We prove a unique continuation property for the fractional Laplacian $ (-\Delta)^s $ when $ s \in (-n/2, \infty)\setminus \mathbb{Z} $ where $ n\geq 1 $. In addition, we study Poincaré-type inequalities for the operator $ (-\Delta)^s $ when $ s\geq 0 $. We apply the results to show that one can uniquely recover, up to a gauge, electric and magnetic potentials from the Dirichlet-to-Neumann map associated to the higher order fractional magnetic Schrödinger equation. We also study the higher order fractional Schrödinger equation with singular electric potential. In both cases, we obtain a Runge approximation property for the equation. Furthermore, we prove a uniqueness result for a partial data problem of the $ d $-plane Radon transform in low regularity. Our work extends some recent results in inverse problems for more general operators.
Citation: |
[1] |
H. Abels, Pseudodifferential and Singular Integral Operators, De Gruyter Graduate Lectures, De Gruyter, Berlin, 2012, An introduction with applications.
![]() ![]() |
[2] |
A. Abouelaz, The $d$-plane Radon transform on the torus $\Bbb T^n$, Fract. Calc. Appl. Anal., 14 (2011), 233-246.
doi: 10.2478/s13540-011-0014-8.![]() ![]() ![]() |
[3] |
F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. J. Toledo-Melero, Nonlocal Diffusion Problems, vol. 165 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI; Real Sociedad Matemática Española, Madrid, 2010.
doi: 10.1090/surv/165.![]() ![]() ![]() |
[4] |
H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, vol. 343 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer, Heidelberg, 2011.
doi: 10.1007/978-3-642-16830-7.![]() ![]() ![]() |
[5] |
A. Behzadan and M. Holst, Multiplication in Sobolev spaces, revisited, arXiv: 1512.07379.
![]() |
[6] |
J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Grundlehren der Mathematischen Wissenschaften, No. 223. Springer-Verlag, Berlin-New York, 1976.
![]() ![]() |
[7] |
C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, vol. 20 of Lecture Notes of the Unione Matematica Italiana, Springer, [Cham]; Unione Matematica Italiana, Bologna, 2016.
doi: 10.1007/978-3-319-28739-3.![]() ![]() ![]() |
[8] |
X. Cabré and 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.
doi: 10.1016/j.anihpc.2013.02.001.![]() ![]() ![]() |
[9] |
L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306.![]() ![]() ![]() |
[10] |
X. Cao, Y.-H. Lin and H. Liu, Simultaneously recovering potentials and embedded obstacles for anisotropic fractional Schrödinger operators, Inverse Probl. Imaging, 13 (2019), 197-210.
doi: 10.3934/ipi.2019011.![]() ![]() ![]() |
[11] |
M. Cekić, Y.-H. Lin and A. Rüland, The Calderón problem for the fractional Schrödinger equation with drift, Calc. Var. Partial Differential Equations, 59 (2020), Paper No. 91, 46pp.
doi: 10.1007/s00526-020-01740-6.![]() ![]() ![]() |
[12] |
S. N. Chandler-Wilde, D. P. Hewett and A. Moiola, Sobolev spaces on non-Lipschitz subsets of $\Bbb{R}^n$ with application to boundary integral equations on fractal screens, Integral Equations Operator Theory, 87 (2017), 179-224.
doi: 10.1007/s00020-017-2342-5.![]() ![]() ![]() |
[13] |
M. Courdurier, F. Noo, M. Defrise and H. Kudo, Solving the interior problem of computed tomography using a priori knowledge, Inverse Problems, 24 (2008), 065001, 27pp.
doi: 10.1088/0266-5611/24/6/065001.![]() ![]() ![]() |
[14] |
G. Covi, An inverse problem for the fractional Schrödinger equation in a magnetic field, Inverse Problems, 36 (2020), 045004, 24pp.
doi: 10.1088/1361-6420/ab661a.![]() ![]() ![]() |
[15] |
G. Covi, Inverse problems for a fractional conductivity equation, Nonlinear Anal., 193 (2020), 111418, 18pp.
doi: 10.1016/j.na.2019.01.008.![]() ![]() ![]() |
[16] |
G. Covi, K. Mönkkönen, J. Railo and G. Uhlmann, The higher order fractional Calderón problem for linear local operators: uniqueness, arXiv: 2008.10227.
