclid.apde/1513731431
Digital Object Identifier
doi:10.2140/apde.2013.6.1601
Mathematical Reviews number (MathSciNet)
MR3148062
Zentralblatt MATH identifier
1319.47038
Subjects
Primary: 35J15: Second-order elliptic equations 35J57: Boundary value problems for second-order elliptic systems 35J75: Singular elliptic equations
Keywords
Carleman estimate elliptic operator nonsmooth coefficient quasimode
Citation
Le Rousseau, Jérôme; Lerner, Nicolas. Carleman estimates for anisotropic elliptic operators with jumps at an interface. Anal. PDE 6 (2013), no. 7, 1601--1648. doi:10.2140/apde.2013.6.1601. https://projecteuclid.org/euclid.apde/1513731431
References
- S. Alinhac, “Non-unicité pour des opérateurs différentiels à caractéristiques complexes simples”, Ann. Sci. École Norm. Sup. $(4)$ 13:3 (1980), 385–393. Mathematical Reviews (MathSciNet): MR83b:35004b
Zentralblatt MATH: 0456.35002
Digital Object Identifier: doi:10.24033/asens.1387 - S. Alinhac and P. Gérard, Pseudo-differential operators and the Nash–Moser theorem, Graduate Studies in Mathematics 82, Amer. Math. Soc., Providence, RI, 2007.
- H. Bahouri, “Dépendance non linéaire des données de Cauchy pour les solutions des équations aux dérivées partielles”, J. Math. Pures Appl. $(9)$ 66:2 (1987), 127–138.
- V. Barbu, “Exact controllability of the superlinear heat equation”, Appl. Math. Optim. 42:1 (2000), 73–89. Mathematical Reviews (MathSciNet): MR2001i:93010
Zentralblatt MATH: 0964.93046
Digital Object Identifier: doi:10.1007/s002450010004 - C. Bardos, G. Lebeau, and J. Rauch, “Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary”, SIAM J. Control Optim. 30:5 (1992), 1024–1065. Mathematical Reviews (MathSciNet): MR94b:93067
Zentralblatt MATH: 0786.93009
Digital Object Identifier: doi:10.1137/0330055 - A. Benabdallah, Y. Dermenjian, and J. Le Rousseau, “Carleman estimates for the one-dimensional heat equation with a discontinuous coefficient and applications to controllability and an inverse problem”, J. Math. Anal. Appl. 336:2 (2007), 865–887. Mathematical Reviews (MathSciNet): MR2008k:35205
Zentralblatt MATH: 1189.35349
Digital Object Identifier: doi:10.1016/j.jmaa.2007.03.024 - A. Benabdallah, Y. Dermenjian, and J. Le Rousseau, “Carleman estimates for stratified media”, J. Funct. Anal. 260:12 (2011), 3645–3677. Mathematical Reviews (MathSciNet): MR2012c:35050
Zentralblatt MATH: 1218.35238
Digital Object Identifier: doi:10.1016/j.jfa.2011.02.007 - A. L. Bukhgeim and M. V. Klibanov, “Global uniqueness of class of multidimensional inverse problems”, Sov. Math. Dokl. 24 (1981), 244–247. Zentralblatt MATH: 0497.35082
- P. Buonocore and P. Manselli, “Nonunique continuation for plane uniformly elliptic equations in Sobolev spaces”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. $(4)$ 29:4 (2000), 731–754.
- N. Burq, “Décroissance de l'énergie locale de l'équation des ondes pour le problème extérieur et absence de résonance au voisinage du réel”, Acta Math. 180:1 (1998), 1–29. Mathematical Reviews (MathSciNet): MR99j:35119
Zentralblatt MATH: 0918.35081
Digital Object Identifier: doi:10.1007/BF02392877
Project Euclid: euclid.acta/1485891128 - A.-P. Calderón, “Uniqueness in the Cauchy problem for partial differential equations”, Amer. J. Math. 80 (1958), 16–36. Mathematical Reviews (MathSciNet): MR21:3675
Zentralblatt MATH: 0080.30302
Digital Object Identifier: doi:10.2307/2372819 - T. Carleman, “Sur un problème d'unicité pur les systèmes d'équations aux dérivées partielles à deux variables indépendantes”, Ark. Mat., Astr. Fys. 26:17 (1939), 9.
