H-半变分不等式及非凸约束问题

Based on hemivariational inequality theory, this project deals with doubly nonlinear problems with nonconvex energy functionals or nonconvex constraints. By means of the theory of nonsmooth analysis, they could be transformed into nonlinear evolutionary hemivariational inequalities. The content is divided into two parts: the first part is concerned with the existence and convergence of solutions to the elliptic-parabolic hemivariational inequalities generated by nonconvex energy functionals. By Clarke's generalized gradient of the distance function defined on the star-shaped sets, in the second part, we construct the approximated hemivariational inequalities of the nonconvex probems and study the existence and convergence of their solutions. As the scientific frontier of hemivariational inequalities and doubly nonlinear equations, this project is not only of great theoretic importance in developing new methods to nonconvex problems, but also have vital application value in engineering and scientific problems such as heat transfer with phase change and the porous media seepage.

本项目主要结合H-半变分不等式理论研究具有非凸能量泛函或非凸约束的双重非线性问题。基于非光滑分析理论，它们可以转化为非线性发展型H-半变分不等式。研究内容分为2部分：1. 研究非凸能量泛函导出的椭圆-抛物型H-半变分不等式初值和和周期问题解的存在性和收敛性；2. 通过非凸星形集上距离函数的Clarke广义梯度，构造非凸约束型双重非线性问题的H-半变分不等式逼近，然后建立非凸约束问题解的存在性和收敛性定理。该课题属于H-半变分不等式和双重非线性问题的科学前沿，不仅对发展非凸问题的研究方法具有重要的理论意义，而且在相变热传导、多空介质渗流等工程科技问题中具有重要应用价值。

双重非线性问题普遍存在物理和工程科学中。本项目主要结合非线性泛函分析、偏微分方程、H-半变分不等式等理论方法与技术，研究具有非凸能量泛函的双重非线性问题以及非凸约束型问题。基于非光滑分析理论，我们将它们转化为非线性发展型H-半变分不等式进行研究。主要研究成果如下： 1.建立了由非凸能量泛函导出的椭圆-抛物型H-半变分不等式解的存在性和收敛性定理，并提高了解的正则性；2.建立了二阶非线性发展型H-半变分不等式解的存在性和收敛性定理；3. 通过非凸星形集上距离函数的Clarke广义梯度，构造了非凸约束型非线性问题的H-半变分不等式逼近，证明了非凸约束问题解的存在性。本课题的研究成果对发展非凸问题的研究方法具有重要的理论意义，而且在相变热传导、多孔介质渗流等工程科技问题中有一定应用价值。