Variational principles and their applications in modern analysis

变分方法为描述和解决对科学家和工程师至关重要的数学问题提供了高效而优雅的工具，它们的应用范围很广，包括优化理论、非光滑分析、经济学、控制理论以及图像处理。这并不令人惊讶，因为科学和工程中的许多问题都可以重塑为变分问题。然而，众所周知，并不是每个方程都符合标准的欧拉-拉格朗日变分结构。近年来，人们致力于发展一种超越这种标准变分结构的变分法，它可以应用于大量的偏微分方程组和发展方程，其中许多不符合经典的欧拉-拉格朗日框架。凸自对偶和非凸自对偶理论的诞生就是这些研究的结果。加权能量耗散泛函是这一方向的另一种新方法。我的目的是进一步推广这些变分原理，以处理超出经典范围的更大一类方程。我还想研究它们在反问题、蒙格质量输运和弹性问题中的应用。

Variational methods provide efficient and elegant tools for formulating and solving mathematical problems that are of key importance to scientists and engineers, and their applications span a wide range that includes optimization theory, non-smooth analysis, economics, control theory, as well as image processing. This is not surprising, since many problems in science and engineering can be recast as variational problems. However, it is known that not every equation admits the standard Euler-Lagrange variational structure. In recent years, considerable effort has been devoted to the development of a calculus of variations that would go beyond this standard variational structure and that could be applied to a large number of partial differential systems and evolution equations, many of which do not fit within the classical Euler-Lagrange framework. The birth of the theory of convex and non-convex self-duality has been the result of some of these studies. The Weighted energy-dissipation functionals is another novel approach in this direction. My aim is to extend further these variatioal principles to deal with a larger class of equations beyond the classical range. I would also like to investigate their applications to inverse problems, Monge's mass transport and problems in elasticity.