Stability estimates for the inverse fractional conductivity problem
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866910593608318976 |
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| author | Covi, Giovanni Railo, Jesse Tyni, Teemu Zimmermann, Philipp |
| author_facet | Covi, Giovanni Railo, Jesse Tyni, Teemu Zimmermann, Philipp |
| contents | We study the stability of an inverse problem for the fractional conductivity equation on bounded smooth domains. We obtain a logarithmic stability estimate for the inverse problem under suitable a priori bounds on the globally defined conductivities. The argument has three main ingredients: 1. the logarithmic stability of the related inverse problem for the fractional Schrödinger equation by Rüland and Salo; 2. the Lipschitz stability of the exterior determination problem; 3. utilizing and identifying nonlocal analogies of Alessandrini's work on the stability of the classical Calderón problem. The main contribution of the article is the resolution of the technical difficulties related to the last mentioned step. Furthermore, we show the optimality of the logarithmic stability estimates, following the earlier works by Mandache on the instability of the inverse conductivity problem, and by Rüland and Salo on the analogous problem for the fractional Schrödinger equation. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2210_01875 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Stability estimates for the inverse fractional conductivity problem Covi, Giovanni Railo, Jesse Tyni, Teemu Zimmermann, Philipp Analysis of PDEs Functional Analysis Primary 35R30, Secondary 26A33, 42B37, 46F12 We study the stability of an inverse problem for the fractional conductivity equation on bounded smooth domains. We obtain a logarithmic stability estimate for the inverse problem under suitable a priori bounds on the globally defined conductivities. The argument has three main ingredients: 1. the logarithmic stability of the related inverse problem for the fractional Schrödinger equation by Rüland and Salo; 2. the Lipschitz stability of the exterior determination problem; 3. utilizing and identifying nonlocal analogies of Alessandrini's work on the stability of the classical Calderón problem. The main contribution of the article is the resolution of the technical difficulties related to the last mentioned step. Furthermore, we show the optimality of the logarithmic stability estimates, following the earlier works by Mandache on the instability of the inverse conductivity problem, and by Rüland and Salo on the analogous problem for the fractional Schrödinger equation. |
| title | Stability estimates for the inverse fractional conductivity problem |
| topic | Analysis of PDEs Functional Analysis Primary 35R30, Secondary 26A33, 42B37, 46F12 |
| url | https://arxiv.org/abs/2210.01875 |