Transmission problems in composite media and overdetermined problems with transmission conditions

2018 Fiscal Year Final Research Report

• Project/Area Number: 16K13768
• Research Category: Grant-in-Aid for Challenging Exploratory Research
• Allocation Type: Multi-year Fund
• Research Field: Mathematical analysis
• Research Institution: Tohoku University
• Principal Investigator: SAKAGUCHI SHIGERU 東北大学, 情報科学研究科, 教授 (50215620)
• Research Collaborator: CAVALLINA Lorenzo
• Project Period (FY): 2016-04-01 – 2019-03-31
• Keywords: 複合媒質 / 伝送条件 / 優決定問題 / 中性導体 / 近似中性導体 / 同心球 / コンフォーカル楕円体 / 伝送問題

Outline of Final Research Achievements

We consider composite media with an inclusion having different conductivity. Since the electrical conduction and heat conduction in composite media depend on their conductivities, on the interface between two different media the electric current and heat flow satisfy the transmission conditions like refraction of light. One of our main results deals with the steady electrical conduction on the composite media, and it is shown that there exist infinitely many non-symmetric composite media which are weakly neutral to and having little influence on external uniform electric fields. Another deals with the steady heat conduction, and we show that if both temperature and heat flow on the boundary of the spherical composite media are constant, then the inclusion must be a concentric ball, and moreover, we show that there exist infinitely many non-symmetric composite media on each of whose boundaries both temperature and heat flow are constant.

Free Research Field

偏微分方程式論

Academic Significance and Societal Importance of the Research Achievements

一つの物質からなる単一媒質上の電気伝導や熱伝導の数理モデルは最も単純で多くの研究がなされてきた。一方, 外部一様電場に全く影響を与えない中性導体は1962年に Hashin, Shtrikman らが同心球からなる２相の複合媒質によって構成し, 複合媒質の重要性が認識された。もちろん, 物理的・工学的に自然な複合媒質（複合材料）の数理モデルの重要性は言うまでもない。本研究成果の学術的・社会的意義は単一媒質上の数理モデルに対応する複合媒質上の数理モデルは一見同様に見えるが本質的に異なることを電気伝導や熱伝導の単純な数理モデルで数学解析を用いて示したことにある。