Construction of the theory of variation applicable to shape optimization problems and fracture, whose application to problems in engineering.

研究结果如下：(1)在广义j积分（gj积分）的理论中，有一个结果叫做主定理，即能量对奇异点摄动的变化表示为gj积分。在本研究中，Kimura教授证明了主要定理在广义非线性问题中是成立的。(2)结合Azegami教授提出的gj积分法和h1 -梯度法，可以从理论上和数值上解决具有奇异点的形状优化问题。(3)通过数学思维和程序设计，证明了巴黎第六大学Jacques-Louis Lions实验室F.Hecht教授开发的FreeFem++是边值问题的有限元求解器。

The results of research are the following. (1) In theory of generalized J-integral (GJ-integral), there is the result called Main Theorem, that is, the variation of energies with respect to the perturbation of the singular points is expressed as GJ-integral. In this research, it is proved that Main Theorem hold in wide nonlinear problems with Professor Kimura. (2) Combining GJ-integral and H1-gradient method proposed by Professor Azegami, we could show that the shape optimization problems with singularities are solved theoretically and numerically. (3) It was shown that FreeFem++ developed by Professor F.Hecht at the Laboratory Jacques-Louis Lions in Paris VI University is the finite element solver for boundary value problems by mathematical thinking and programming.