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All issues Volume 54 / No 2 (March-April 2020) ESAIM: M2AN, 54 2 (2020) 493-529 References
Open Access
Issue
ESAIM: M2AN
Volume 54, Number 2, March-April 2020
Page(s) 493 - 529
DOI https://doi.org/10.1051/m2an/2019073
Published online 18 February 2020
  1. G. Alessandrini, L. Rondi, E. Rosset and S. Vessella, The stability for the Cauchy problem for elliptic equations. Inverse Prob. 25 (2009) 123004. [CrossRef] [Google Scholar]
  2. M. Azaez, F.B. Belgacem and H. El Fekih, On Cauchy’s problem: II. Completion, regularization and approximation. Inverse Prob. 22 (2006) 1307. [CrossRef] [Google Scholar]
  3. F.B. Belgacem, Why is the Cauchy problem severely ill-posed?. Inverse Prob. 23 (2007) 823–836. [CrossRef] [Google Scholar]
  4. L. Bourgeois, A mixed formulation of quasi-reversibility to solve the Cauchy problem for Laplace’s equation. Inverse Prob. 21 (2005) 1087–1104. [CrossRef] [Google Scholar]
  5. L. Bourgeois, About stability and regularization of ill-posed elliptic Cauchy problems: the case of C1,1 domains. M2AN 44 (2010) 715–735. [CrossRef] [EDP Sciences] [Google Scholar]
  6. L. Bourgeois and J. Dardé, About stability and regularization of ill-posed elliptic Cauchy problems: the case of Lipschitz domains. Appl. Anal. 89 (2010) 1745–1768. [Google Scholar]
  7. L. Bourgeois and A. Recoquillay, A mixed formulation of the Tikhonov regularization and its application to inverse PDE problems. ESAIM: M2AN 52 (2018) 123–145. [CrossRef] [EDP Sciences] [Google Scholar]
  8. H. Brezis, Analyse fonctionnelle: théorie et applications, Editions Dunod (1999). [Google Scholar]
  9. E. Burman, Stabilized finite element methods for nonsymmetric, noncoercive, and ill-posed problems. Part I: Elliptic equations, SIAM J. Sci. Comput. 35 (2013) A2752–A2780. [Google Scholar]
  10. E. Burman, Stabilized finite element methods for nonsymmetric, noncoercive, and ill-posed problems. Part II: Hyperbolic equations, SIAM J. Sci. Comput. 36 (2014) A1911–A1936. [Google Scholar]
  11. E. Burman, A stabilized nonconforming finite element method for the elliptic Cauchy problem, Math. Comput. 86 (2017) 75–96. [Google Scholar]
  12. E. Burman, P. Hansbo and M.G. Larson, Solving ill-posed control problems by stabilized finite element methods: an alternative to Tikhonov regularization, Inverse Prob. 34 (2018) 035004. [CrossRef] [Google Scholar]
  13. E. Burman, M.G. Larson and L. Oksanen, Primal-dual mixed finite element methods for the elliptic Cauchy problem, SIAM J. Numer. Anal. 56 (2018) 3480–3509. [Google Scholar]
  14. L. Chesnel and P. Ciarlet Jr, T-coercivity and continuous Galerkin methods: application to transmission problems with sign changing coefficients, Numer. Math. 124 (2013) 1–29. [Google Scholar]
  15. L. Chesnel, X. Claeys and S.A. Nazarov, A curious instability phenomenon for a rounded corner in presence of a negative material, Asymp. Anal. 88 (2014) 43–74. [Google Scholar]
  16. L. Chesnel, X. Claeys and S.A. Nazarov, Oscillating behaviour of the spectrum for a plasmonic problem in a domain with a rounded corner, M2AN 52 (2018) 1285–1313. [CrossRef] [EDP Sciences] [Google Scholar]
  17. P.G. Ciarlet, The finite element method for elliptic problems. In: Vol. 4 of Studies in Mathematics and its Applications. North-Holland Publishing Co., Amsterdam-New York-Oxford (1978). [Google Scholar]
  18. M. Costabel and M. Dauge, A singularly perturbed mixed boundary value problem, Comm. Partial Differ. Equ. 21 (1996) 1919–1949. [CrossRef] [Google Scholar]
  19. J. Dardé, Iterated quasi-reversibility method applied to elliptic and parabolic data completion problems, Inverse Probl. Imaging 10 (2016) 379–407. [CrossRef] [Google Scholar]
