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

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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 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 & Imaging, 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. Google Scholar
[2]	A. P. Calderón, Seminar in numerical analysis and its applications to continuum physics (Río de Janeiro: Soc. Brasileira de Matemática), (1980), 65–73. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[7]	G. Covi, Inverse problems for a fractional conductivity equation, Nonlinear Analysis, 193 (2020), 111418. doi: 10.1016/j. na. 2019.01.008. Google Scholar
[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. Google Scholar
[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). Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[26]	R. -Y. Lai and Y. -H. Lin, Inverse problems for fractional semilinear elliptic equations, arXiv: 2004.00549, (2020). Google Scholar
[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. Google Scholar
[28]	R. -Y. Lai and T. Zhou, Partial data inverse problems for nonlinear magnetic Schrödinger equations, arXiv: 2007.02475, (2020). Google Scholar
[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. Google Scholar
[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. Google Scholar
[31]	L. Li, The Calderón problem for the fractional magnetic operator, Inverse Problems, 36 (2020), 075003. doi: 10.1088/1361-6420/ab8445. Google Scholar
[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. Google Scholar
[33]	L. Li, A semilinear inverse problem for the fractional magnetic Laplacian, arXiv: 2005.06714, (2020). Google Scholar
[34]	Y. H. Lin, Monotonicity-based inversion of fractional semilinear elliptic equations with power type nonlinearities, arXiv: 2005.07163, (2020). Google Scholar
[35]	W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, 2020. Google Scholar
[36]	M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations, Springer-Verlag, New York, 13, 2004. Google Scholar
[37]	X. Ros-Oton, Nonlocal elliptic equations in bounded domains: A survey, Publicacions Matemètiques., 60 (2016), 3–26. doi: 10.5565/PUBLMAT_60116_01. Google Scholar
[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. Google Scholar
[39]	A. Rüland, On single measurement stability for the fractional Calderón problem, arXiv: 2007.13624, (2020). Google Scholar
[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. Google Scholar
[41]	Z. Sun, On a quasilinear inverse boundary value problem, Math. Z., 221 (1996), 293-305. doi: 10.1007/BF02622117. Google Scholar
[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. Google Scholar
[43]	Z. Sun, An inverse boundary-value problem for semilinear elliptic equations, Electronic Journal of Differential Equations (EJDE)[electronic only], 37 (2010), 1-5. Google Scholar
[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. Google Scholar
[45]	G. Uhlmann, Electrical impedance tomography and Calderón's problem, Inverse Problems, 25 (2009), 123011. doi: 10.1088/0266-5611/25/12/123011. Google Scholar

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. Google Scholar
[2]	A. P. Calderón, Seminar in numerical analysis and its applications to continuum physics (Río de Janeiro: Soc. Brasileira de Matemática), (1980), 65–73. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[7]	G. Covi, Inverse problems for a fractional conductivity equation, Nonlinear Analysis, 193 (2020), 111418. doi: 10.1016/j. na. 2019.01.008. Google Scholar
[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. Google Scholar
[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). Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[26]	R. -Y. Lai and Y. -H. Lin, Inverse problems for fractional semilinear elliptic equations, arXiv: 2004.00549, (2020). Google Scholar
[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. Google Scholar
[28]	R. -Y. Lai and T. Zhou, Partial data inverse problems for nonlinear magnetic Schrödinger equations, arXiv: 2007.02475, (2020). Google Scholar
[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. Google Scholar
[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. Google Scholar
[31]	L. Li, The Calderón problem for the fractional magnetic operator, Inverse Problems, 36 (2020), 075003. doi: 10.1088/1361-6420/ab8445. Google Scholar
[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. Google Scholar
[33]	L. Li, A semilinear inverse problem for the fractional magnetic Laplacian, arXiv: 2005.06714, (2020). Google Scholar
[34]	Y. H. Lin, Monotonicity-based inversion of fractional semilinear elliptic equations with power type nonlinearities, arXiv: 2005.07163, (2020). Google Scholar
[35]	W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, 2020. Google Scholar
[36]	M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations, Springer-Verlag, New York, 13, 2004. Google Scholar
[37]	X. Ros-Oton, Nonlocal elliptic equations in bounded domains: A survey, Publicacions Matemètiques., 60 (2016), 3–26. doi: 10.5565/PUBLMAT_60116_01. Google Scholar
[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. Google Scholar
[39]	A. Rüland, On single measurement stability for the fractional Calderón problem, arXiv: 2007.13624, (2020). Google Scholar
[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. Google Scholar
[41]	Z. Sun, On a quasilinear inverse boundary value problem, Math. Z., 221 (1996), 293-305. doi: 10.1007/BF02622117. Google Scholar
[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. Google Scholar
[43]	Z. Sun, An inverse boundary-value problem for semilinear elliptic equations, Electronic Journal of Differential Equations (EJDE)[electronic only], 37 (2010), 1-5. Google Scholar
[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. Google Scholar
[45]	G. Uhlmann, Electrical impedance tomography and Calderón's problem, Inverse Problems, 25 (2009), 123011. doi: 10.1088/0266-5611/25/12/123011. Google Scholar

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