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Inverse problems for fractional equations with a minimal number of measurements

  • *Corresponding author: Yi-Hsuan Lin

    *Corresponding author: Yi-Hsuan Lin 
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  • In this paper, we study several inverse problems associated with a fractional differential equation of the following form:

    $ (-\Delta)^s u(x)+\sum\limits_{k = 0}^N a^{(k)}(x) [u(x)]^k = 0, \ \ 0<s<1, \ N\in\mathbb{N}\cup\{0\}\cup\{\infty\}, $

    which is given in a bounded domain $ \Omega\subset\mathbb{R}^n $, $ n\geq 1 $. For any finite $ N $, we show that $ a^{(k)}(x) $, $ k = 0, 1, \ldots, N $, can be uniquely determined by $ N+1 $ different pairs of Cauchy data in $ \Omega_e: = \mathbb{R}^n\backslash\overline{\Omega} $. If $ N = \infty $, the uniqueness result is established by using infinitely many pairs of Cauchy data. The results are highly intriguing in that it generally does not hold true in the local case, namely $ s = 1 $, even for the simplest case when $ N = 0 $, a fortiori $ N\geq 1 $. The nonlocality plays a key role in establishing the uniqueness result, and we do not utilize any linearization techniques. We also establish several other unique determination results by making use of a minimal number of measurements. Moreover, in the process we derive a novel comparison principle for nonlinear fractional differential equations as a significant byproduct.

    Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.


