Numerical Analysis of Selected Variational and Quasi-variational Inequalities

单边和约束现象在科学和工程中普遍存在。控制物理过程的量经常受到约束：无论是机械的、物理的、热力学的还是实际性质的。这些例子可以是不可穿透性条件（两个物体不能同时在同一个地方），质量不能为负的事实，熵不能减少，或者简单地说，我们不能产生超过固定数量的力。此外，许多这些约束可能取决于兴趣本身的数量（例如摩擦）。最后一个例子是机械接触，事实上，这是我们能对另一个物体施加机械作用的唯一机制。这些例子表明，推导出描述这些现象的真实模型在实际应用中是至关重要的。然而，这些模型通常是非线性的。发展和分析近似解决这些问题的数值技术具有实际意义。控制约束现象的模型被称为变分不等式和拟变分不等式。本建议旨在研究一组变分和拟变分不等式的数值技术，这些不等式以前没有被考虑过，也不适用于经典的思想和技术。它们包括：非局部算子的变分不等式，退化抛物方程的演化问题以及准变分不等式的时间离散化技术的一般研究。由于所提出的问题具有广泛的应用范围，因此本项目将产生的数值方法将引起广泛实践者的兴趣。例如，分数扩散障碍问题出现在控制理论、流体力学甚至金融学中；将考虑在成像和材料科学中找到应用的退化抛物方程；准变分不等式的典型例子是摩擦，但它们也出现在博弈论中——当试图找到纳什点时——以及在控制理论中处理冲动控制时。

Unilateral and constrained phenomena are ubiquitous in science and engineering. The quantities that govern a physical process are often subject to constraints: Be it of mechanical, physical, thermodynamical or practical nature. Examples of these can be impenetrability conditions (two bodies cannot be at the same place at the same time), the fact that mass cannot be negative and that entropy cannot decrease or simply the fact that we are not able to produce more than a fixed amount of forcing. In addition, many of these constraints might depend on the quantities of interest themselves (e.g. friction). One final example is mechanical contact which is, in fact, the only mechanism through which we can exert any mechanical action on another body. These examples show that the derivation of realistic models for the description of these phenomena is of fundamental importance in applications. However these models will be, as a rule, nonlinear. The development and analysis of numerical techniques for approximating the solution of these problems is of practical relevance.The models that govern constrained phenomena carry the name of variational and quasi-variational inequalities. This proposal aims at the study of numerical techniques for a collection of variational and quasi-variational inequalities which have not been considered before and to which the classical ideas and techniques do not apply. They include: variational inequalities for nonlocal operators, evolution problems for degenerate parabolic equations and the general study of time discretization techniques for quasi-variational inequalities. The numerical methods that will result from this project will be of interest to a wide range of practitioners, since the proposed problems have a wide range of applications. For instance, obstacle problems with fractional diffusion appear in control theory, fluid mechanics and even finance; the degenerate parabolic equations that will be considered find applications in imaging and materials science; the prototypical example of a quasi-variational inequality is friction, but they also arise in game theory - when trying to find Nash points - and in control theory when dealing with impulse controls.