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

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April 2022, 16(2): 305-323. doi: 10.3934/ipi.2021051

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

Ru-Yu Lai ^1, and Laurel Ohm ^2,

1.	School of Mathematics, University of Minnesota, USA
2.	Courant Institute, New York University, USA

Received February 2021 Revised May 2021 Published April 2022 Early access July 2021

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.

Keywords: Inverse problems, fractional Laplacian, nonlinear perturbations.

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

Citation: Ru-Yu Lai, Laurel Ohm. Inverse problems for the fractional Laplace equation with lower order nonlinear perturbations. Inverse Problems and Imaging, 2022, 16 (2) : 305-323. doi: 10.3934/ipi.2021051

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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