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

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

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.

Citation: Angkana Rüland, Eva Sincich. Lipschitz stability for the finite dimensional fractional Calderón problem with finite Cauchy data. Inverse Problems and 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.

[2]

G. AlessandriniM. V. de HoopR. 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. AlessandriniM. 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. BarcelóE. 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. BerettaM. 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. DipierroO. 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. FabesH. 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. GhoshY.-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

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.

[2]

G. AlessandriniM. V. de HoopR. 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. AlessandriniM. 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. BarcelóE. 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. BerettaM. 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. DipierroO. 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. FabesH. 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. GhoshY.-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

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