- A. Doubova, A. Osses, and J.-P. Puel, “Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients”, ESAIM Control Optim. Calc. Var. 8 (2002), 621–661. Mathematical Reviews (MathSciNet): MR2004c:93011
Zentralblatt MATH: 1092.93006
Digital Object Identifier: doi:10.1051/cocv:2002047 - E. Fernández-Cara and E. Zuazua, “Null and approximate controllability for weakly blowing up semilinear heat equations”, Ann. Inst. H. Poincaré Anal. Non Linéaire 17:5 (2000), 583–616. Mathematical Reviews (MathSciNet): MR2001j:93009
Zentralblatt MATH: 0970.93023
Digital Object Identifier: doi:10.1016/S0294-1449(00)00117-7 - N. Filonov, “Second-order elliptic equation of divergence form having a compactly supported solution”, J. Math. Sci. $($New York$)$ 106:3 (2001), 3078–3086. Mathematical Reviews (MathSciNet): MR2003h:35047
Zentralblatt MATH: 0991.35020
Digital Object Identifier: doi:10.1023/A:1011379807662 - A. V. Fursikov and O. Y. Imanuvilov, Controllability of evolution equations, Lecture Notes Series 34, Seoul National University, Seoul, 1996.
- L. H örmander, “On the uniqueness of the Cauchy problem”, Math. Scand. 6 (1958), 213–225. Mathematical Reviews (MathSciNet): MR21:3674
Zentralblatt MATH: 0088.30201
Digital Object Identifier: doi:10.7146/math.scand.a-10546 - L. H örmander, Linear partial differential operators, Die Grundlehren der mathematischen Wissenschaften 116, Academic, New York, 1963.
- L. H örmander, The analysis of linear partial differential operators, III: Pseudodifferential operators, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 274, Springer, Berlin, 1985.
- L. H örmander, The analysis of linear partial differential operators, IV: Fourier integral operators, Grundlehren der Mathematischen Wissenschaften 275, Springer, Berlin, 1985.
- O. Y. Imanuvilov and J.-P. Puel, “Global Carleman estimates for weak solutions of elliptic nonhomogeneous Dirichlet problems”, Int. Math. Res. Not. 2003:16 (2003), 883–913. Mathematical Reviews (MathSciNet): MR2004c:35081
Zentralblatt MATH: 1146.35340
Digital Object Identifier: doi:10.1155/S107379280321117X - O. Imanuvilov, V. Isakov, and M. Yamamoto, “An inverse problem for the dynamical Lamé system with two sets of boundary data”, Comm. Pure Appl. Math. 56:9 (2003), 1366–1382. Mathematical Reviews (MathSciNet): MR2004i:35308
Zentralblatt MATH: 1044.35105
Digital Object Identifier: doi:10.1002/cpa.10097 - V. Isakov, Inverse problems for partial differential equations, Applied Mathematical Sciences 127, Springer, New York, 1998.
- D. Jerison and C. E. Kenig, “Unique continuation and absence of positive eigenvalues for Schrödinger operators”, Ann. of Math. $(2)$ 121:3 (1985), 463–494. Mathematical Reviews (MathSciNet): MR87a:35058
Zentralblatt MATH: 0593.35119
Digital Object Identifier: doi:10.2307/1971205 - D. Jerison and G. Lebeau, “Nodal sets of sums of eigenfunctions”, pp. 223–239 in Harmonic analysis and partial differential equations (Chicago, 1996), edited by M. Christ et al., University of Chicago Press, Chicago, 1999.
- F. John, “Continuous dependence on data for solutions of partial differential equations with a presribed bound”, Comm. Pure Appl. Math. 13 (1960), 551–585. Mathematical Reviews (MathSciNet): MR24:A317
Zentralblatt MATH: 0097.08101
Digital Object Identifier: doi:10.1002/cpa.3160130402 - C. E. Kenig, J. Sj östrand, and G. Uhlmann, “The Calderón problem with partial data”, Ann. of Math. $(2)$ 165:2 (2007), 567–591. Mathematical Reviews (MathSciNet): MR2008k:35498
Zentralblatt MATH: 1127.35079
Digital Object Identifier: doi:10.4007/annals.2007.165.567 - H. Koch and D. Tataru, “Carleman estimates and unique continuation for second-order elliptic equations with nonsmooth coefficients”, Comm. Pure Appl. Math. 54:3 (2001), 339–360. Mathematical Reviews (MathSciNet): MR2001m:35075
Zentralblatt MATH: 1033.32025
Digital Object Identifier: doi:10.1002/1097-0312(200103)54:3<339::AID-CPA3>3.0.CO;2-D - H. Koch and D. Tataru, “Sharp counterexamples in unique continuation for second order elliptic equations”, J. Reine Angew. Math. 542 (2002), 133–146.
