2022 Fiscal Year Final Research Report
Dynamical analysis of elliptic overdetermined problems
| Project/Area Number |
16K17628
|
|---|
| Research Category |
Grant-in-Aid for Young Scientists (B)
|
| Allocation Type | Multi-year Fund |
|---|
| Research Field |
Mathematical analysis
|
|---|
| Research Institution | Tokyo Institute of Technology |
|---|
Principal Investigator |
|
|---|
| Project Period (FY) |
2016-04-01 – 2023-03-31
|
| Keywords | 過剰決定問題 / 自由境界問題 / 発展方程式 / 陰関数定理 |
|---|
| Outline of Final Research Achievements |
This research established a unified functional analytic method of deriving the existence, uniqueness and quantitative stability estimates of solutions (unknown domains) of general overdetermined problems including Bernoulli's free boundary problem and Serrin's problem as special cases. In particular, a classification of solutions, previously known for some special cases, are extended to general overdetermined problems. More importantly, this research clarifies a deep connection between the classification and qualitative behavior of solutions in terms of related evolution equations. Namely, it turns out that the ellipticity (resp. hyperbolicity) of a solution corresponds to the parabolic (reversed parabolic) character of the evolution equation.
|
| Free Research Field |
偏微分方程式
|
| Academic Significance and Societal Importance of the Research Achievements |
本研究成果である過剰決定問題の解の分類およびその発展方程式論的解析は，過去の研究に新たな視点を与えるもので，従来各論的に取り扱われてきた過剰決定問題を統一的に解析する手法を与えるものである．これらは研究が盛んな自由境界問題や形状最適化問題などの領域形状に関する変分問題における未解決問題（例：Flucher-Rumpf予想，Schiffer予想）の解決に向けても有効な手段を与えるものと期待される．
|