![]() |
[17] |
A. D'Agnolo and M. Eastwood, Radon and Fourier transforms for $\mathcal{D}$-modules, Adv. Math., 180 (2003), 452-485.
doi: 10.1016/S0001-8708(03)00011-2.![]() ![]() ![]() |
[18] |
S. Dipierro, O. Savin and E. Valdinoci, All functions are locally $s$-harmonic up to a small error, J. Eur. Math. Soc., 19 (2017), 957-966.
doi: 10.4171/JEMS/684.![]() ![]() ![]() |
[19] |
Q. Du, M. Gunzburger, R. B. Lehoucq and K. Zhou, Analysis and approximation of nonlocal diffusion problems with volume constraints, SIAM Rev., 54 (2012), 667-696.
doi: 10.1137/110833294.![]() ![]() ![]() |
[20] |
Q. Du, M. Gunzburger, R. B. Lehoucq and K. Zhou, A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws, Math. Models Methods Appl. Sci., 23 (2013), 493-540.
doi: 10.1142/S0218202512500546.![]() ![]() ![]() |
[21] |
G. Eskin, Global uniqueness in the inverse scattering problem for the Schrödinger operator with external Yang-Mills potentials, Comm. Math. Phys., 222 (2001), 503-531.
doi: 10.1007/s002200100522.![]() ![]() ![]() |
[22] |
L. C. Evans, Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics, 2nd edition, American Mathematical Society, Providence, RI, 2010.
doi: 10.1090/gsm/019.![]() ![]() ![]() |
[23] |
M. M. Fall and V. Felli, Unique continuation property and local asymptotics of solutions to fractional elliptic equations, Comm. Partial Differential Equations, 39 (2014), 354-397.
doi: 10.1080/03605302.2013.825918.![]() ![]() ![]() |
[24] |
V. Felli and A. Ferrero, Unique continuation principles for a higher order fractional Laplace equation, Nonlinearity, 33 (2020), 4133-4190.
doi: 10.1088/1361-6544/ab8691.![]() ![]() ![]() |
[25] |
G. B. Folland and A. Sitaram, The uncertainty principle: a mathematical survey, J. Fourier Anal. Appl., 3 (1997), 207-238.
doi: 10.1007/BF02649110.![]() ![]() |
[26] |
J. Frikel and E. T. Quinto, Limited data problems for the generalized Radon transform in $\Bbb R^n$, SIAM J. Math. Anal., 48 (2016), 2301-2318.
doi: 10.1137/15M1045405.![]() ![]() ![]() |
[27] |
M. A. García-Ferrero and A. Rüland, Strong unique continuation for the higher order fractional Laplacian, Math. Eng., 1 (2019), 715-774.
doi: 10.3934/mine.2019.4.715.![]() ![]() ![]() |
[28] |
T. Ghosh, A. Rüland, M. Salo and G. Uhlmann, Uniqueness and reconstruction for the fractional Calderón problem with a single measurement, J. Funct. Anal., 279 (2020), 108505, 42pp.
doi: 10.1016/j.jfa.2020.108505.![]() ![]() ![]() |
[29] |
T. Ghosh, M. Salo and G. Uhlmann, The Calderón problem for the fractional Schrödinger equation, Anal. PDE, 13 (2020), 455-475.
doi: 10.2140/apde.2020.13.455.![]() ![]() ![]() |
[30] |
F. O. Goncharov, An iterative inversion of weighted radon transforms along hyperplanes, Inverse Problems, 33 (2017), 124005, 20pp.
doi: 10.1088/1361-6420/aa91a4.![]() ![]() ![]() |
[31] |
F. O. Goncharov and R. G. Novikov, An example of non-uniqueness for Radon transforms with continuous positive rotation invariant weights, J. Geom. Anal., 28 (2018), 3807-3828.
