Carleman Estimates for Some Non-Smooth Anisotropic Media

References

• Agelas , L. , Di Pietro , D. , Droniou , J. ( 2010 ). The G method for heterogeneous anisotropic diffusion on general meshes . M2AN Math. Model. Numer. Anal. 44 : 597 – 625 .
   Web of Science ®Google Scholar
• Alessandrini , G. , Escauriaza , L. ( 2008 ). Null-controllability of one-dimensional parabolic equations . ESAIM Control Optim. Calc. Var. 14 ( 2 ): 284 – 293 .
   Web of Science ®Google Scholar
• Barbu , V. ( 2000 ). Exact controllability of the superlinear heat equation . Appl. Math. Optim. 42 : 73 – 89 .
   Web of Science ®Google Scholar
• Bellassoued , M. ( 2003 ). Carleman estimates and distribution of resonances for the transparent obstacle and application to the stabilization . Asymptotic Anal. 35 : 257 – 279 .
   Web of Science ®Google Scholar
• Benabdallah , A. , Dermenjian , Y. , Le Rousseau , J. ( 2006 ). Carleman estimates for the one-dimensional heat equation with a discontinuous coefficient and applications . Comptes Rendus Mécanique 334 : 582 – 586 .
   Web of Science ®Google Scholar
• Benabdallah , A. , Dermenjian , Y. , Le Rousseau , J. ( 2007 ). 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 : 865 – 887 .
   Web of Science ®Google Scholar
• Benabdallah , A. , Dermenjian , Y. , Le Rousseau , J. ( 2011 ). Carleman estimates for stratified media . J. Funct. Anal. 260 : 3645 – 3677 .
   Web of Science ®Google Scholar
• Brezis , H. ( 1983 ). Analyse Fonctionnelle [Functional Analysis] . Paris : Masson .
   Google Scholar
• Bukhgeim , A.L. , Klibanov , M.V. ( 1981 ). Global uniqueness of class of multidimensional inverse problems . Soviet Math. Dokl. 24 : 244 – 247 .
   Google Scholar
• Carleman , T. ( 1939 ). Sur une problème d'unicité pour les systèmes d’équations aux dérivées partielles à deux variables indépendantes. [On a uniqueness problem for systems of equations with partial derivatives with two independent variables] . Ark. Mat. Astr. Fys. 26B : 1 – 9 .
   Google Scholar
• Doubova , A. , Osses , A. , Puel , J.P. ( 2002 ). Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients . ESAIM Control Optim. Calc. Var. 8 : 621 – 661 .
   Web of Science ®Google Scholar
• Fernández-Cara , E. , Zuazua , E. (2000). Null and approximate controllability for weakly blowing up semilinear heat equations. Ann. Inst. H. Poincaré, Analyse Non lin. 17:583–616.
   Web of Science ®Google Scholar
• Fursikov , A. , Imanuvilov , O.Y. ( 1996 ). Controllability of Evolution Equations . Lecture Notes , vol. 34. Seoul : Seoul National University .
   Google Scholar
• Hörmander , L. ( 1963 ). Linear Partial Differential Operators . Berlin : Springer-Verlag .
   Google Scholar
• Hörmander , L. ( 1963 ). The Analysis of Linear Partial Differential Operators, vol. IV. Berlin: Springer-Verlag .
   Google Scholar
• Le Rousseau , J. ( 2007 ). Carleman estimates and controllability results for the one-dimensional heat equation with BV coefficients . J. Differential Equations 233 : 417 – 447 .
   Web of Science ®Google Scholar
• Le Rousseau , J. , Lerner , N. ( 2010 ). Carleman estimates for elliptic operators with jumps at an interface: Anisotropic case and sharp geometric conditions. Journées “Équations aux Dérivées Partielles”. 13:1–23 .
   Google Scholar
• Le Rousseau , J. , Lerner , N. Carleman estimates for anisotropic elliptic operators with jumps at an interface . Anal. PDE , to appear
   Google Scholar
• Le Rousseau , J. , Robbiano , L. ( 2010 ). 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. Rational Mech. Anal. 105 : 953 – 990 .
   Web of Science ®Google Scholar
• Le Rousseau , J. , Robbiano , L. ( 2011 ). Local and global Carleman estimates for parabolic operators with coefficients with jumps at interfaces . Invent. Math. 183 : 245 – 336 .
   Web of Science ®Google Scholar
• Lebeau , G. , Robbiano , L. ( 1995 ). Contrôle exact de l’équation de la chaleur [Exact control of the heat equation] . Comm. Part. Diff. Eqs. 20 : 335 – 356 .
   Web of Science ®Google Scholar
• Miller , K. ( 1974 ). Nonunique continuation for uniformly parabolic and elliptic equations in self-adjoint divergence form with Hölder continuous coefficients . Arch. Rational Mech. Anal. 54 : 105 – 117 .
   Web of Science ®Google Scholar
• Pliś , A. ( 1963 ). On non-uniqueness in Cauchy problem for an elliptic second order differential equation . Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 11 : 95 – 100 .
   Google Scholar
• Poisson , O. ( 2008 ). Carleman estimates for the heat equation with discontinuous diffusion coefficients . Applicable Anal. 87 : 1129 – 1144 .
   Web of Science ®Google Scholar
• Zuily , C. ( 1983 ). Uniqueness and Non Uniqueness in the Cauchy Problem . Boston : Birkhauser .
   Google Scholar