On a variational study of nonlinear Dirac equations on compact spin manifolds

2014 Fiscal Year Final Research Report

• Project/Area Number: 22540222
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Global analysis
• Research Institution: Tokyo Institute of Technology
• Principal Investigator: ISOBE Takeshi 東京工業大学, 理工学研究科, 准教授 (10262255)
• Project Period (FY): 2010-04-01 – 2015-03-31
• Keywords: 変分法 / モース理論 / ディラック方程式 / 超対称性シグマモデル / 山辺問題

Outline of Final Research Achievements

We studied nonlinear Dirac equations on compact spin manifolds arising from geometry and physics via variational method. We obtain the following results: 1) For nonlinear Dirac equations obtained as 0-the order nonlinear perturbations of the Dirac operator, we proved the existence and multiplicity of solutions under assuming that the nonlinear term is subcritical. 2) For the case where the nonlinearity has critical growth, we prove a global compactness and the existence of solutions for the associated variational problem. 3) For the Dirac-geodesics problem (which is the 1-dimensional version of Dirac-harmonic maps), we prove the existence of solutions via infinite dimensional Linking theory. 4) For the spinorial Yamabe problem, we proved the existence of a solution via variational argument.

Free Research Field

解析学、大域解析学