doi: 10.1007/s12220-018-0001-y.![]() ![]() ![]() |
[32] |
F. O. Goncharov and R. G. Novikov, An example of non-uniqueness for the weighted Radon transforms along hyperplanes in multidimensions, Inverse Problems, 34 (2018), 054001, 6pp.
doi: 10.1088/1361-6420/aab24d.![]() ![]() ![]() |
[33] |
F. B. Gonzalez, On the range of the Radon $d$-plane transform and its dual, Trans. Amer. Math. Soc., 327 (1991), 601-619.
doi: 10.2307/2001816.![]() ![]() ![]() |
[34] |
B. Harrach and Y.-H. Lin, Monotonicity-based inversion of the fractional Schrödinger equation I. Positive potentials, SIAM J. Math. Anal., 51 (2019), 3092-3111.
doi: 10.1137/18M1166298.![]() ![]() ![]() |
[35] |
B. Harrach and Y.-H. Lin, Monotonicity-based inversion of the fractional Schödinger equation II. General potentials and stability, SIAM J. Math. Anal., 52 (2020), 402-436.
doi: 10.1137/19M1251576.![]() ![]() ![]() |
[36] |
H. Heck, X. Li and J.-N. Wang, Identification of viscosity in an incompressible fluid, Indiana Univ. Math. J., 56 (2007), 2489-2510.
doi: 10.1512/iumj.2007.56.3037.![]() ![]() ![]() |
[37] |
S. Helgason, Integral Geometry and Radon transforms, Springer, New York, 2011.
doi: 10.1007/978-1-4419-6055-9.![]() ![]() ![]() |
[38] |
A. Homan and H. Zhou, Injectivity and stability for a generic class of generalized Radon transforms, J. Geom. Anal., 27 (2017), 1515-1529.
doi: 10.1007/s12220-016-9729-4.![]() ![]() ![]() |
[39] |
L. Hörmander, The Analysis of Linear Partial Differential Operators. I, 2nd edition, Springer Study Edition, Springer-Verlag, Berlin, 1990, Distribution theory and Fourier analysis.
doi: 10.1007/978-3-642-61497-2.![]() ![]() ![]() |
[40] |
J. Horváth, Topological Vector Spaces and Distributions. Vol. I, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1966.
![]() |
[41] |
J. Ilmavirta, On Radon transforms on tori, J. Fourier Anal. Appl., 21 (2015), 370-382.
doi: 10.1007/s00041-014-9374-x.![]() ![]() ![]() |
[42] |
J. Ilmavirta and K. Mönkkönen, Unique continuation of the normal operator of the x-ray transform and applications in geophysics, Inverse Problems, 36 (2020), 045014, 23pp.
doi: 10.1088/1361-6420/ab6e75.![]() ![]() ![]() |
[43] |
E. Katsevich, A. Katsevich and G. Wang, Stability of the interior problem with polynomial attenuation in the region of interest, Inverse Problems, 28 (2012), 065022, 28pp.
doi: 10.1088/0266-5611/28/6/065022.![]() ![]() ![]() |
[44] |
E. Klann, E. T. Quinto and R. Ramlau, Wavelet methods for a weighted sparsity penalty for region of interest tomography, Inverse Problems, 31 (2015), 025001, 22pp.
doi: 10.1088/0266-5611/31/2/025001.![]() ![]() ![]() |
[45] |
V. P. Krishnan and E. T. Quinto, Microlocal Analysis in Tomography, in Handbook of Mathematical Methods in Imaging (ed. O. Scherzer), Springer, New York, 2015,847–902.
doi: 10.1007/978-1-4939-0790-8_36.![]() ![]() ![]() |
[46] |
N. V. Krylov, All functions are locally $s$-harmonic up to a small error, J. Funct. Anal., 277 (2019), 2728-2733.
doi: 10.1016/j.jfa.2019.02.012.![]() ![]() ![]() |
[47] |
M. Kwaśnicki, Ten equivalent definitions of the fractional Laplace operator, Fract. Calc. Appl. Anal., 20 (2017), 7-51.
doi: 10.1515/fca-2017-0002.![]() ![]() ![]() |
[48] |
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.![]() ![]() ![]() |
[49] |
R.-Y. Lai and Y.-H. Lin, Inverse problems for fractional semilinear elliptic equations, arXiv: 2004.00549.
