Lipschitz stability for the finite dimensional fractional Calderón problem with finite Cauchy data

Angkana Rüland; Eva Sincich

doi:10.3934/ipi.2019046

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October 2019, 13(5): 1023-1044. doi: 10.3934/ipi.2019046

Lipschitz stability for the finite dimensional fractional Calderón problem with finite Cauchy data

Angkana Rüland ^1, and Eva Sincich ^2,,

1.	Max-Planck Institute for Mathematics in the Sciences, Inselstraße 22, 04103 Leipzig, Germany
2.	Dipartimento di Matematica e Geoscienze Università degli Studi di Trieste, via Valerio 12/1, 34127 Trieste, Italy

^* Corresponding author: Eva Sincich

Received November 2018 Revised April 2019 Published July 2019

In this note we discuss the conditional stability issue for the finite dimensional Calderón problem for the fractional Schrödinger equation with a finite number of measurements. More precisely, we assume that the unknown potential $ q \in L^{\infty}(\Omega) $ in the equation $ ((- \Delta)^s+ q)u = 0 \mbox{ in } \Omega\subset \mathbb{R}^n $ satisfies the a priori assumption that it is contained in a finite dimensional subspace of $ L^{\infty}(\Omega) $. Under this condition we prove Lipschitz stability estimates for the fractional Calderón problem by means of finitely many Cauchy data depending on $ q $. We allow for the possibility of zero being a Dirichlet eigenvalue of the associated fractional Schrödinger equation. Our result relies on the strong Runge approximation property of the fractional Schrödinger equation.

Keywords: Lipschitz stability, finite dimensional fractional Calderón problem, finite Cauchy data.

Mathematics Subject Classification: Primary: 35R30, 35A35, 35S15.

Citation: Angkana Rüland, Eva Sincich. Lipschitz stability for the finite dimensional fractional Calderón problem with finite Cauchy data. Inverse Problems & Imaging, 2019, 13 (5) : 1023-1044. doi: 10.3934/ipi.2019046

References:

