Project/Area Number  09640177 
Research Category 
GrantinAid for Scientific Research (C).

Section  一般 
Research Field 
解析学

Research Institution  Science University of Tokyo 
Principal Investigator 
TACHIKAWA Atsushi Science University of Tokyo(S.U.T.), Faculty of Science and Technology, Associate Professor, 理工学部, 助教授 (50188257)

CoInvestigator(Kenkyūbuntansha) 
NAGASAWA Takeyuki Tohoku Univ., Mathematical Institute, Associate Professor, 理学研究科, 助教授 (70202223)
細尾 敏男 東京理科大学, 理工学部, 助手 (30130339)
浜畑 芳紀 東京理科大学, 理工学部, 講師 (90260645)
TANAKA Ryuichi Science University of Tokyo(S.U.T.), Fac.of Sci.and Tech.nology, Lecturer, 理工学部, 講師 (10112898)
吾郷 孝視 東京理科大学, 理工学部, 教授 (60112893)
KOTANI Kouichi Science University of Tokyo(S.U.T.), Fac.of Sci.and Tech.nology, Assistant, 理工学部, 助手 (80183341)
TAMIYA Takanori Science University of Tokyo(S.U.T.), Fac.of Sci.and Tech.nology, Lecturer, 理工学部, 講師 (60183472)
KOBAYASHI takao Science University of Tokyo(S.U.T.), Fac.of Sci.and Tech.nology, Associate Professor, 理工学部, 助教授 (90178319)

Project Fiscal Year 
1997 – 1999

Project Status 
Completed(Fiscal Year 1999)

Budget Amount *help 
¥2,800,000 (Direct Cost : ¥2,800,000)
Fiscal Year 1999 : ¥900,000 (Direct Cost : ¥900,000)
Fiscal Year 1998 : ¥800,000 (Direct Cost : ¥800,000)
Fiscal Year 1997 : ¥1,100,000 (Direct Cost : ¥1,100,000)

Keywords  Variational Method / Discretization / Harmonic Maps / 変分法 / 差分法 / 調和写像 
Research Abstract 
The aim of this research is to study the solutions of the nonlinear partial differential equations which are closely related to variational problems by using calculus of variations and timediscretization schemes. More precisely, for some variational functional F(u) we treat the partial differential equation ィイD7∂u(/)∂tィエD7(the EulerLanguage equation of F) = 0. To construct weak solutions for the above equation we proceed as follows. We consider the functional GィイD2nィエD2(u) = ∫ィイD7uuィイD2nィエD21(/)2hィエD7dx+F(u), and define uィイD2nィエD2 as a minimizer of GィイD2nィエD2(u) successively. Combining {uィイD2nィエD2} by line segments we construct a approximate solution uィイD2hィエD2(x, t). Finally, under some conditions, we prove that uィイD2hィエD2(x, t) converge to a weak solution. On the other hand, id the limit of the sequence of maps {uィイD2hィエD2}converges to some map u(x, t), u(x, t) will be closely related to minimizing movement which is a new notion introduced by E.De Giorgi. Using the above method
… More
, Tachikawa constructed weak solutions of the heattype equations for harmonic maps (EellsSampson equation) from noncompact Riemannian manifolds into the ndimensional spheres (ィイD7∂u(/)∂tィエD7ΔuuDuィイD12ィエD1 = 0). Moreover, he proved that the weak solutions are minimizing movements of the energy functionals. Related to the above problem, Nagasawa and Tachikawa studied harmonic maps between noncompact complete Riemannian manifolds. Especially, they considered harmonic maps with a certain nondegeneracy condition and get the following nonexistence result. "Let N be a Handamard manifold whose sectional curvatures at a point p do not decay faster than distィイD12ィエD1"(p, pィイD20ィエD2) for some fixed point pィイD20ィエD2. Then there is no entire harmonic maps from RィイD1mィエD1 into N which satisfies a certain nondegeneracy condition." Nagasawa constructed a weak solution of the NavierStrokes equation on a Riemannian manifold using the above method. Moreover, he sharpened the energy estimates on the weak solutions constructed as above and got a new partial regularity estimates. He constructed weak solutions of the hyperbolic Ginzburg Landau equations too and studied them numerically. Less
