Analysis of a two-phase overdetermined problem of Serrin type: from local to global

2022 Fiscal Year Final Research Report

• Project/Area Number: 20K22298
• 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: Tohoku University
• Principal Investigator: Cavallina Lorenzo 東北大学, 理学研究科, 助教 (40881264)
• Project Period (FY): 2020-09-11 – 2023-03-31
• Keywords: 複合媒質 / 形状最適化問題 / 優決定問題 / 過剰決定問題 / 楕円型偏微分方程式 / 分岐解析 / 対称性 / 陰関数定理

Outline of Final Research Achievements

In general, there are infinitely many functions that satisfy a given partial differential equation in a domain. On the other hand, if we also specify the behavior on the boundary, there will be a unique solution. In this study, we consider the mathematical model of a composite-medium given by an "overdetermined problem" for partial differential equations where two boundary conditions are imposed simultaneously. This is an extension to the composite-medium case of the overdetermined problem introduced by Serrin in 1971 for the case of a single medium, and one of its characteristics is that it is solvable only for some special domains. In the homogeneous-medium case, balls are the only domains that allow solutions. On the other hand, in the composite-medium case, there are also non spherically-symmetric domains that allow solutions. In this study, we consider the geometric properties (smoothness, symmetry breaking, etc.) of such composite media.

Free Research Field

偏微分方程式論

Academic Significance and Societal Importance of the Research Achievements

本研究では、複合媒質の形状最適化に由来する問題を扱う。具体的には、ねじりに対する抵抗を表す「ねじり剛性」を最大化する（すなわち、最大限に頑丈な）長い棒の断面について、本研究で扱う優決定問題は解を許すことが知られている。逆にいえば、優決定問題が解を持たない場合、与えられた形状はねじり剛性を最大化しないこととなる。単一媒体の場合、最適形状の断面が円形であることは知られていたが、複合媒質の場合の最適形状については未解決であった。本研究では、複合媒質中の介在物の形状が複合媒質全体の最適形状を決定することを明らかとし、その結果、回転対称でない最適形状の族を構築することに成功した。