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  • [1] S. BhattacharyyaT. Ghosh and G. Uhlmann, Inverse problems for the fractional Laplacian with lower order non-local perturbations, Transactions of the American Mathematical Society, 374 (2021), 3053-3075.  doi: 10.1090/tran/8151.
    [2] E. L. K. Blåsten and H. Liu, On corners scattering stably and stable shape determination by a single far-field pattern, Indiana University Mathematics Journal, 70 (2021), 907-947.  doi: 10.1512/iumj.2021.70.8411.
    [3] E. L. K. Blåsten and H. Liu, Scattering by curvatures, radiationless sources, transmission eigenfunctions, and inverse scattering problems, SIAM Journal on Mathematical Analysis, 53 (2021), 3801-3837.  doi: 10.1137/20M1384002.
    [4] X. CaoY.-H. Lin and H. Liu, Simultaneously recovering potentials and embedded obstacles for anisotropic fractional Schrödinger operators, Inverse Problems and Imaging, 13 (2019), 197-210.  doi: 10.3934/ipi.2019011.
    [5] X. Cao and H. Liu, Determining a fractional Helmholtz equation with unknown source and scattering potential, Communications in Mathematical Sciences, 17 (2019), 1861-1876.  doi: 10.4310/CMS.2019.v17.n7.a5.
    [6] M. Cekic, Y.-H. Lin and A. Rüland, The Calderón problem for the fractional Schrödinger equation with drift, Cal. Var. Partial Differential Equations, 59 (2020), Paper No. 91, 46 pp. doi: 10.1007/s00526-020-01740-6.
    [7] Y. DengJ. Li and H. Liu, On identifying magnetized anomalies using geomagnetic monitoring, Archive for Rational Mechanics and Analysis, 231 (2019), 153-187.  doi: 10.1007/s00205-018-1276-7.
    [8] Y. DengJ. Li and H. Liu, On identifying magnetized anomalies using geomagnetic monitoring within a magnetohydrodynamic model, Archive for Rational Mechanics and Analysis, 235 (2020), 691-721.  doi: 10.1007/s00205-019-01429-x.
    [9] E. Di NezzaG. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bulletin des Sciences Mathématiques, 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.
    [10] M. M. Fall and V. Felli, Unique continuation property and local asymptotics of solutions to fractional elliptic equations, Communications in Partial Differential Equations, 39 (2014), 354-397.  doi: 10.1080/03605302.2013.825918.
    [11] A. Feizmohammadi, T. Ghosh, K. Krupchyk and G. Uhlmann, Fractional anisotropic Calderón problem on closed Riemannian manifolds, arXiv preprint, arXiv: 2112.03480.
    [12] A. Feizmohammadi, T. Liimatainen and Y.-H. Lin, An inverse problem for a semilinear elliptic equation on conformally transversally anisotropic manifolds, arXiv preprint, arXiv: 2112.08305.
    [13] A. Feizmohammadi and L. Oksanen, An inverse problem for a semi-linear elliptic equation in Riemannian geometries, Journal of Differential Equations, 269 (2020), 4683-4719.  doi: 10.1016/j.jde.2020.03.037.
    [14] T. Ghosh, A non-local inverse problem with boundary response, Revista Matemática Iberoamericana, 38 (2022), 2011-2032.  doi: 10.4171/RMI/1323.
    [15] T. GhoshY.-H. Lin and J. Xiao, The Calderón problem for variable coefficients nonlocal elliptic operators, Communications in Partial Differential Equations, 42 (2017), 1923-1961.  doi: 10.1080/03605302.2017.1390681.
    [16] T. Ghosh, A. Rüland, M. Salo and G. Uhlmann, Uniqueness and reconstruction for the fractional Calderón problem with a single measurement, Journal of Functional Analysis, 108505. doi: 10.1016/j.jfa.2020.108505.
    [17] T. GhoshM. Salo and G. Uhlmann, The Calderón problem for the fractional Schrödinger equation, Analysis & PDE, 13 (2020), 455-475.  doi: 10.2140/apde.2020.13.455.
    [18] T. Ghosh and G. Uhlmann, The Calderón problem for nonlocal operators, arXiv preprint, arXiv: 2110.09265.
    [19] B. Harrach and Y.-H. Lin, Monotonicity-based inversion of the fractional Schrödinger equation Ⅰ. Positive potentials, SIAM Journal on Mathematical Analysis, 51 (2019), 3092-3111.  doi: 10.1137/18M1166298.
    [20] B. Harrach and Y.-H. Lin, Monotonicity-based inversion of the fractional Schödinger equation Ⅱ. General potentials and stability, SIAM Journal on Mathematical Analysis, 52 (2020), 402-436.  doi: 10.1137/19M1251576.
    [21] B. Harrach and Y.-H. Lin, Simultaneous recovery of piecewise analytic coefficients in a semilinear elliptic equation, Nonlinear Analysis, 228 (2023), 113188.  doi: 10.1016/j.na.2022.113188.
    [22] O. Yu. ImanuvilovG. Uhlmann and M. Yamamoto, The Calderón problem with partial data in two dimensions, Journal of the American Mathematical Society, 23 (2010), 655-691.  doi: 10.1090/S0894-0347-10-00656-9.
    [23] C. Kenig and M. Salo, The Calderón problem with partial data on manifolds and applications, Analysis & PDE, 6 (2014), 2003-2048.  doi: 10.2140/apde.2013.6.2003.
    [24] C. E. KenigJ. Sjöstrand and G. Uhlmann, The Calderón problem with partial data, Annals of Mathematics, 165 (2007), 567-591.  doi: 10.4007/annals.2007.165.567.
