New developments on evolution equations and variational problems by geometric conservation laws

2015 Fiscal Year Final Research Report

• Project/Area Number: 25800072
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Mathematical analysis
• Research Institution: Kyushu University
• Principal Investigator: Onodera Michiaki 九州大学, マス・フォア・インダストリ研究所, 助教 (70614999)
• Project Period (FY): 2013-04-01 – 2016-03-31
• Keywords: ポテンシャル論 / 発展方程式 / 変分問題 / 過剰決定問題

Outline of Final Research Achievements

An inverse problem in potential theory asks if, for a given potential, there is a unique surface which exactly induces the same gravitational potential as the given one. Although the problem has a variational structure from which the existence of a desired surface follows, the uniqueness question had not been clarified for decades because of the lack of information on the shape of the functional corresponding to the variational structure.
My research shows that the corresponding surface is actually unique if the given potential is close to a radially symmetric one. This is the first result asserting the uniqueness for asymmetric situation, to the best of my knowledge. The proof is based on the consideration of a parametrized auxiliary problem which produces a family of surfaces satisfying an evolution equation. I clarified the dynamical structure of the evolution equation and derived the uniqueness conclusion together with a quantitative estimate for the shape of the surface.

Free Research Field

偏微分方程式