Home Mathematics Determining both leading coefficient and source in a nonlocal elliptic equation
Article
Licensed
Unlicensed Requires Authentication

Determining both leading coefficient and source in a nonlocal elliptic equation

  • Yi-Hsuan Lin ORCID logo EMAIL logo
Published/Copyright: January 13, 2025

Abstract

In this short note, we investigate an inverse (source) problem associated with a nonlocal elliptic equation ( - σ ) s u = F that is given in a bounded open set Ω n , for n 3 and 0 < s < 1 . We demonstrate both the leading coefficient σ and the source F can be determined uniquely by using the exterior Dirichlet-to-Neumann (DN) map in Ω e := n Ω ¯ . The result is intriguing in that analogous theory cannot be true for the local case generally, that is, s = 1 . The key ingredients to prove the uniqueness are based on the unique continuation principle for nonlocal elliptic operators and the reduction from the nonlocal to the local via the Stinga–Torrea extension problem.

MSC 2020: 35R30; 26A33; 35J70

Award Identifier / Grant number: 112-2628-M-A49-003

Award Identifier / Grant number: 113-2628-M-A49-003

Funding statement: Yi-Hsuan Lin is partially supported by the National Science and Technology Council (NSTC) Taiwan, under the projects 112-2628-M-A49-003 and 113-2628-M-A49-003. Yi-Hsuan Lin is also a Humboldt research fellowship for experienced researchers.

References

[1] S. Bhattacharyya, T. Ghosh and G. Uhlmann, Inverse problems for the fractional-Laplacian with lower order non-local perturbations, Trans. Amer. Math. Soc. 374 (2021), no. 5, 3053–3075. 10.1090/tran/8151Search in Google Scholar

[2] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 (2007), no. 7–9, 1245–1260. 10.1080/03605300600987306Search in Google Scholar

[3] X. Cao, Y.-H. Lin and H. Liu, Simultaneously recovering potentials and embedded obstacles for anisotropic fractional Schrödinger operators, Inverse Probl. Imaging 13 (2019), no. 1, 197–210. 10.3934/ipi.2019011Search in Google Scholar

[4] M. Cekić, Y.-H. Lin and A. Rüland, The Calderón problem for the fractional Schrödinger equation with drift, Calc. Var. Partial Differential Equations 59 (2020), no. 3, Paper No. 91. 10.1007/s00526-020-01740-6Search in Google Scholar

[5] G. Covi, T. Ghosh, A. Rüland and G. Uhlmann, A reduction of the fractional Calderón problem to the local Calderón problem by means of the Caffarelli–Silvestre extension, preprint (2023), https://arxiv.org/abs/2305.04227. Search in Google Scholar

[6] G. Covi, K. Mönkkönen and J. Railo, Unique continuation property and Poincaré inequality for higher order fractional Laplacians with applications in inverse problems, Inverse Probl. Imaging 15 (2021), no. 4, 641–681. 10.3934/ipi.2021009Search in Google Scholar

[7] G. Covi, K. Mönkkönen, J. Railo and G. Uhlmann, The higher order fractional Calderón problem for linear local operators: Uniqueness, Adv. Math. 399 (2022), Article ID 108246. 10.1016/j.aim.2022.108246Search in Google Scholar

[8] G. Covi, J. Railo, T. Tyni and P. Zimmermann, Stability estimates for the inverse fractional conductivity problem, SIAM J. Math. Anal. 56 (2024), no. 2, 2456–2487. 10.1137/22M1533542Search in Google Scholar

[9] G. Covi, J. Railo and P. Zimmermann, The global inverse fractional conductivity problem, preprint (2022), https://arxiv.org/abs/2204.04325. Search in Google Scholar

[10] E. B. Davies, Heat Kernels and Spectral Theory, Cambridge Tracts in Math. 92, Cambridge University, Cambridge, 1990. Search in Google Scholar

[11] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), no. 5, 521–573. 10.1016/j.bulsci.2011.12.004Search in Google Scholar

[12] A. Feizmohammadi, Fractional Calderón problem on a closed Riemannian manifold, Trans. Amer. Math. Soc. 377 (2024), no. 4, 2991–3013. 10.1090/tran/9106Search in Google Scholar

[13] A. Feizmohammadi, T. Ghosh, K. Krupchyk and G. Uhlmann, Fractional anisotropic Calderón problem on closed Riemannian manifolds, preprint (2021), https://arxiv.org/abs/2112.03480; to appear in J. Differential Geom. Search in Google Scholar

[14] T. Ghosh, Y.-H. Lin and J. Xiao, The Calderón problem for variable coefficients nonlocal elliptic operators, Comm. Partial Differential Equations 42 (2017), no. 12, 1923–1961. 10.1080/03605302.2017.1390681Search in Google Scholar

[15] T. Ghosh, A. Rüland, M. Salo and G. Uhlmann, Uniqueness and reconstruction for the fractional Calderón problem with a single measurement, J. Funct. Anal. 279 (2020), no. 1, Article ID 108505. 10.1016/j.jfa.2020.108505Search in Google Scholar

[16] T. Ghosh, M. Salo and G. Uhlmann, The Calderón problem for the fractional Schrödinger equation, Anal. PDE 13 (2020), no. 2, 455–475. 10.2140/apde.2020.13.455Search in Google Scholar

[17] T. Ghosh and G. Uhlmann, The Calderón problem for nonlocal operators, preprint (2021), https://arxiv.org/abs/2110.09265. Search in Google Scholar

[18] B. Harrach and Y.-H. Lin, Monotonicity-based inversion of the fractional Schrödinger equation I. Positive potentials, SIAM J. Math. Anal. 51 (2019), no. 4, 3092–3111. 10.1137/18M1166298Search in Google Scholar

