双重非线性问题中的H-半变分不等式

本项目结合相变热传导，研究由非光滑、非凸能量泛函导出的双重非线性发展型H-半变分不等式，解决双重非线性问题中的非单调、多值型Clarke广义梯度边界问题。项目选题属于H-半变分不等式和双重非线性问题的科学前沿。研究内容如下: 1) 发展非单调、多值扰动型偏微分包含的数学方法，研究非线性发展型H-半变分不等式的多值解、最优控制以及解的性态问题，证明 H-半变分不等式表述的非局部问题解的存在性与正则性； 2）拟将热传导问题中的单调型温度控制理论推广到相变热传导中的非单调型温度控制并应用于金属浇铸的相变热传导模型，为工程应用提供理论指导；3）研究非线性耦合系统中的Clarke广义梯度型多值边界问题，证明弱解的存在性并应用于热敏电阻问题。该项目是崭新的研究课题，不仅对发展非线性问题的研究方法具有重要的理论意义，而且在科技和工程问题中具有重要的应用价值。

Based on the heat transfer with phase change, this project deals with nonlinear evolutionary hemivariational inequalities with doubly nonlinear operators, which are generated by nonsmooth and nonconvex energy functionals. It solves doubly nonlinear problems with nonmonotone and multivaled boundary conditions of Clarke’s type. As the scientific frontier of hemivariational inequalities and doubly nonlinear partial differential inclusions, this project, firstly, aims to develop new methods in nonmonotone and multivalued partial differential inclusions to study the problems concerned with multi-solutions, optimal control and the behavior of the solutions in long time to nonlinear evolutionary hemivariational inequality. It will also prove the solvability and regularity of nonlocal problems described by hemivariational inequalities. Secondly, it aims to generalize the temperature control theory of monotone type in classical heat transfer to nonmonotone type in phase change heat transfer, which will be applied to the model of heat transfer problems with phase change, and which could provide theoretic guidance for engineering applications. Thirdly, it will study existence of weak solutions for nonlinear coupled systems with multivalued boundary conditions of Clarke's generalized gradient type, which will be applied to thermistor problems. This project is a new and meaningful subject. It is of great theoretic significance to developing the research methods to nonlinear problems, and of important application value in the fields of scientific technology and engineering problems.