Monotonicity and local uniqueness for an isotropic nonlocal elliptic equation
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866918159157559296 |
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| author | Lin, Yi-Hsuan |
| author_facet | Lin, Yi-Hsuan |
| contents | We extend monotonicity-based inversion methods to an inverse coefficient problem for the isotropic nonlocal elliptic equation
\[
(-\nabla \cdot σ\nabla)^s u = 0 \quad \text{in } Ω\subset \mathbb{R}^n,
\]
where $0 < s < 1$, $n \geq 3$, and $Ω$ is a bounded open set. We establish a monotonicity relation between the leading coefficient $σ$ and the (partial) exterior Dirichlet-to-Neumann (DN) map. Our main result shows that a monotonicity ordering of the coefficients implies a corresponding ordering of the DN maps. Furthermore, we construct localized potentials for the nonlocal equation, which yield a local uniqueness result for the fractional inverse problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_10408 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Monotonicity and local uniqueness for an isotropic nonlocal elliptic equation Lin, Yi-Hsuan Analysis of PDEs We extend monotonicity-based inversion methods to an inverse coefficient problem for the isotropic nonlocal elliptic equation \[ (-\nabla \cdot σ\nabla)^s u = 0 \quad \text{in } Ω\subset \mathbb{R}^n, \] where $0 < s < 1$, $n \geq 3$, and $Ω$ is a bounded open set. We establish a monotonicity relation between the leading coefficient $σ$ and the (partial) exterior Dirichlet-to-Neumann (DN) map. Our main result shows that a monotonicity ordering of the coefficients implies a corresponding ordering of the DN maps. Furthermore, we construct localized potentials for the nonlocal equation, which yield a local uniqueness result for the fractional inverse problem. |
| title | Monotonicity and local uniqueness for an isotropic nonlocal elliptic equation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2510.10408 |