一类非线性椭圆-抛物耦合系统的可解性和灵敏度分析

变分和H-半变分不等式广泛应用于物理学、力学、工程科学中的非光滑能量泛函或多值次微分边界问题。然而，越来越多的实际问题被发现不能单纯地用变分/H-半变分不等式描述，而是由它们和微分方程的耦合系统刻画，由此产生了一个新的研究领域—微分变分/H-半变分不等式。本项目以带混合边界条件的非稳态热敏电阻问题为驱动，研究一类由椭圆型偏微分方程与抛物型变分/H-半变分不等式耦合的非线性系统，它是一类新型偏微分变分/H-半变分不等式。拟结合非线性分析、偏微分方程、非光滑分析、集值分析、最优控制等理论，研究该系统解的存在性、唯一（或多解）性及其最优控制问题的灵敏度分析。解决这些问题需发展一些新的研究方法与技术，具有较好的理论及应用价值。此外，作为天元访问学者项目，项目的实施将有助于加强中国科学院数学与系统科学研究院与西部民族地区高校的合作与交流，培养西部地区青年学术骨干。

Variational and hemivariational inequalities are widely used in the problems of non-smooth energy functionals or subdifferential boundary in physics, mechanics and engineering science. However, it has been found that more and more practical problems can't be described simply by the variational/hemivariational inequalities, but by the coupling system of them and differential equations, which gives rise to a new research field--differential variational/hemivariational inequalities. Driven by the unsteady thermistor problem with mixed boundary conditions, this project studies a class of nonlinear systems coupled by elliptic partial differential equations and parabolic variational/hemivariational inequalities, which is a new class of partial differential variational/ hemivariational inequalities. By nonlinear analysis, partial differential equation, non-smooth analysis, set-valued analysis and optimal control theory, the existence, uniqueness (or multiplicity) of the solution to the system and the sensitivity analysis of the optimal control problem are studied. To solve these problems, we need to develop some new research methods and technologies, which have significant theoretical and application value. In addition, as a Tianyuan Visiting Scholar program, the implementation of the project will help strengthen the academic cooperation and exchanges between the Academy of Mathematics and Systems Sciences of the Chinese Academy of Sciences and universities in the western minority areas, and cultivate young academic backbones in the western region.