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Lipschitz stability for the finite dimensional fractional Calderón problem with finite Cauchy data

  • * Corresponding author: Eva Sincich

    * Corresponding author: Eva Sincich
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  • In this note we discuss the conditional stability issue for the finite dimensional Calderón problem for the fractional Schrödinger equation with a finite number of measurements. More precisely, we assume that the unknown potential $ q \in L^{\infty}(\Omega) $ in the equation $ ((- \Delta)^s+ q)u = 0 \mbox{ in } \Omega\subset \mathbb{R}^n $ satisfies the a priori assumption that it is contained in a finite dimensional subspace of $ L^{\infty}(\Omega) $. Under this condition we prove Lipschitz stability estimates for the fractional Calderón problem by means of finitely many Cauchy data depending on $ q $. We allow for the possibility of zero being a Dirichlet eigenvalue of the associated fractional Schrödinger equation. Our result relies on the strong Runge approximation property of the fractional Schrödinger equation.

    Mathematics Subject Classification: Primary: 35R30, 35A35, 35S15.


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  • [1] G. S. Alberti and M. Santacesaria, Calderón's Inverse Problem with a Finite Number of Measurements, arXiv: 1803.04224, 2018.
    [2] G. AlessandriniM. V. de HoopR. Gaburro and E. Sincich, Lipschitz stability for the electrostatic inverse boundary value problem with piecewise linear conductivities, Journal de Mathématiques Pures et Appliquées, 107 (2016), 638-664.  doi: 10.1016/j.matpur.2016.10.001.
    [3] G. AlessandriniM. V. de Hoop and R. Gaburro, Romina and Eva Sincich, Lipschitz stability for a piecewise linear Schrödinger potential from local Cauchy data, Asymptotic Analysis, 108 (2018), 115-149.  doi: 10.3233/ASY-171457.
    [4] G. Alessandrini and V. Isakov, Analiticity and uniqueness for the inverse conductivity problem, Rend. Ist. Mat. Univ. Trieste, 28 (1996), 351-369. 
    [5] G. Alessandrini, L. Rondi, E. Rosset and S. Vessella, The stability for the Cauchy problem for elliptic equations, Inverse Problems, 25 (2009), 123004, 47pp. doi: 10.1088/0266-5611/25/12/123004.
    [6] G. Alessandrini and S. Vessella, Lipschitz stability for the inverse conductivity problem, Advances in Applied Mathematics, 35 (2005), 207-241.  doi: 10.1016/j.aam.2004.12.002.
    [7] B. BarcelóE. Fabes and J. Keun Seo, The inverse conductivity problem with one measurement: Uniqueness for convex polyhedra, Proc. Amer. Math. Soc., 122 (1994), 183-189.  doi: 10.1090/S0002-9939-1994-1195476-6.
    [8] E. BerettaM. V. de Hoop and L. Qiu, Lipschitz stability for an inverse boundary value problem for a Schrödinger-type equation, SIAM Journal on Mathematical Analysis, 45 (2013), 679-699.  doi: 10.1137/120869201.
    [9] E. Beretta and E. Francini, Lipschitz stability for the electrical impedance tomography problem: The complex case, Comm. Partial Differential Equations, 36 (2011), 1723-1749.  doi: 10.1080/03605302.2011.552930.
    [10] E. Beretta, E. Francini and S. Vessella, Lipschitz stability estimates for polygonal conductivity inclusions from boundary measurements, arXiv: 1901.01152.
    [11] E. Blåsten and H. Liu, Recovering piecewise constant refractive indices by a single far-field pattern, arXiv: 1705.00815, 2017.
    [12] X. Cao and H. Liu, Determining a fractional Helmholtz system with unknown source and medium parameter, arXiv preprint, arXiv: 1803.09538, 2018.
    [13] X. Cao, Y.-H. Lin and H. Liu, Simultaneously recovering potentials and embedded obstacles for anisotropic fractional Schrödinger operators, AIMS, 13 (2019), 197–210, arXiv: 1712.00937. doi: 10.3934/ipi.2019011.
    [14] S. DipierroO. Savin and E. Valdinoci, All functions are locally s-harmonic up to a small error, J. Eur. Math. Soc. (JEMS), 19 (2017), 957-966.  doi: 10.4171/JEMS/684.
    [15] E. FabesH. Kang and J. K. Seo, Inverse conductivity problem with one measurement: Error estimates and approximate identification for perturbed disks, SIAM J. Math. Anal., 30 (1999), 699-720.  doi: 10.1137/S0036141097324958.
