双重非线性问题中的H-半变分不等式

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.

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。

H-半变分不等式源于物理学、力学和工程学中的非光滑、非凸能量泛函问题，它对应于非单调、多值的本构关系或边界条件。本课题研究非线性H-半变分不等式及其耦合系统。在双重非线性发展型H-半变分不等式的可解性、最优控制，不等式问题在相变热传导和接触力学问题中应用，以及耦合型Clarke次微分包含可解性等方面取得了一些较好的研究成果。证明了双重非线性发展型H-半变分不等式（微分包含）解的存在性、多解性、解关于参数变量的上半连续性；结合相变热传导，研究了双重非线性H-半变分不等式的最优控制，证明了最优解的存在性；建立了一类变分—H-半变分不等式最优控制问题的最优系统；研究了含Clarke次微分的椭圆型和抛物型偏微分包含耦合系统，证明了这两类耦合系统弱解的存在性并研究了解的性质。该课题发展了非线性、非凸问题的研究方法，对解决力学和工程学中的非光滑、非凸问题具有较重要的意义与价值。