![]() |
[50] |
R.-Y. Lai, Y.-H. Lin and A. Rüland, The Calderón problem for a space-time fractional parabolic equation, SIAM J. Math. Anal., 52 (2020), 2655-2688.
doi: 10.1137/19M1270288.![]() ![]() ![]() |
[51] |
N. Laskin, Fractional Quantum Mechanics, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2018.
doi: 10.1142/10541.![]() ![]() ![]() |
[52] |
N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.
doi: 10.1016/S0375-9601(00)00201-2.![]() ![]() ![]() |
[53] |
L. Li, A semilinear inverse problem for the fractional magnetic Laplacian, arXiv: 2005.06714.
![]() |
[54] |
L. Li, The Calderón problem for the fractional magnetic operator, Inverse Problems, 36 (2020), 075003, 14pp.
doi: 10.1088/1361-6420/ab8445.![]() ![]() ![]() |
[55] |
L. Li, Determining the magnetic potential in the fractional magnetic Calderón problem, arXiv: 2006.10150.
![]() |
[56] |
V. G. Maz'ya and T. O. Shaposhnikova, Theory of Sobolev Multipliers, vol. 337 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 2009, With applications to differential and integral operators.
![]() ![]() |
[57] |
S. R. McDowall, An electromagnetic inverse problem in chiral media, Trans. Amer. Math. Soc., 352 (2000), 2993-3013.
doi: 10.1090/S0002-9947-00-02518-6.![]() ![]() ![]() |
[58] |
W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000.
![]() ![]() |
[59] |
R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 77pp.
doi: 10.1016/S0370-1573(00)00070-3.![]() ![]() ![]() |
[60] |
D. Mitrea, Distributions, Partial Differential Equations, and Harmonic Analysis, Universitext, Springer, New York, 2013.
doi: 10.1007/978-1-4614-8208-6.![]() ![]() ![]() |
[61] |
G. Nakamura, Z. Q. Sun and G. Uhlmann, Global identifiability for an inverse problem for the Schrödinger equation in a magnetic field, Math. Ann., 303 (1995), 377-388.
doi: 10.1007/BF01460996.![]() ![]() ![]() |
[62] |
G. Nakamura and T. Tsuchida, Uniqueness for an inverse boundary value problem for Dirac operators, Comm. Partial Differential Equations, 25 (2000), 1327-1369.
doi: 10.1080/03605300008821551.![]() ![]() ![]() |
[63] |
G. Nakamura and G. Uhlmann, Global uniqueness for an inverse boundary problem arising in elasticity, Invent. Math., 118 (1994), 457-474.
doi: 10.1007/BF01231541.![]() ![]() ![]() |
[64] |
F. Natterer, The Mathematics of Computerized Tomography, vol. 32 of Classics in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2001, Reprint of the 1986 original.
doi: 10.1137/1.9780898719284.![]() ![]() ![]() |
[65] |
T. Ozawa, On critical cases of Sobolev's inequalities, J. Funct. Anal., 127 (1995), 259-269.
doi: 10.1006/jfan.1995.1012.![]() ![]() ![]() |
[66] |
E. T. Quinto, Singularities of the X-ray transform and limited data tomography in ${\mathbf{R}}^2$ and ${\mathbf{R}}^3$, SIAM J. Math. Anal., 24 (1993), 1215-1225.
doi: 10.1137/0524069.![]() ![]() ![]() |
[67] |
E. T. Quinto, Artifacts and visible singularities in limited data X-ray tomography, Sens Imaging, 18 (2017), 9.
doi: 10.1007/s11220-017-0158-7.![]() ![]() |
[68] |
J. Railo, Fourier analysis of periodic Radon transforms, J. Fourier Anal. Appl., 26 (2020), Paper No. 64, 27pp.
doi: 10.1007/s00041-020-09775-1.![]() ![]() ![]() |
[69] |
A. G. Ramm and A. I. Katsevich, The Radon Transform and Local Tomography, CRC Press, Boca Raton, FL, 1996.
![]() ![]() |
[70] |
T. Reichelt, A comparison theorem between Radon and Fourier-Laplace transforms for D-modules, Ann. Inst. Fourier (Grenoble), 65 (2015), 1577-1616.
