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Inverse problems for the fractional Laplace equation with lower order nonlinear perturbations

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  • We study the inverse problem for the fractional Laplace equation with multiple nonlinear lower order terms. We show that the direct problem is well-posed and the inverse problem is uniquely solvable. More specifically, the unknown nonlinearities can be uniquely determined from exterior measurements under suitable settings.

    Mathematics Subject Classification: Primary: 35R30; Secondary: 35R11.


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  • [1] S. BhattacharyaT. Ghosh and G. Uhlmann, Inverse problem for fractional-{L}aplacian with lower order non-local perturbations, Trans. Amer. Math. Soc., 374 (2021), 3053-3075.  doi: 10.1090/tran/8151.
    [2] A. P. Calderón, Seminar in numerical analysis and its applications to continuum physics (Río de Janeiro: Soc. Brasileira de Matemática), (1980), 65–73.
    [3] 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.
    [4] C. CârsteaG. Nakamura and M. Vashisth, Reconstruction for the coefficients of a quasilinear elliptic partial differential equation, Applied Mathematics Letters, 98 (2019), 121-127.  doi: 10.1016/j.aml.2019.06.009.
    [5] 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.
    [6] G. Covi, An inverse problem for the fractional Schrödinger equation in a magnetic field, Inverse Problems, 36 (2020), 045004. doi: 10.1088/1361-6420/ab661a.
    [7] G. Covi, Inverse problems for a fractional conductivity equation, Nonlinear Analysis, 193 (2020), 111418. doi: 10.1016/j. na. 2019.01.008.
    [8] G. CoviK. Mönkkönen and J. Railo, Unique continuation property and Poincaré inequality for higher order fractional Laplacians with applications in inverse problems, Inverse Problems and Imaging, 15 (2021), 641-681.  doi: 10.3934/ipi.2021009.
    [9] G. Covi, K. Mönkkönen, J. Railo and G. Uhlmann, The higher order fractional Calderón problem for linear local operators: Uniqueness, arXiv: 2008.10227, (2020).
    [10] E. Di Nezza, G. 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.
    [11] 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.
    [12] 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.
    [13] 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, 279 (2020), 108505. doi: 10.1016/j. jfa. 2020.108505.
    [14] 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.
    [15] 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.
    [16] B. Harrach and Y.-H. Lin, Monotonicity-based inversion of the fractional Schrödinger equation II. General potentials and stability, SIAM Journal on Mathematical Analysis, 52 (2020), 402-436.  doi: 10.1137/19M1251576.
    [17] D. Hervas and Z. Sun, An inverse boundary value problem for quasilinear elliptic equations, Communications in Partial Differential Equations, 27 (2002), 2449-2490.  doi: 10.1081/PDE-120016164.
    [18] V. Isakov, On uniqueness in inverse problems for semilinear parabolic equations, Archive for Rational Mechanics and Analysis, 124 (1993), 1-12.  doi: 10.1007/BF00392201.
    [19] V. Isakov, Uniqueness of recovery of some quasilinear partial differential equations, Communications in Partial Differential Equations, 26 (2001), 1947-1973.  doi: 10.1081/PDE-100107813.
    [20] V. Isakov and A. I. Nachman, Global uniqueness for a two-dimensional elliptic inverse problem, Trans. Amer. Math. Soc., 347 (1995), 3375-3390.  doi: 10.2307/2155015.
    [21] V. Isakov and J. Sylvester, Global uniqueness for a semilinear elliptic inverse problem, Communications on Pure and Applied Mathematics, 47 (1994), 1403-1410.  doi: 10.1002/cpa.3160471005.
    [22] H. Kang and G. Nakamura, Identification of nonlinearity in a conductivity equation via the Dirichlet-to-Neumann map, Inverse Problems, 18 (2002), 1079-1088.  doi: 10.1088/0266-5611/18/4/309.
    [23] K. Krupchyk and G. Uhlmann, Partial data inverse problems for semilinear elliptic equations with gradient nonlinearities, Math. Res. Lett., 27 (2020), 1801-1824.  doi: 10.4310/MRL.2020.v27.n6.a10.
    [24] 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.
    [25] 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.
    [26] R. -Y. Lai and Y. -H. Lin, Inverse problems for fractional semilinear elliptic equations, arXiv: 2004.00549, (2020).
    [27] R.-Y. LaiY.-H. Lin and A. Rüland, The Calderón problem for a space-time fractional parabolic equation, SIAM Journal on Mathematical Analysis, 52 (2020), 2655-2688.  doi: 10.1137/19M1270288.
    [28] R. -Y. Lai and T. Zhou, Partial data inverse problems for nonlinear magnetic Schrödinger equations, arXiv: 2007.02475, (2020).
    [29] M. LassasT. LiimatainenY.-H. Lin and M. Salo, Partial data inverse problems and simultaneous recovery of boundary and coefficients for semilinear elliptic equations, Rev. Mat. Iberoam., 37 (2021), 1553-1580.  doi: 10.4171/rmi/1242.
    [30] M. LassasT. LiimatainenY.-H. Lin and M. Salo, Inverse problems for elliptic equations with power type nonlinearities, J. Math. Pures Appl., 145 (2021), 44-82.  doi: 10.1016/j.matpur.2020.11.006.
    [31] L. Li, The Calderón problem for the fractional magnetic operator, Inverse Problems, 36 (2020), 075003. doi: 10.1088/1361-6420/ab8445.
    [32] L. Li, Determining the magnetic potential in the fractional magnetic Calderón problem, Communications in Partial Differential Equations, 46 (2021), 1017-1026.  doi: 10.1080/03605302.2020.1857406.
    [33] L. Li, A semilinear inverse problem for the fractional magnetic Laplacian, arXiv: 2005.06714, (2020).
    [34] Y. H. Lin, Monotonicity-based inversion of fractional semilinear elliptic equations with power type nonlinearities, arXiv: 2005.07163, (2020).
    [35] W. McLeanStrongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, 2020. 
    [36] M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations, Springer-Verlag, New York, 13, 2004.
    [37] X. Ros-Oton, Nonlocal elliptic equations in bounded domains: A survey, Publicacions Matemètiques., 60 (2016), 3–26. doi: 10.5565/PUBLMAT_60116_01.
    [38] X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, Journal de Mathématiques Pures et Appliquées, 101 (2014), 275–302. doi: 10.1016/j. matpur. 2013.06.003.
    [39] A. Rüland, On single measurement stability for the fractional Calderón problem, arXiv: 2007.13624, (2020).
    [40] 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.
    [41] Z. Sun, On a quasilinear inverse boundary value problem, Math. Z., 221 (1996), 293-305.  doi: 10.1007/BF02622117.
    [42] Z. Sun, Inverse boundary value problems for a class of semilinear elliptic equations, Advances in Applied Mathematics, 32 (2004), 791-800.  doi: 10.1016/j.aam.2003.06.001.
    [43] Z. Sun, An inverse boundary-value problem for semilinear elliptic equations, Electronic Journal of Differential Equations (EJDE)[electronic only], 37 (2010), 1-5. 
    [44] Z. Sun and G. Uhlmann, Inverse problems in quasilinear anisotropic media, Amer. J. Math., 119 (1997), 771-797.  doi: 10.1353/ajm.1997.0027.
    [45] G. Uhlmann, Electrical impedance tomography and Calderón's problem, Inverse Problems, 25 (2009), 123011. doi: 10.1088/0266-5611/25/12/123011.
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