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2022-12-25

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cc-by-nc-nd (c) Elsevier, 2022
Please use this identifier to cite or link to this item: https://hdl.handle.net/2445/192988

The Calderón problem for nonlocal Schrödinger equations with homogeneous, directionally antilocal principal symbols

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https://doi.org/10.1016/j.jde.2022.09.009

Abstract

In this article we consider direct and inverse problems for $\alpha$-stable, elliptic nonlocal operators whose kernels are possibly only supported on cones and which satisfy the structural condition of directional antilocality as introduced by Y. Ishikawa in the 80s. We consider the Dirichlet problem for these operators on the respective "domain of dependence of the operator" and in several, adapted function spaces. This formulation allows one to avoid natural "gauges" which would else have to be considered in the study of the associated inverse problems. Exploiting the directional antilocality of these operators we complement the investigation of the direct problem with infinite data and single measurement uniqueness results for the associated inverse problems. Here, due to the only directional antilocality, new geometric conditions arise on the measurement domains. We discuss both the setting of symmetric and a particular class of non-symmetric nonlocal elliptic operators, and contrast the corresponding results for the direct and inverse problems. In particular for only "one-sided operators" new phenomena emerge both in the direct and inverse problems: For instance, it is possible to study the problem in data spaces involving local and nonlocal data, the unique continuation property may not hold in general and further restrictions on the measurement set for the inverse problem arise.

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Problemes inversos (Equacions diferencials), Operadors integrals, Operadors diferencials

Subject (English)

Inverse problems (Differential equations), Integral operators, Differential operators

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Articles publicats en revistes (Matemàtiques i Informàtica)

Citation

COVI, Giovanni, GARCÍA-FERRERO, María Ángeles and RÜLAND, Angkana. The Calderón problem for nonlocal Schrödinger equations with homogeneous, directionally antilocal principal symbols. Journal of Differential Equations. 2022. Vol. 341, num. 79-149. ISSN 0022-0396. [consulted: 10 of August of 2026]. Available at: https://hdl.handle.net/2445/192988

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