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Variational source conditions for inverse Robin and flux problems by partial measurements
Inverse problems for the fractional Laplace equation with lower order nonlinear perturbations
1. | School of Mathematics, University of Minnesota, USA |
2. | Courant Institute, New York University, USA |
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.
References:
[1] |
S. Bhattacharya, T. 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. Cao, Y.-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. Cârstea, G. 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. Covi, K. 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. Ghosh, Y.-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. Ghosh, M. 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. Lai, Y.-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. Lassas, T. Liimatainen, Y.-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. Lassas, T. Liimatainen, Y.-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. McLean, Strongly 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. |
show all references
References:
[1] |
S. Bhattacharya, T. 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. Cao, Y.-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. Cârstea, G. 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. Covi, K. 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. Ghosh, Y.-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. Ghosh, M. 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. Lai, Y.-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. Lassas, T. Liimatainen, Y.-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. Lassas, T. Liimatainen, Y.-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. McLean, Strongly 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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