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Identifiability of diffusion coefficients for source terms of non-uniform sign
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 |
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.
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 |
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. 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 |
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