[1]	G. S. Alberti and M. Santacesaria, Calderón's Inverse Problem with a Finite Number of Measurements, arXiv: 1803.04224, 2018. Google Scholar
[2]	G. Alessandrini, M. V. de Hoop, R. Gaburro and E. Sincich, Lipschitz stability for the electrostatic inverse boundary value problem with piecewise linear conductivities, Journal de Mathématiques Pures et Appliquées, 107 (2016), 638-664. doi: 10.1016/j.matpur.2016.10.001. Google Scholar
[3]	G. Alessandrini, M. V. de Hoop and R. Gaburro, Romina and Eva Sincich, Lipschitz stability for a piecewise linear Schrödinger potential from local Cauchy data, Asymptotic Analysis, 108 (2018), 115-149. doi: 10.3233/ASY-171457. Google Scholar
[4]	G. Alessandrini and V. Isakov, Analiticity and uniqueness for the inverse conductivity problem, Rend. Ist. Mat. Univ. Trieste, 28 (1996), 351-369. Google Scholar
[5]	G. Alessandrini, L. Rondi, E. Rosset and S. Vessella, The stability for the Cauchy problem for elliptic equations, Inverse Problems, 25 (2009), 123004, 47pp. doi: 10.1088/0266-5611/25/12/123004. Google Scholar
[6]	G. Alessandrini and S. Vessella, Lipschitz stability for the inverse conductivity problem, Advances in Applied Mathematics, 35 (2005), 207-241. doi: 10.1016/j.aam.2004.12.002. Google Scholar
[7]	B. Barceló, E. Fabes and J. Keun Seo, The inverse conductivity problem with one measurement: Uniqueness for convex polyhedra, Proc. Amer. Math. Soc., 122 (1994), 183-189. doi: 10.1090/S0002-9939-1994-1195476-6. Google Scholar
[8]	E. Beretta, M. V. de Hoop and L. Qiu, Lipschitz stability for an inverse boundary value problem for a Schrödinger-type equation, SIAM Journal on Mathematical Analysis, 45 (2013), 679-699. doi: 10.1137/120869201. Google Scholar
[9]	E. Beretta and E. Francini, Lipschitz stability for the electrical impedance tomography problem: The complex case, Comm. Partial Differential Equations, 36 (2011), 1723-1749. doi: 10.1080/03605302.2011.552930. Google Scholar
[10]	E. Beretta, E. Francini and S. Vessella, Lipschitz stability estimates for polygonal conductivity inclusions from boundary measurements, arXiv: 1901.01152. Google Scholar
[11]	E. Blåsten and H. Liu, Recovering piecewise constant refractive indices by a single far-field pattern, arXiv: 1705.00815, 2017. Google Scholar
[12]	X. Cao and H. Liu, Determining a fractional Helmholtz system with unknown source and medium parameter, arXiv preprint, arXiv: 1803.09538, 2018. Google Scholar
[13]	X. Cao, Y.-H. Lin and H. Liu, Simultaneously recovering potentials and embedded obstacles for anisotropic fractional Schrödinger operators, AIMS, 13 (2019), 197–210, arXiv: 1712.00937. doi: 10.3934/ipi.2019011. Google Scholar
[14]	S. Dipierro, O. Savin and E. Valdinoci, All functions are locally s-harmonic up to a small error, J. Eur. Math. Soc. (JEMS), 19 (2017), 957-966. doi: 10.4171/JEMS/684. Google Scholar
[15]	E. Fabes, H. Kang and J. K. Seo, Inverse conductivity problem with one measurement: Error estimates and approximate identification for perturbed disks, SIAM J. Math. Anal., 30 (1999), 699-720. doi: 10.1137/S0036141097324958. Google Scholar
[16]	M. Moustapha Fall and V. Felli, Unique continuation property and local asymptotics of solutions to fractional elliptic equations, Communication in Partial Differential Equations, 39 (2014), 354-397. doi: 10.1080/03605302.2013.825918. Google Scholar
[17]	A. Friedman and V. Isakov, On the uniqueness in the inverse conductivity problem with one measurement, Indiana Univ. Math J., 38 (1989), 563-579. doi: 10.1512/iumj.1989.38.38027. Google Scholar
[18]	R. Gaburro and E. Sincich, Lipschitz stability for the inverse conductivity problem for a conformal class of anisotropic conductivities, Inverse Problems, 31 (2015), 015008, 26pp. doi: 10.1088/0266-5611/31/1/015008. Google Scholar