  20. J. Dardé, A. Hannukainen and N. Hyvönen, An #-based mixed quasi-reversibility method for solving elliptic Cauchy problems, SIAM J. Numer. Anal. 51 (2013) 2123–2148. [Google Scholar]
  21. P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, London, 1985. [Google Scholar]
  22. P. Grisvard, Singularities in boundary value problems. In: Vol. 22 of Recherches en Mathématiques Appliquées [Research in Applied Mathematics]. Masson, Paris; Springer-Verlag, Berlin (1992). [Google Scholar]
  23. J. Hadamard, Sur les problèmes aux dérivées partielles et leur signification physique, Princeton Univ. Bull. (1902) 49–52. [Google Scholar]
  24. A.M. Il’in, Matching of asymptotic expansions of solutions of boundary value problems. In: Vol. 102 of Translations of Mathematical Monographs. AMS, Providence, RI (1992). [Google Scholar]
  25. T. Johansson, An iterative procedure for solving a Cauchy problem for second order elliptic equations, Math. Nachr. 272 (2004) 46–54. [CrossRef] [Google Scholar]
  26. A. Kirsch, The Robin problem for the Helmholtz equation as a singular perturbation problem, Numer. Func. Anal. Opt. 8 (1985) 1–20. [CrossRef] [MathSciNet] [Google Scholar]
  27. M.V. Klibanov and F. Santosa, A computational quasi-reversibility method for Cauchy problems for Laplace’s equation, SIAM J. Appl. Math. 51 (1991) 1653–1675. [Google Scholar]
  28. V.A. Kondratiev, Boundary-value problems for elliptic equations in domains with conical or angular points, Trans. Moscow Math. Soc. 16 (1967) 227–313. [Google Scholar]
  29. V.A. Kozlov, V.G. Maz’ya and J. Rossmann, Elliptic boundary value problems in domains with point singularities. In: Vol. 52 of Mathematical Surveys and Monographs. AMS, Providence, 1997. [Google Scholar]
  30. V.A. Kozlov, V.G. Maz’ya and J. Rossmann, Spectral problems associated with corner singularities of solutions to elliptic equations. In: Vol. 85 of Mathematical Surveys and Monographs, AMS, Providence, 2001. [Google Scholar]
  31. R. Lattès and J.-L. Lions, Méthode de quasi-réversibilité et applications. Travaux et Recherches Mathématiques, No. 15. Dunod, Paris (1967). [Google Scholar]
  32. V.G. Maz’ya and B.A. Plamenevskiĭ, On the coefficients in the asymptotics of solutions of elliptic boundary value problems with conical points. Math. Nachr. 76 (1977) 29–60. Engl. transl. Amer. Math. Soc. Transl. 123 (1984) 57–89. [CrossRef] [MathSciNet] [Google Scholar]
  33. V.G. Maz’ya, S.A. Nazarov and B.A. Plamenevskiĭ, In: Vol. 1 of Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains. Birkhäuser, Basel (2000). Translated from the original German 1991 edition. [Google Scholar]
  34. S.A. Nazarov and B.A. Plamenevskiĭ, Elliptic problems in domains with piecewise smooth boundaries. In: Vol. 13 of Expositions in Mathematics. De Gruyter, Berlin, Germany, 1994. [Google Scholar]
  35. S.A. Nazarov and N. Popoff, Self-adjoint and skew-symmetric extensions of the Laplacian with singular Robin boundary condition. C. R. Acad. Sci. Paris Ser. I 356 (2018) 927–932. [CrossRef] [Google Scholar]
  36. S.A. Nazarov, N. Popoff and J. Taskinen, Plummeting and blinking eigenvalues of the Robin Laplacian in a cuspidal domain. Preprint arXiv:1809.10963 (2018). [Google Scholar]
  37. S. Nicaise, Regularity of the solutions of elliptic systems in polyhedral domains. Bull. Belg. Math. Soc. Simon Stevin 4 (1997) 411–429. [CrossRef] [Google Scholar]
  38. L.E. Payne, Bounds in the Cauchy problem for the Laplace equation. Arch. Ration. Mech. Anal. 5 (1960) 35–45. [Google Scholar]
  39. L.E. Payne, On a priori bounds in the Cauchy problem for elliptic equations. SIAM J. Math. Anal. 1 (1970) 82–89. [CrossRef] [MathSciNet] [Google Scholar]
  40. K.-D. Phung, Remarques sur l’observabilité pour l’équation de Laplace. ESAIM Control Optim. Calc. Var. 9 (2003) 621–635. [CrossRef] [Google Scholar]

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