Symmetry of solutions for elliptic partial differential equations

2015 Fiscal Year Final Research Report

• Project/Area Number: 24540179
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Saga University
• Principal Investigator: Kajikiya Ryuji 佐賀大学, 工学(系)研究科(研究院), 教授 (10183261)
• Project Period (FY): 2012-04-01 – 2016-03-31
• Keywords: 楕円型偏微分方程式 / 変分法 / 最小エネルギー解 / 解の対称性 / 群不変性

Outline of Final Research Achievements

We study elliptic partial differential equations in symmetric domains. Let H and G be closed subgroups of the orthogonal group such that H is a closed subgroup of G and the domain is G invariant. Then we show the existence of a positive solution which is H invariant but G non-invariant under suitable assumptions of H, G and the coefficient function of the equation. Such a solution is obtained as a least energy solution. Here a least energy solution is a solution which is a minimizer of the Rayleigh quotient. Our theorem ensures the existence of various solutions which has a weak symmetry but does not have a strong symmetry.

Free Research Field

楕円型偏微分方程式