[19]	B. Gebauer, Localized potentials in electrical impedance tomography, AIMS, 2 (2018), 251-369. doi: 10.3934/ipi.2008.2.251. Google Scholar
[20]	T. Ghosh, Y.-H. Lin and J. Xiao, The Calderón problem for variable coefficients nonlocal elliptic operators, Communication in Partial Differential Equations, 42 (2017), 1923-1961. doi: 10.1080/03605302.2017.1390681. Google Scholar
[21]	T. Ghosh, A. Rüland, M. Salo and G. Uhlmann, Uniqueness and reconstruction for the fractional Calderón problem with a single measurement, arXiv: 1801.04449, 2018. Google Scholar
[22]	T. Ghosh, M. Salo and G. Uhlmann, The Calderón problem for the fractional Schrödinger equation, to appear in Analysis and PDE. Google Scholar
[23]	G. Grubb, Fractional Laplacians on domains, a development of Hörmander's theory of $\mu$-transmission pseudodifferential operators, Advances in Mathematics, 268 (2015), 478-528. doi: 10.1016/j.aim.2014.09.018. Google Scholar
[24]	B. Harrach and Y.-H. Lin, Monotonicity-based inversion of the fractional Schrödinger equation, arXiv: 1711.05641, 2017. Google Scholar
[25]	B. Harrach, V. Pojola and M. Salo, Monotonicity and local uniqueness for the Helmholtz equation, arXiv: 1709.08756, 2017. Google Scholar
[26]	M. Ikehata, On reconstruction in the inverse conductivity problem with one measurement, Inverse Problems, 16 (2000), 785-793. doi: 10.1088/0266-5611/16/3/314. Google Scholar
[27]	M. Ikehata, On reconstruction from a partial knowledge of the Neumann-to-Dirichlet operator, Inverse Problems, 17 (2001), 45-51. doi: 10.1088/0266-5611/17/1/304. Google Scholar
[28]	M. Ikehata, Extraction formulae for an inverse boundary value problem for the equation $\nabla\cdot(\sigma-i\omega\epsilon)\nabla u = 0$, Inverse Problems, 18 (2002), 1281-1290. doi: 10.1088/0266-5611/18/5/304. Google Scholar
[29]	V. Isakov and J. Powell, On the inverse conductivity problem with one measurement, Inverse Problems, 6 (1990), 311-318. doi: 10.1088/0266-5611/6/2/011. Google Scholar
[30]	H. Liu and C.-H. Tsou, Stable determination of polygonal inclusions in Calderón's problem by a single partial boundary measurement, arXiv: 1902.04462. 2019. Google Scholar
[31]	N. Mandache, Exponential instability in an inverse problem for the Schrödinger equation, Inverse Problems, 17 (2001), 1435-1444. doi: 10.1088/0266-5611/17/5/313. Google Scholar
[32]	W. McLean, Strongly Elliptic Systems and Boundary Integral Equations,, Cambridge University Press, Cambridge, 2000. Google Scholar
[33]	L. Rondi, A remark on a paper by Alessandrini and Vessella, Advances in Applied Mathematics, 36 (2006), 67-69. doi: 10.1016/j.aam.2004.12.003. Google Scholar
[34]	R.-Y. Lai and Y.-H. Lin, Global uniqueness for the semilinear fractional Schrödinger equation, Proceedings of the American Mathematical Society, 147 (2019), 1189-1199. Google Scholar
[35]	A. Rüland, Unique continuation for fractional Schrödinger equations with rough potentials, Communications In Partial Differential Equations, 40 (2015), 77-114. doi: 10.1080/03605302.2014.905594. Google Scholar
[36]	A. Rüland and M. Salo, The fractional Calderón problem: Low regularity and stability, to appear in Nonlinear Analysis. Google Scholar
[37]	A. Rüland and M. Salo, Exponential instability in the fractional Calderón problem, Inverse Problems, 34 (2018), 045003, 21pp. doi: 10.1088/1361-6420/aaac5a. Google Scholar
[38]	M. Salo, The fractional Calderón problem, Journés Équations aux Dérivées Partielles, 7 (2017), 8pp, arXiv: 1711.06103. doi: 10.5802/jedp.657. Google Scholar
[39]	J. K. Seo, On the uniqueness in the inverse conductivity problem, J. Fourier Anal. Appl., 2 (1996), 227-235. doi: 10.1007/s00041-001-4030-7. Google Scholar
[40]	H. Yu, Unique continuation for fractional orders of elliptic equations, Annals of PDE (2017) 3: 16. https://doi.org/10.1007/s40818-017-0033-9 Google Scholar

