Linear stability analysis of overdetermined problems with non-constant data

Michiaki Onodera; Michiaki Onodera

doi:10.3934/mine.2023048

Mathematics in Engineering

2023, Volume 5, Issue 3: 1-18. doi: 10.3934/mine.2023048

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Linear stability analysis of overdetermined problems with non-constant data

Michiaki Onodera ^,

Department of Mathematics, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8551, Japan

Received: 17 March 2022 Revised: 16 July 2022 Accepted: 19 July 2022 Published: 09 August 2022

We study an overdetermined problem that arises as the Euler-Lagrange equation of a weighted variational problem in elasticity. Based on a detailed linear analysis by spherical harmonics, we prove the existence and local uniqueness as well as an optimal stability estimate for the shape of a domain allowing the solvability of the overdetermined problem. Our linear analysis reveals that the solution structure is strongly related to the choice of parameters in the problem. In particular, the global uniqueness holds for the pair of the parameters lying in a triangular region.
- overdetermined problem,
- stability estimate,
- symmetry,
- spherical harmonics,
- implicit function theorem
Citation: Michiaki Onodera. Linear stability analysis of overdetermined problems with non-constant data[J]. Mathematics in Engineering, 2023, 5(3): 1-18. doi: 10.3934/mine.2023048

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Abstract

We study an overdetermined problem that arises as the Euler-Lagrange equation of a weighted variational problem in elasticity. Based on a detailed linear analysis by spherical harmonics, we prove the existence and local uniqueness as well as an optimal stability estimate for the shape of a domain allowing the solvability of the overdetermined problem. Our linear analysis reveals that the solution structure is strongly related to the choice of parameters in the problem. In particular, the global uniqueness holds for the pair of the parameters lying in a triangular region.

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