Carleman estimates for elliptic operators with jumps at an interface: Anisotropic case and sharp geometric conditions

Le Rousseau, Jérôme; Lerner, Nicolas

Journées équations aux dérivées partielles (2010), article no. 13, 23 p.

We consider a second-order selfadjoint 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 moreover prove that the conditions imposed on the weight function are necessary.

Mots clés : Carleman estimate, elliptic operator, non-smooth coefficient, sharp condition, quasi-mode

[1] Alinhac, S. Non-unicité pour des opérateurs différentiels à caractéristiques complexes simples, Ann. Sci. École Norm. Sup. (4), Volume 13 (1980) no. 3, pp. 385-393 | Numdam | MR | Zbl

[2] Benabdallah, A.; Dermenjian, Y.; Rousseau, J. Le Carleman estimates for the one-dimensional heat equation with a discontinuous coefficient and applications to controllability and an inverse problem, J. Math. Anal. Appl., Volume 336 (2007), pp. 865-887 | MR | Zbl

[3] Buonocore, P.; Manselli, P. Nonunique continuation for plane uniformly elliptic equations in Sobolev spaces, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), Volume 29 (2000) no. 4, pp. 731-754 | Numdam | MR | Zbl

[4] Calderón, A.-P. Uniqueness in the Cauchy problem for partial differential equations., Amer. J. Math., Volume 80 (1958), pp. 16-36 | MR | Zbl

[5] Carleman, T. Sur un problème d’unicité pur les systèmes d’équations aux dérivées partielles à deux variables indépendantes, Ark. Mat., Astr. Fys., Volume 26 (1939) no. 17, pp. 9 | Zbl

[6] Doubova, A.; Osses, A.; Puel, J.-P. Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients, ESAIM Control Optim. Calc. Var., Volume 8 (2002), pp. 621-661 (A tribute to J. L. Lions) | Numdam | MR | Zbl

[7] Hörmander, L. On the uniqueness of the Cauchy problem, Math. Scand., Volume 6 (1958), pp. 213-225 | MR | Zbl

[8] Hörmander, L. Linear partial differential operators, Die Grundlehren der mathematischen Wissenschaften, Bd. 116, Academic Press Inc., Publishers, New York, 1963 | MR | Zbl

[9] Hörmander, L. The analysis of linear partial differential operators. IV, Grundlehren der Mathematischen Wissenschaften, 275, Springer-Verlag, Berlin, 1994 (Fourier integral operators) | MR | Zbl

[10] Imanuvilov, O. Yu.; Puel, J.-P. Global Carleman estimates for weak solutions of Elliptic nonhomogeneous Dirichlet problems, Int. Math. Res. Not., Volume 16 (2003), pp. 883-913 | MR | Zbl

[11] Le Rousseau, J. Carleman estimates and controllability results for the one-dimensional heat equation with $BV$ coefficients, J. Differential Equations, Volume 233 (2007), pp. 417-447 | MR | Zbl

[12] Le Rousseau, J.; Lebeau, G. On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations, Preprint (2009)

[13] Le Rousseau, J.; Lerner, N. Carleman estimates for elliptic operators with jumps at an interface: Anisotropic case and sharp geometric conditions, in prep. (2010)

[14] Le Rousseau, J.; Robbiano, L. Carleman estimate for elliptic operators with coefficents with jumps at an interface in arbitrary dimension and application to the null controllability of linear parabolic equations, Arch. Rational Mech. Anal., Volume 195 (2010), pp. 953-990 | MR

[15] Le Rousseau, J.; Robbiano, L. Local and global Carleman estimates for parabolic operators with coefficients with jumps at interfaces, Invent. Math, to appear (2010), pp. 92 pages

[16] Lerner, N. Metrics on the Phase Space and Non-Selfadjoint Pseudo-Differential Operators, Pseudo-Differential Operators, Vol. 3, Birkhäuser, Basel, 2010 | MR | Zbl

[17] Miller, K. Nonunique continuation for uniformly parabolic and elliptic equations in self-adjoint divergence form with Hölder continuous coefficients, Arch. Rational Mech. Anal., Volume 54 (1974), pp. 105-117 | MR | Zbl

[18] Pliś, A. On non-uniqueness in Cauchy problem for an elliptic second order differential equation, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., Volume 11 (1963), pp. 95-100 | MR | Zbl

[19] Schulz, F. On the unique continuation property of elliptic divergence form equations in the plane, Math. Z., Volume 228 (1998) no. 2, pp. 201-206 | MR | Zbl