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

Angkana Rüland; Eva Sincich

doi:10.3934/ipi.2019046

Article Contents

2019, Volume 13, Issue 5: 1023-1044. Doi: 10.3934/ipi.2019046

This issue Previous Article Identifiability of diffusion coefficients for source terms of non-uniform sign Next Article Determining rough first order perturbations of the polyharmonic operator

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
^* Corresponding author: Eva Sincich

Received: November 2018

Revised: April 2019

Published: July 2019

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Abstract

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.

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References

[1]	G. S. Alberti and M. Santacesaria, Calderón's Inverse Problem with a Finite Number of Measurements, arXiv: 1803.04224, 2018.
[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.
[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.
[4]	G. Alessandrini and V. Isakov, Analiticity and uniqueness for the inverse conductivity problem, Rend. Ist. Mat. Univ. Trieste, 28 (1996), 351-369.
[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.
[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.
[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.
[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.
[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.
[10]	E. Beretta, E. Francini and S. Vessella, Lipschitz stability estimates for polygonal conductivity inclusions from boundary measurements, arXiv: 1901.01152.
[11]	E. Blåsten and H. Liu, Recovering piecewise constant refractive indices by a single far-field pattern, arXiv: 1705.00815, 2017.
[12]	X. Cao and H. Liu, Determining a fractional Helmholtz system with unknown source and medium parameter, arXiv preprint, arXiv: 1803.09538, 2018.
[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.
[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.
[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.
[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.
[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.
[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.
[19]	B. Gebauer, Localized potentials in electrical impedance tomography, AIMS, 2 (2018), 251-369. doi: 10.3934/ipi.2008.2.251.
[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.
[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.
[22]	T. Ghosh, M. Salo and G. Uhlmann, The Calderón problem for the fractional Schrödinger equation, to appear in Analysis and PDE.
[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.
[24]	B. Harrach and Y.-H. Lin, Monotonicity-based inversion of the fractional Schrödinger equation, arXiv: 1711.05641, 2017.
[25]	B. Harrach, V. Pojola and M. Salo, Monotonicity and local uniqueness for the Helmholtz equation, arXiv: 1709.08756, 2017.
[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.
[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.
[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.
[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.
[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.
[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.
[32]	W. McLean, Strongly Elliptic Systems and Boundary Integral Equations,, Cambridge University Press, Cambridge, 2000.
[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.
[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.
[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.
[36]	A. Rüland and M. Salo, The fractional Calderón problem: Low regularity and stability, to appear in Nonlinear Analysis.
[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.
[38]	M. Salo, The fractional Calderón problem, Journés Équations aux Dérivées Partielles, 7 (2017), 8pp, arXiv: 1711.06103. doi: 10.5802/jedp.657.
[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.
[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