show all references

References:

[1]	G. S. Alberti and M. Santacesaria, Calderón's Inverse Problem with a Finite Number of Measurements, arXiv: 1803.04224, 2018. Google Scholar
[2]	G. Alessandrini, M. V. de Hoop, R. Gaburro and E. Sincich, Lipschitz stability for the electrostatic inverse boundary value problem with piecewise linear conductivities, Journal de Mathématiques Pures et Appliquées, 107 (2016), 638-664. doi: 10.1016/j.matpur.2016.10.001. Google Scholar
[3]	G. Alessandrini, M. V. de Hoop and R. Gaburro, Romina and Eva Sincich, Lipschitz stability for a piecewise linear Schrödinger potential from local Cauchy data, Asymptotic Analysis, 108 (2018), 115-149. doi: 10.3233/ASY-171457. Google Scholar
[4]	G. Alessandrini and V. Isakov, Analiticity and uniqueness for the inverse conductivity problem, Rend. Ist. Mat. Univ. Trieste, 28 (1996), 351-369. Google Scholar
[5]	G. Alessandrini, L. Rondi, E. Rosset and S. Vessella, The stability for the Cauchy problem for elliptic equations, Inverse Problems, 25 (2009), 123004, 47pp. doi: 10.1088/0266-5611/25/12/123004. Google Scholar
[6]	G. Alessandrini and S. Vessella, Lipschitz stability for the inverse conductivity problem, Advances in Applied Mathematics, 35 (2005), 207-241. doi: 10.1016/j.aam.2004.12.002. Google Scholar
[7]	B. Barceló, E. Fabes and J. Keun Seo, The inverse conductivity problem with one measurement: Uniqueness for convex polyhedra, Proc. Amer. Math. Soc., 122 (1994), 183-189. doi: 10.1090/S0002-9939-1994-1195476-6. Google Scholar
[8]	E. Beretta, M. V. de Hoop and L. Qiu, Lipschitz stability for an inverse boundary value problem for a Schrödinger-type equation, SIAM Journal on Mathematical Analysis, 45 (2013), 679-699. doi: 10.1137/120869201. Google Scholar
[9]	E. Beretta and E. Francini, Lipschitz stability for the electrical impedance tomography problem: The complex case, Comm. Partial Differential Equations, 36 (2011), 1723-1749. doi: 10.1080/03605302.2011.552930. Google Scholar
[10]	E. Beretta, E. Francini and S. Vessella, Lipschitz stability estimates for polygonal conductivity inclusions from boundary measurements, arXiv: 1901.01152. Google Scholar
[11]	E. Blåsten and H. Liu, Recovering piecewise constant refractive indices by a single far-field pattern, arXiv: 1705.00815, 2017. Google Scholar
[12]	X. Cao and H. Liu, Determining a fractional Helmholtz system with unknown source and medium parameter, arXiv preprint, arXiv: 1803.09538, 2018. Google Scholar
[13]	X. Cao, Y.-H. Lin and H. Liu, Simultaneously recovering potentials and embedded obstacles for anisotropic fractional Schrödinger operators, AIMS, 13 (2019), 197–210, arXiv: 1712.00937. doi: 10.3934/ipi.2019011. Google Scholar
[14]	S. Dipierro, O. Savin and E. Valdinoci, All functions are locally s-harmonic up to a small error, J. Eur. Math. Soc. (JEMS), 19 (2017), 957-966. doi: 10.4171/JEMS/684. Google Scholar
[15]	E. Fabes, H. Kang and J. K. Seo, Inverse conductivity problem with one measurement: Error estimates and approximate identification for perturbed disks, SIAM J. Math. Anal., 30 (1999), 699-720. doi: 10.1137/S0036141097324958. Google Scholar
[16]	M. Moustapha Fall and V. Felli, Unique continuation property and local asymptotics of solutions to fractional elliptic equations, Communication in Partial Differential Equations, 39 (2014), 354-397. doi: 10.1080/03605302.2013.825918. Google Scholar
[17]	A. Friedman and V. Isakov, On the uniqueness in the inverse conductivity problem with one measurement, Indiana Univ. Math J., 38 (1989), 563-579. doi: 10.1512/iumj.1989.38.38027. Google Scholar
[18]	R. Gaburro and E. Sincich, Lipschitz stability for the inverse conductivity problem for a conformal class of anisotropic conductivities, Inverse Problems, 31 (2015), 015008, 26pp. doi: 10.1088/0266-5611/31/1/015008. Google Scholar
[19]	B. Gebauer, Localized potentials in electrical impedance tomography, AIMS, 2 (2018), 251-369. doi: 10.3934/ipi.2008.2.251. Google Scholar
