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Journées équations aux dérivées partielles

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Mikko Salo
The fractional Calderón problem
Journées équations aux dérivées partielles (2017), Exp. No. 7, 8 p., doi: 10.5802/jedp.657
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Résumé - Abstract

We review recent progress in the fractional Calderón problem, where one tries to determine an unknown coefficient in a fractional Schrödinger equation from exterior measurements of solutions. This equation enjoys remarkable uniqueness and approximation properties, which turn out to yield strong results in related inverse problems.


[ARRV09] G. Alessandrini, L. Rondi, E. Rosset, S. Vessella, The stability for the Cauchy problem for elliptic equations, Inverse Problems 25 (2009), 123004.
[BV16] C. Bucur, E. Valdinoci, Non-local diffusion and applications. Lecture Notes of the Unione Matematica Italiana 20, Springer, 2016.
[CS14] X. Cabre, Y. Sire, Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire 31 (2014), 23–53.
[CS07] L. Caffarelli, L. Silvestre, An extension problem related to the fractional Laplacian, Comm. PDE 32 (2007), 1245–1260.
[CLL17] X. Cao, Y.-H. Lin, H. Liu, Simultaneously recovering potentials and embedded obstacles for anisotropic fractional Schrödinger operators, arXiv:1712.00937.
[CNYY09] J. Cheng, J. Nakagawa, M. Yamamoto, T. Yamazaki, Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation, Inverse Problems 25 (2009), 115002.
[DKN17] T. Daudé, N. Kamran, F. Nicoleau, On the hidden mechanism behind non-uniqueness for the anisotropic Calderón problem with data on disjoint sets, arXiv:1701.09056.
[DSV16] S. Dipierro, O. Savin, E. Valdinoci, Local approximation of arbitrary functions by solutions of nonlocal equations, arXiv:1609.04438.
[DSV17] S. Dipierro, O. Savin, E. Valdinoci, All functions are locally s-harmonic up to a small error, Journal of EMS 19 (2017), 957–966.
[FJK82] E. Fabes, D. Jerison, C. Kenig, The Wiener test for degenerate elliptic equations, Ann. Inst. Fourier 32 (1982), 151–182.
[FKS82] E. Fabes, C. Kenig, R. Serapioni, The local regularity of solutions of degenerate elliptic equations, Communications in Statistics – Theory and Methods 7 (1982), 77–116.
[FF14] M. Moustapha Fall, V. Felli, Unique continuation property and local asymptotics of solutions to fractional elliptic equations, Comm. PDE 39 (2014), 354–397.
[GLX17] T. Ghosh, Y.-H. Lin, J. Xiao, The Calderón problem for variable coefficients nonlocal elliptic operators, Comm. PDE 42 (2017), no. 12, 1923–1961.
[GRSU18] T. Ghosh, A. Rüland, M. Salo, G. Uhlmann, Uniqueness and reconstruction for the fractional Calderón problem with a single measurement, arXiv:1801.04449.
[GSU16] T. Ghosh, M. Salo, G. Uhlmann, The Calderón problem for the fractional Schrödinger equation, arXiv:1609.09248.
[Gr15] G. Grubb, Fractional Laplacians on domains, a development of Hörmander’s theory of $\mu $-transmission pseudodifferential operators, Adv. Math. 268 (2015), 478–528.
[HL17] B. Harrach, Y.-H. Lin, Monotonicity-based inversion of the fractional Schrödinger equation, arXiv:1711.05641.
[Hö65] L. Hörmander, Boundary problems for classical pseudo-differential operators. Unpublished lecture notes at Inst. Adv. Study, Princeton, 1965.
[JR15] B. Jin, W. Rundell, A tutorial on inverse problems for anomalous diffusion processes, Inverse Problems 31 (2015), 035003.
[LL17] R.-Y. Lai, Y.-H. Lin, Global uniqueness for the semilinear fractional Schrödinger equation, arXiv:1710.07404.
[La56] P.D. Lax, A stability theorem for solutions of abstract differential equations, and its application to the study of the local behavior of solutions of elliptic equations, Comm. Pure Appl. Math. 9 (1956), 747–766.
[LR95] G. Lebeau, L. Robbiano, Contrôle exact de léquation de la chaleur, Comm. PDE 20 (1995), 335–356.
[LL12] J. Le Rousseau, G. Lebeau, On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations, ESAIM: Control, Optimisation and Calculus of Variations 18 (2012), 712–747.
[Ma56] B. Malgrange, Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution, Ann. Inst. Fourier, Grenoble 6 (1955–1956), 271–355.
[Mc00] W. McLean, Strongly elliptic systems and boundary integral equations. Cambridge University Press, 2000.
[Ri38] M. Riesz, Integrales de Riemann-Liouville et potentiels, Acta Sci. Math. (Szeged) 9:1-1 (1938-40), 1–42.
[Ro95] L. Robbiano, Fonction de coût et contrôle des solutions des équations hyperboliques, Asymptotic Anal. 10 (1995), 95–115.
[Ro16] X. Ros-Oton, Nonlocal elliptic equations in bounded domains: a survey, Publ. Mat. 60 (2016), 3–26.
[Rü15] A. Rüland, Unique continuation for fractional Schrödinger equations with rough potentials, Comm. PDE 40 (2015), 77–114.
[RS17a] A. Rüland, M. Salo, The fractional Calderón problem: low regularity and stability, arXiv:1708.06294.
[RS17b] A. Rüland, M. Salo, Quantitative approximation properties for the fractional heat equation, arXiv:1708.06300.
[RS17c] A. Rüland, M. Salo, Quantitative Runge approximation and inverse problems, IMRN (to appear).
[RS17d] A. Rüland, M. Salo, Exponential instability in the fractional Calderón problem, Inverse Problems (to appear).
[SY11] K. Sakamoto, M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl. 382 (2011), 426–447.
[Uh14] G. Uhlmann, Inverse problems: seeing the unseen, Bull. Math. Sci. 4 (2014), 209–279.
[Ve93] R. Verch, Antilocality and a Reeh-Schlieder theorem on manifolds, Lett. Math. Phys. 28 (1993), 143–154.
[Yu17] H. Yu, Unique continuation for fractional orders of elliptic equations, Annals of PDE 3 (2017), 16.
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