Inverse problems for degenerate hyperbolic partial differential equations on manifolds

2023 Fiscal Year Final Research Report

• Project/Area Number: 22K20340
• Research Category: Grant-in-Aid for Research Activity Start-up
• Allocation Type: Multi-year Fund
• Review Section: 0201:Algebra, geometry, analysis, applied mathematics,and related fields
• Research Institution: Kyushu University
• Principal Investigator: Takase Hiroshi 九州大学, マス・フォア・インダストリ研究所, 助教 (60963204)
• Project Period (FY): 2022-08-31 – 2024-03-31
• Keywords: 逆問題 / 非適切問題 / 偏微分方程式

Outline of Final Research Achievements

He studied an inverse problem for a system of hyperbolic partial differential equations written in terms of the Laplace-Beltrami operator on a Lorentzian manifold. He established a weighted energy estimate, the Carleman estimate, for this system and proved a global Lipschitz stability when the observation data is taken on a part of the boundary of the Lorentzian manifold. Furthermore, he proved a global Lipschitz stability for an inverse source problem of determining the source term of a one-dimensional wave equation with an inverse square potential.

Free Research Field

偏微分方程式の逆問題解析

Academic Significance and Societal Importance of the Research Achievements

重力波を記述するような曲がった空間上における波動方程式の未知波源項を決定する逆問題に対し，安定性評価を証明した．これにより，境界における解の観測誤差が小さければ，未知の波源項同士の差も小さく，未知量が安定的に決定できることが分かる．