[20]	T. Ghosh, Y.-H. Lin and J. Xiao, The Calderón problem for variable coefficients nonlocal elliptic operators, Communication in Partial Differential Equations, 42 (2017), 1923-1961. doi: 10.1080/03605302.2017.1390681. Google Scholar
[21]	T. Ghosh, A. Rüland, M. Salo and G. Uhlmann, Uniqueness and reconstruction for the fractional Calderón problem with a single measurement, arXiv: 1801.04449, 2018. Google Scholar
[22]	T. Ghosh, M. Salo and G. Uhlmann, The Calderón problem for the fractional Schrödinger equation, to appear in Analysis and PDE. Google Scholar
[23]	G. Grubb, Fractional Laplacians on domains, a development of Hörmander's theory of $\mu$-transmission pseudodifferential operators, Advances in Mathematics, 268 (2015), 478-528. doi: 10.1016/j.aim.2014.09.018. Google Scholar
[24]	B. Harrach and Y.-H. Lin, Monotonicity-based inversion of the fractional Schrödinger equation, arXiv: 1711.05641, 2017. Google Scholar
[25]	B. Harrach, V. Pojola and M. Salo, Monotonicity and local uniqueness for the Helmholtz equation, arXiv: 1709.08756, 2017. Google Scholar
[26]	M. Ikehata, On reconstruction in the inverse conductivity problem with one measurement, Inverse Problems, 16 (2000), 785-793. doi: 10.1088/0266-5611/16/3/314. Google Scholar
[27]	M. Ikehata, On reconstruction from a partial knowledge of the Neumann-to-Dirichlet operator, Inverse Problems, 17 (2001), 45-51. doi: 10.1088/0266-5611/17/1/304. Google Scholar
[28]	M. Ikehata, Extraction formulae for an inverse boundary value problem for the equation $\nabla\cdot(\sigma-i\omega\epsilon)\nabla u = 0$, Inverse Problems, 18 (2002), 1281-1290. doi: 10.1088/0266-5611/18/5/304. Google Scholar
[29]	V. Isakov and J. Powell, On the inverse conductivity problem with one measurement, Inverse Problems, 6 (1990), 311-318. doi: 10.1088/0266-5611/6/2/011. Google Scholar
[30]	H. Liu and C.-H. Tsou, Stable determination of polygonal inclusions in Calderón's problem by a single partial boundary measurement, arXiv: 1902.04462. 2019. Google Scholar
[31]	N. Mandache, Exponential instability in an inverse problem for the Schrödinger equation, Inverse Problems, 17 (2001), 1435-1444. doi: 10.1088/0266-5611/17/5/313. Google Scholar
[32]	W. McLean, Strongly Elliptic Systems and Boundary Integral Equations,, Cambridge University Press, Cambridge, 2000. Google Scholar
[33]	L. Rondi, A remark on a paper by Alessandrini and Vessella, Advances in Applied Mathematics, 36 (2006), 67-69. doi: 10.1016/j.aam.2004.12.003. Google Scholar
[34]	R.-Y. Lai and Y.-H. Lin, Global uniqueness for the semilinear fractional Schrödinger equation, Proceedings of the American Mathematical Society, 147 (2019), 1189-1199. Google Scholar
[35]	A. Rüland, Unique continuation for fractional Schrödinger equations with rough potentials, Communications In Partial Differential Equations, 40 (2015), 77-114. doi: 10.1080/03605302.2014.905594. Google Scholar
[36]	A. Rüland and M. Salo, The fractional Calderón problem: Low regularity and stability, to appear in Nonlinear Analysis. Google Scholar
[37]	A. Rüland and M. Salo, Exponential instability in the fractional Calderón problem, Inverse Problems, 34 (2018), 045003, 21pp. doi: 10.1088/1361-6420/aaac5a. Google Scholar
[38]	M. Salo, The fractional Calderón problem, Journés Équations aux Dérivées Partielles, 7 (2017), 8pp, arXiv: 1711.06103. doi: 10.5802/jedp.657. Google Scholar
[39]	J. K. Seo, On the uniqueness in the inverse conductivity problem, J. Fourier Anal. Appl., 2 (1996), 227-235. doi: 10.1007/s00041-001-4030-7. Google Scholar
[40]	H. Yu, Unique continuation for fractional orders of elliptic equations, Annals of PDE (2017) 3: 16. https://doi.org/10.1007/s40818-017-0033-9 Google Scholar

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Lipschitz stability for the finite dimensional fractional Calderón problem with finite Cauchy data

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