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）理论中，有一个结果称为主要定理，即能量随奇点扰动的变化表示为GJ积分。在本研究中，与木村教授一起证明了主要定理在广泛的非线性问题中成立。(2)结合GJ-积分和Azegami教授提出的H1-梯度方法，我们可以从理论和数值上证明具有奇异性的形状优化问题是可以解决的。(3)结果表明，由巴黎第六大学Jacques-Louis Lions实验室的F.Hecht教授开发的FreeFem++是一种通过数学思维和编程实现的边值问题有限元求解器。

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.