    [25] P.-Z. KowY.-H. Lin and J.-N. Wang, The Calderón problem for the fractional wave equation: Uniqueness and optimal stability, SIAM Journal on Mathematical Analysis, 54 (2022), 3379-3419.  doi: 10.1137/21M1444941.
    [26] K. Krupchyk and G. Uhlmann, Partial data inverse problems for semilinear elliptic equations with gradient nonlinearities, Mathematical Research Letters, 27 (2020), 1801-1824.  doi: 10.4310/MRL.2020.v27.n6.a10.
    [27] K. Krupchyk and G. Uhlmann, A remark on partial data inverse problems for semilinear elliptic equations, Proc. Amer. Math. Soc., 148 (2020), 681-685.  doi: 10.1090/proc/14844.
    [28] Y. KurylevM. Lassas and G. Uhlmann, Inverse problems for Lorentzian manifolds and non-linear hyperbolic equations, Invent. Math., 212 (2018), 781-857.  doi: 10.1007/s00222-017-0780-y.
    [29] R.-Y. Lai and Y.-H. Lin, Global uniqueness for the fractional semilinear Schrödinger equation, Proc. Amer. Math. Soc., 147 (2019), 1189-1199.  doi: 10.1090/proc/14319.
    [30] R.-Y. Lai and Y.-H. Lin, Inverse problems for fractional semilinear elliptic equations, Nonlinear Analysis, 216 (2022), 112699.  doi: 10.1016/j.na.2021.112699.
    [31] M. LassasT. LiimatainenY.-H. Lin and M. Salo, Partial data inverse problems and simultaneous recovery of boundary and coefficients for semilinear elliptic equations, Revista Matemática Iberoamericana, 37 (2020), 1553-1580.  doi: 10.4171/rmi/1242.
    [32] M. LassasT. LiimatainenY.-H. Lin and M. Salo, Inverse problems for elliptic equations with power type nonlinearities, Journal de Mathématiques Pures et Appliquées, 145 (2021), 44-82.  doi: 10.1016/j.matpur.2020.11.006.
    [33] J. LiH. Liu and S. Ma, Determining a random Schrödinger equation with unknown source and potential, SIAM Journal on Mathematical Analysis, 51 (2019), 3465-3491.  doi: 10.1137/18M1225276.
    [34] J. LiH. Liu and S. Ma, Determining a random Schrödinger operator: Both potential and source are random, Communications in Mathematical Physics, 381 (2021), 527-556.  doi: 10.1007/s00220-020-03889-9.
    [35] T. Liimatainen and Y.-H. Lin, Uniqueness results and gauge breaking for inverse source problems of semilinear elliptic equations, arXiv preprint, arXiv: 2204.11774.
    [36] T. LiimatainenY.-H. LinM. Salo and T. Tyni, Inverse problems for elliptic equations with fractional power type nonlinearities, Journal of Differential Equations, 306 (2022), 189-219.  doi: 10.1016/j.jde.2021.10.015.
    [37] Y.-H. Lin, Monotonicity-based inversion of fractional semilinear elliptic equations with power type nonlinearities, Calculus of Variations and Partial Differential Equations, 61 (2022), 1-30.  doi: 10.1007/s00526-022-02299-0.
    [38] Y.-H. Lin, H. Liu and X. Liu, Determining a nonlinear hyperbolic system with unknown sources and nonlinearity, arXiv preprint, arXiv: 2107.10219.
    [39] Y.-H. LinH. LiuX. Liu and S. Zhang, Simultaneous recoveries for semilinear parabolic systems, Inverse Problems, 38 (2022), 115006. 
    [40] H. Liu, On local and global structures of transmission eigenfunctions and beyond, Journal of Inverse and Ill-posed Problems, 30 (2022), 287-305.  doi: 10.1515/jiip-2020-0099.
    [41] H. LiuL. Rondi and J. Xiao, Mosco convergence for $ h $(curl) spaces, higher integrability for Maxwell's equations, and stability in direct and inverse EM scattering problems, Journal of the European Mathematical Society, 21 (2019), 2945-2993.  doi: 10.4171/JEMS/895.
    [42] H. Liu and G. Uhlmann, Determining both sound speed and internal source in thermo-and photo-acoustic tomography, Inverse Problems, 31 (2015), 105005.  doi: 10.1088/0266-5611/31/10/105005.
    [43] H. Liu and J. Zou, Uniqueness in an inverse acoustic obstacle scattering problem for both sound-hard and sound-soft polyhedral scatterers, Inverse Problems, 22 (2006), 515-524.  doi: 10.1088/0266-5611/22/2/008.
    [44] H. Liu and J. Zou, On uniqueness in inverse acoustic and electromagnetic obstacle scattering problems, Journal of Physics: Conference Series, 124 (2008), 012006.  doi: 10.1088/1742-6596/124/1/012006.
    [45] W. McLeanStrongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, 2000. 
    [46] Q. MengZ. BaiH. Diao and H. Liu, Effective medium theory for embedded obstacles in elasticity with applications to inverse problems, SIAM Journal on Applied Mathematics, 82 (2022), 720-749.  doi: 10.1137/21M1431369.
    [47] M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations, vol. 13, Springer Science & Business Media, 2006.
    [48] A. Rüland and M. Salo, Exponential instability in the fractional Calderón problem, Inverse Problems, 34 (2018), 045003.  doi: 10.1088/1361-6420/aaac5a.
    [49] A. Rüland and M. Salo, The fractional Calderón problem: Low regularity and stability, Nonlinear Analysis, 193 (2020), 111529.  doi: 10.1016/j.na.2019.05.010.
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