- J. Le Rousseau, “Carleman estimates and controllability results for the one-dimensional heat equation with BV coefficients”, J. Differential Equations 233:2 (2007), 417–447. Mathematical Reviews (MathSciNet): MR2009c:35190
Zentralblatt MATH: 1128.35020
Digital Object Identifier: doi:10.1016/j.jde.2006.10.005 - J. Le Rousseau and G. Lebeau, “On Carleman estimates for elliptic and parabolic operators: Applications to unique continuation and control of parabolic equations”, ESAIM Control Optim. Calc. Var. 18:3 (2012), 712–747. Mathematical Reviews (MathSciNet): MR3041662
Zentralblatt MATH: 06111871
Digital Object Identifier: doi:10.1051/cocv/2011168 - J. Le Rousseau and L. Robbiano, “Carleman estimate for elliptic operators with coefficients with jumps at an interface in arbitrary dimension and application to the null controllability of linear parabolic equations”, Arch. Ration. Mech. Anal. 195:3 (2010), 953–990. Mathematical Reviews (MathSciNet): MR2011a:93018
Zentralblatt MATH: 1202.35336
Digital Object Identifier: doi:10.1007/s00205-009-0242-9 - J. Le Rousseau and L. Robbiano, “Local and global Carleman estimates for parabolic operators with coefficients with jumps at interfaces”, Invent. Math. 183:2 (2011), 245–336. Mathematical Reviews (MathSciNet): MR2012k:35208
Zentralblatt MATH: 1218.35054
Digital Object Identifier: doi:10.1007/s00222-010-0278-3 - G. Lebeau, “Équation des ondes amorties”, pp. 73–109 in Algebraic and geometric methods in mathematical physics (Kaciveli, 1993), edited by A. Boutet de Monvel and V. Marchenko, Math. Phys. Stud. 19, Kluwer, Dordrecht, 1996.
- G. Lebeau and L. Robbiano, “Contrôle exact de l'équation de la chaleur”, Comm. Partial Differential Equations 20:1-2 (1995), 335–356. Mathematical Reviews (MathSciNet): MR95m:93045
Zentralblatt MATH: 0819.35071
Digital Object Identifier: doi:10.1080/03605309508821097 - G. Lebeau and L. Robbiano, “Stabilisation de l'équation des ondes par le bord”, Duke Math. J. 86:3 (1997), 465–491. Mathematical Reviews (MathSciNet): MR98c:35104
Zentralblatt MATH: 0884.58093
Digital Object Identifier: doi:10.1215/S0012-7094-97-08614-2
Project Euclid: euclid.dmj/1077242846 - G. Lebeau and E. Zuazua, “Null-controllability of a system of linear thermoelasticity”, Arch. Rational Mech. Anal. 141:4 (1998), 297–329. Mathematical Reviews (MathSciNet): MR99f:93013
Zentralblatt MATH: 1064.93501
Digital Object Identifier: doi:10.1007/s002050050078 - N. Lerner, Metrics on the phase space and non-selfadjoint pseudo-differential operators, Pseudo-Differential Operators: Theory and Applications 3, Birkhäuser, Basel, 2010.
- N. Mandache, “On a counterexample concerning unique continuation for elliptic equations in divergence form”, Math. Phys. Anal. Geom. 1:3 (1998), 273–292. Mathematical Reviews (MathSciNet): MR2000c:35033
Zentralblatt MATH: 0920.35034
Digital Object Identifier: doi:10.1023/A:1009745125885 - K. Miller, “Nonunique continuation for uniformly parabolic and elliptic equations in self-adjoint divergence form with Hölder continuous coefficients”, Arch. Rational Mech. Anal. 54 (1974), 105–117. Mathematical Reviews (MathSciNet): MR49:7566
Zentralblatt MATH: 0289.35046
Digital Object Identifier: doi:10.1007/BF00247634 - A. Pliś, “On non-uniqueness in Cauchy problem for an elliptic second order differential equation”, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 11 (1963), 95–100.
- J. Rauch and M. Taylor, “Exponential decay of solutions to hyperbolic equations in bounded domains”, Indiana Univ. Math. J. 24 (1974), 79–86. Mathematical Reviews (MathSciNet): MR50:13906
Zentralblatt MATH: 0281.35012
Digital Object Identifier: doi:10.1512/iumj.1975.24.24004 - F. Schulz, “On the unique continuation property of elliptic divergence form equations in the plane”, Math. Z. 228:2 (1998), 201–206. Mathematical Reviews (MathSciNet): MR99e:35035
Zentralblatt MATH: 0905.35020
Digital Object Identifier: doi:10.1007/PL00004610 - C. D. Sogge, “Oscillatory integrals and unique continuation for second order elliptic differential equations”, J. Amer. Math. Soc. 2:3 (1989), 491–515. Mathematical Reviews (MathSciNet): MR91d:35037
Zentralblatt MATH: 0703.35027
Digital Object Identifier: doi:10.1090/S0894-0347-1989-0999662-3
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