Strong uniqueness principle for fractional polyharmonic operators and applications to inverse problems
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arXiv
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| Natura: | Preprint |
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2023
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| _version_ | 1866916733347954688 |
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| author | Lin, Ching-Lung Liu, Hongyu Lo, Catharine W. K. |
| author_facet | Lin, Ching-Lung Liu, Hongyu Lo, Catharine W. K. |
| contents | In this work, we are concerned with inverse problems involving poly-fractional operators, where the poly-fractional operator is of the form
\[P( (-Δ_g)^s)u := \sum_{i=1}^M α_i(-Δ_{g_i})^{s_i}u\]
for $s=(s_1,\dots,s_M)$, $0<s_1<\cdots<s_M<\infty$, $s_M\in\mathbb{R}_+\backslash\mathbb{Z}$, $g=(g_1,\dots,g_M)$. There are three major contributions in this work that are new to the literature. First, we propose equations involving such poly-fractional operators $P$, which have not been previously considered in the general setting. Such equations arise naturally from the superposition of multiple stochastic processes with different scales, including classical random walks and Lévy flights. Secondly, we give novel results for the unique continuation properties for fractional polyharmonic $u$, in the sense that $u$ satisfies $\tilde{P}((-Δ_{\tilde{g}})^{\tilde{s}})=0$ in a bounded Lipschitz domain $Ω$ for some $\tilde{P}$. With these results in hand, we consider the inverse problems for $P$, and proved the uniqueness in recovering the potential, the source function in the semilinear case, and the coefficients associated to the non-isotropy of the fractional operator. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_00744 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Strong uniqueness principle for fractional polyharmonic operators and applications to inverse problems Lin, Ching-Lung Liu, Hongyu Lo, Catharine W. K. Analysis of PDEs Primary 35R30, secondary 35R11, 26A33 In this work, we are concerned with inverse problems involving poly-fractional operators, where the poly-fractional operator is of the form \[P( (-Δ_g)^s)u := \sum_{i=1}^M α_i(-Δ_{g_i})^{s_i}u\] for $s=(s_1,\dots,s_M)$, $0<s_1<\cdots<s_M<\infty$, $s_M\in\mathbb{R}_+\backslash\mathbb{Z}$, $g=(g_1,\dots,g_M)$. There are three major contributions in this work that are new to the literature. First, we propose equations involving such poly-fractional operators $P$, which have not been previously considered in the general setting. Such equations arise naturally from the superposition of multiple stochastic processes with different scales, including classical random walks and Lévy flights. Secondly, we give novel results for the unique continuation properties for fractional polyharmonic $u$, in the sense that $u$ satisfies $\tilde{P}((-Δ_{\tilde{g}})^{\tilde{s}})=0$ in a bounded Lipschitz domain $Ω$ for some $\tilde{P}$. With these results in hand, we consider the inverse problems for $P$, and proved the uniqueness in recovering the potential, the source function in the semilinear case, and the coefficients associated to the non-isotropy of the fractional operator. |
| title | Strong uniqueness principle for fractional polyharmonic operators and applications to inverse problems |
| topic | Analysis of PDEs Primary 35R30, secondary 35R11, 26A33 |
| url | https://arxiv.org/abs/2307.00744 |