Viscosity Solutions: Beyond Well-Posedness Theory

该项目涉及一些自然出现在物理学，社会科学和工程学中的非线性偏微分方程，例如，在复合材料，燃烧，博弈论，交通流和优化的研究中出现。所考虑的方程与许多其他数学领域有着深刻的联系，包括变分法、微分对策、动力系统、几何、均匀化理论、反问题和最优控制理论。该项目的主要目标是发现新的基本原理和一般方法，以了解所研究的微分方程解的性质。研究的一个关键对象是均匀化理论，其中模型（例如，物理或社会现象）设置在异质介质中，并且具有许多在小尺度上变化的参数。如果我们缩小并观察宏观尺度，我们通常会看到一个简单的有效（平均）行为。为了实际应用这些模型，深入理解这种有效行为的定性和定量方面是极其重要的。该项目有两个首要主题：（i）超越单纯的适定性，以理解均匀化理论中极限过程和有效方程的精细性质（例如，有效哈密顿量的形状，最佳收敛速率），以及（ii）推进对偶方法和完全非线性椭圆方程的知识（例如，解的表示公式，消失折扣问题，正则性）。首席研究员和他的合作者最近在这些（和相关）主题领域开发了新的方法，这些方法已经为一些开放的问题提供了解决方案。这些方法有望在本项目中得到进一步发展，从而为粘度解决方案领域带来新的视角和见解。

This project concerns some nonlinear partial differential equations that appear naturally in physics, the social sciences, and engineering and that arise, for example, in the study of composite materials, combustion, game theory, traffic flow, and optimization. The equations considered have deep connections with a host of other areas of mathematics, including the calculus of variations, differential games, dynamical systems, geometry, homogenization theory, inverse problems, and optimal control theory. The main goal of the project is to discover new underlying principles and general methods to understand the properties of solutions of the differential equations under investigation. One of the key objects of the research is homogenization theory, in which the models (say, of physical or social phenomena) are set in heterogeneous media and have many parameters varying on a small scale. If one zooms out and looks at the macroscopic scale, one often sees a simple effective (averaging) behavior. To make practical use of the models, it is extremely important to understand deeply the qualitative and quantitative aspects of this effective behavior.The project has two overarching themes: (i) going beyond mere well-posedness in order to understand fine properties of both the limiting process and the effective equation in homogenization theory (e.g., shape of the effective Hamiltonian, optimal rate of convergence), and (ii) advancing knowledge of the duality method and fully nonlinear elliptic equations (e.g., representation formulas of solutions, the vanishing discount problem, regularity). The principal investigator and his collaborators have recently developed new approaches in these (and related) topical areas, approaches that have already provided solutions to some open problems. These methods are expected to be developed further in this project, thereby bringing fresh perspectives on and insights into the field of viscosity solutions.