[19] B. Harrach and Y.-H. Lin, Monotonicity-based inversion of the fractional Schödinger equation II. General potentials and stability, SIAM J. Math. Anal. 52 (2020), no. 1, 402–436. 10.1137/19M1251576Search in Google Scholar

[20] Y. Kian, T. Liimatainen and Y.-H. Lin, On determining and breaking the gauge class in inverse problems for reaction-diffusion equations, Forum Math. Sigma 12 (2024), Paper No. e25. 10.1017/fms.2024.18Search in Google Scholar

[21] P.-Z. Kow, Y.-H. Lin and J.-N. Wang, The Calderón problem for the fractional wave equation: Uniqueness and optimal stability, SIAM J. Math. Anal. 54 (2022), no. 3, 3379–3419. 10.1137/21M1444941Search in Google Scholar

[22] R.-Y. Lai and Y.-H. Lin, Inverse problems for fractional semilinear elliptic equations, Nonlinear Anal. 216 (2022), Article ID 112699. 10.1016/j.na.2021.112699Search in Google Scholar

[23] R.-Y. Lai, Y.-H. Lin and A. Rüland, The Calderón problem for a space-time fractional parabolic equation, SIAM J. Math. Anal. 52 (2020), no. 3, 2655–2688. 10.1137/19M1270288Search in Google Scholar

[24] T. Liimatainen and Y.-H. Lin, Uniqueness results for inverse source problems for semilinear elliptic equations, Inverse Problems 40 (2024), no. 4, Article ID 045030. 10.1088/1361-6420/ad3088Search in Google Scholar

[25] C.-L. Lin, Y.-H. Lin and G. Uhlmann, The Calderón problem for nonlocal parabolic operators, preprint (2022), https://arxiv.org/abs/2209.11157. Search in Google Scholar

[26] C.-L. Lin, Y.-H. Lin and G. Uhlmann, The Calderón problem for nonlocal parabolic operators: A new reduction from the nonlocal to the local, preprint (2023), https://arxiv.org/abs/2308.09654. Search in Google Scholar

[27] Y.-H. Lin, Monotonicity-based inversion of fractional semilinear elliptic equations with power type nonlinearities, Calc. Var. Partial Differential Equations 61 (2022), no. 5, Paper No. 188. 10.1007/s00526-022-02299-0Search in Google Scholar

[28] Y.-H. Lin, A local uniqueness theorem for the fractional Schrödinger equation on closed Riemannian manifolds, preprint (2024), https://arxiv.org/abs/2409.01921. Search in Google Scholar

[29] Y.-H. Lin, The fractional anisotropic Calderón problem for a nonlocal parabolic equation on closed Riemannian manifolds, preprint (2024), https://arxiv.org/abs/2410.17750. Search in Google Scholar

[30] Y.-H. Lin and H. Liu, Inverse problems for fractional equations with a minimal number of measurements, Commun. Anal. Comput. 1 (2023), no. 1, 72–93. 10.3934/cac.2023005Search in Google Scholar

[31] Y.-H. Lin, J. Railo and P. Zimmermann, The Calderón problem for a nonlocal diffusion equation with time-dependent coefficients, preprint (2022), https://arxiv.org/abs/2211.07781. Search in Google Scholar

[32] Y.-H. Lin and P. Zimmermann, Unique determination of coefficients and kernel in nonlocal porous medium equations with absorption term, preprint (2023), https://arxiv.org/abs/2305.16282. Search in Google Scholar

[33] Y.-H. Lin and P. Zimmermann, Approximation and uniqueness results for the nonlocal diffuse optical tomography problem, preprint (2024), https://arxiv.org/abs/2406.06226. Search in Google Scholar

[34] J. Railo and P. Zimmermann, Counterexamples to uniqueness in the inverse fractional conductivity problem with partial data, Inverse Probl. Imaging 17 (2023), no. 2, 406–418. 10.3934/ipi.2022048Search in Google Scholar

[35] J. Railo and P. Zimmermann, Fractional Calderón problems and Poincaré inequalities on unbounded domains, J. Spectr. Theory 13 (2023), no. 1, 63–131. 10.4171/jst/444Search in Google Scholar

[36] J. Railo and P. Zimmermann, Low regularity theory for the inverse fractional conductivity problem, Nonlinear Anal. 239 (2024), Article ID 113418. 10.1016/j.na.2023.113418Search in Google Scholar

[37] A. Rüland, Revisiting the anisotropic fractional Calderón problem using the Caffarelli–Silvestre extension, preprint (2023), https://arxiv.org/abs/2309.00858. Search in Google Scholar

[38] A. Rüland and M. Salo, The fractional Calderón problem: Low regularity and stability, Nonlinear Anal. 193 (2020), Article ID 111529. 10.1016/j.na.2019.05.010Search in Google Scholar

[39] P. R. Stinga and J. L. Torrea, Extension problem and Harnack’s inequality for some fractional operators, Comm. Partial Differential Equations 35 (2010), no. 11, 2092–2122. 10.1080/03605301003735680Search in Google Scholar

[40] J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math. (2) 125 (1987), no. 1, 153–169. 10.2307/1971291Search in Google Scholar

[41] P. Zimmermann, Inverse problem for a nonlocal diffuse optical tomography equation, Inverse Problems 39 (2023), no. 9, Article ID 094001. 10.1088/1361-6420/ace4edSearch in Google Scholar

Received: 2024-08-20
Revised: 2024-10-24
Accepted: 2024-11-11
Published Online: 2025-01-13
Published in Print: 2025-04-01

© 2025 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 29.1.2026 from https://www.degruyterbrill.com/document/doi/10.1515/jiip-2024-0059/html
Scroll to top button