doi: 10.5802/aif.2968.![]() ![]() ![]() |
[71] |
M. Riesz, Intégrales de Riemann-Liouville et potentiels, Acta Sci. Math. Szeged, 9 (1938), 1-42.
![]() |
[72] |
X. Ros-Oton, Nonlocal equations in bounded domains: A survey, Publ. Mat., 60 (2016), 3-26.
doi: 10.5565/PUBLMAT_60116_01.![]() ![]() ![]() |
[73] |
A. Rüland, Unique continuation for fractional Schrödinger equations with rough potentials, Comm. Partial Differential Equations, 40 (2015), 77-114.
doi: 10.1080/03605302.2014.905594.![]() ![]() ![]() |
[74] |
A. Rüland and M. Salo, Exponential instability in the fractional Calderón problem, Inverse Problems, 34 (2018), 045003, 21pp.
doi: 10.1088/1361-6420/aaac5a.![]() ![]() ![]() |
[75] |
A. Rüland and M. Salo, The fractional Calderón problem: Low regularity and stability, Nonlinear Anal., 193 (2020), 111529, 56pp.
doi: 10.1016/j.na.2019.05.010.![]() ![]() ![]() |
[76] |
A. Rüland and M. Salo, Quantitative approximation properties for the fractional heat equation, Math. Control Relat. Fields, 10 (2020), 1-26.
doi: 10.3934/mcrf.2019027.![]() ![]() ![]() |
[77] |
M. Salo, Recovering first order terms from boundary measurements, J. Phys.: Conf. Ser., 73 (2007), 012020.
doi: 10.1088/1742-6596/73/1/012020.![]() ![]() |
[78] |
M. Salo, Calderón problem, 2008, http://users.jyu.fi/~salomi/lecturenotes/calderon_lectures.pdf, Lecture notes.
![]() |
[79] |
M. Salo, Fourier analysis and distribution theory, 2013, http://users.jyu.fi/~salomi/lecturenotes/FA_distributions.pdf, Lecture notes.
![]() |
[80] |
M. Salo, The fractional calderón problem, Journées Équations aux Dérivées Partielles, 2017, 8pp.
doi: 10.5802/jedp.657.![]() ![]() |
[81] |
L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.
doi: 10.1002/cpa.20153.![]() ![]() ![]() |
[82] |
P. Stefanov and G. Uhlmann, Microlocal Analysis and Integral Geometry (working title), 2018, http://www.math.purdue.edu/~stefanov/publications/book.pdf, Draft version.
![]() |
[83] |
F. Trèves, Topological Vector Spaces, Distributions and Kernels, Academic Press, New York-London, 1967.
![]() ![]() |
[84] |
G. Uhlmann, Inverse problems: Seeing the unseen, Bull. Math. Sci., 4 (2014), 209-279.
doi: 10.1007/s13373-014-0051-9.![]() ![]() ![]() |
[85] |
L. Xiaojun, A Note On Fractional Order Poincarés Inequalities, 2012, http://www.bcamath.org/documentos_public/archivos/publicaciones/Poicare_Academie.pdf.
![]() |
[86] |
J. Yang, H. Yu, M. Jiang and G. Wang, High-order total variation minimization for interior tomography, Inverse Problems, 26 (2010), 035013, 29pp.
doi: 10.1088/0266-5611/26/3/035013.![]() ![]() ![]() |
[87] |
R. Yang, On higher order extensions for the fractional Laplacian, arXiv: 1302.4413.
![]() |
[88] |
Y. Ye, H. Yu and G. Wang, Exact interior reconstruction from truncated limited-angle projection data, International Journal of Biomedical Imaging, 2008 (2008), Article ID 427989.
doi: 10.1155/2008/427989.![]() ![]() |
[89] |
H. Yu and G. Wang, Compressed sensing based interior tomography, Phys. Med. Biol., 54 (2009), 2791–2805, https://doi.org/10.1088%2F0031-9155%2F54%2F9%2F014.
![]() |