    [16] M. Moustapha Fall and V. Felli, Unique continuation property and local asymptotics of solutions to fractional elliptic equations, Communication in Partial Differential Equations, 39 (2014), 354-397.  doi: 10.1080/03605302.2013.825918.
    [17] A. Friedman and V. Isakov, On the uniqueness in the inverse conductivity problem with one measurement, Indiana Univ. Math J., 38 (1989), 563-579.  doi: 10.1512/iumj.1989.38.38027.
    [18] R. Gaburro and E. Sincich, Lipschitz stability for the inverse conductivity problem for a conformal class of anisotropic conductivities, Inverse Problems, 31 (2015), 015008, 26pp. doi: 10.1088/0266-5611/31/1/015008.
    [19] B. Gebauer, Localized potentials in electrical impedance tomography, AIMS, 2 (2018), 251-369.  doi: 10.3934/ipi.2008.2.251.
    [20] T. GhoshY.-H. Lin and J. Xiao, The Calderón problem for variable coefficients nonlocal elliptic operators, Communication in Partial Differential Equations, 42 (2017), 1923-1961.  doi: 10.1080/03605302.2017.1390681.
    [21] T. Ghosh, A. Rüland, M. Salo and G. Uhlmann, Uniqueness and reconstruction for the fractional Calderón problem with a single measurement, arXiv: 1801.04449, 2018.
    [22] T. Ghosh, M. Salo and G. Uhlmann, The Calderón problem for the fractional Schrödinger equation, to appear in Analysis and PDE.
    [23] G. Grubb, Fractional Laplacians on domains, a development of Hörmander's theory of $\mu$-transmission pseudodifferential operators, Advances in Mathematics, 268 (2015), 478-528.  doi: 10.1016/j.aim.2014.09.018.
    [24] B. Harrach and Y.-H. Lin, Monotonicity-based inversion of the fractional Schrödinger equation, arXiv: 1711.05641, 2017.
    [25] B. Harrach, V. Pojola and M. Salo, Monotonicity and local uniqueness for the Helmholtz equation, arXiv: 1709.08756, 2017.
    [26] M. Ikehata, On reconstruction in the inverse conductivity problem with one measurement, Inverse Problems, 16 (2000), 785-793.  doi: 10.1088/0266-5611/16/3/314.
    [27] M. Ikehata, On reconstruction from a partial knowledge of the Neumann-to-Dirichlet operator, Inverse Problems, 17 (2001), 45-51.  doi: 10.1088/0266-5611/17/1/304.
    [28] M. Ikehata, Extraction formulae for an inverse boundary value problem for the equation $\nabla\cdot(\sigma-i\omega\epsilon)\nabla u = 0$, Inverse Problems, 18 (2002), 1281-1290.  doi: 10.1088/0266-5611/18/5/304.
    [29] V. Isakov and J. Powell, On the inverse conductivity problem with one measurement, Inverse Problems, 6 (1990), 311-318.  doi: 10.1088/0266-5611/6/2/011.
    [30] H. Liu and C.-H. Tsou, Stable determination of polygonal inclusions in Calderón's problem by a single partial boundary measurement, arXiv: 1902.04462. 2019.
    [31] N. Mandache, Exponential instability in an inverse problem for the Schrödinger equation, Inverse Problems, 17 (2001), 1435-1444.  doi: 10.1088/0266-5611/17/5/313.
    [32] W. McLeanStrongly Elliptic Systems and Boundary Integral Equations,, Cambridge University Press, Cambridge, 2000. 
    [33] L. Rondi, A remark on a paper by Alessandrini and Vessella, Advances in Applied Mathematics, 36 (2006), 67-69.  doi: 10.1016/j.aam.2004.12.003.
    [34] R.-Y. Lai and Y.-H. Lin, Global uniqueness for the semilinear fractional Schrödinger equation, Proceedings of the American Mathematical Society, 147 (2019), 1189-1199.
    [35] A. Rüland, Unique continuation for fractional Schrödinger equations with rough potentials, Communications In Partial Differential Equations, 40 (2015), 77-114.  doi: 10.1080/03605302.2014.905594.
    [36] A. Rüland and M. Salo, The fractional Calderón problem: Low regularity and stability, to appear in Nonlinear Analysis.
    [37] A. Rüland and M. Salo, Exponential instability in the fractional Calderón problem, Inverse Problems, 34 (2018), 045003, 21pp. doi: 10.1088/1361-6420/aaac5a.
    [38] M. Salo, The fractional Calderón problem, Journés Équations aux Dérivées Partielles, 7 (2017), 8pp, arXiv: 1711.06103. doi: 10.5802/jedp.657.
    [39] J. K. Seo, On the uniqueness in the inverse conductivity problem, J. Fourier Anal. Appl., 2 (1996), 227-235.  doi: 10.1007/s00041-001-4030-7.
    [40] H. Yu, Unique continuation for fractional orders of elliptic equations, Annals of PDE (2017) 3: 16. https://doi.org/10.1007/s40818-017-0033-9
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