Partial Differential Equations and Fourier Analysis

偏微分方程和傅里叶分析

• 批准号: 0244991
• 负责人: David Jerison
• 金额: --
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-07-01 至 2008-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0244991&HistoricalAwards=false
• 关键词: Partial; Differential; Equations; Fourier; Analysis

PI: David S. Jerison, MITDMS-0244991 ABSTRACT:The main goal of this project is to understand smoothness (or nonsmoothness) and other quantitative properties of level surfaces of solutions to nonlinear elliptic and parabolic partial differential equations. The PI's will consider semilinear equations that model flame fronts, free boundary problems that are the singular limits of semilinear equations, and other models mentioned in the next paragraph. The regularity for two-dimensional free boundaries in three-space was only recently established. The PI's propose to show that such regularity results extend to a broad class of equations in three space dimensions, including as many physically motivated examples as possible. In view of the strong analogy between the existence and regularity for free boundaries and the corresponding questions about minimal surfaces, it is suspected that regularity will break down in higher dimensions, where one expects to find singular energy-minimizing solutions, analogous to the celebrated examples of the Simons cone and of counterexamples to the Bernstein problem. Finally, the PI's will examine global behavior of level sets. For example, consider a Neumann eigenfunction corresponding to the smallest, nonzero eigenvalue in a convex planar domain. J. Rauch conjectures that all its level curves touch the boundary.This project focuses on problems in nonlinear differential equations in which the boundary is unknown and has to be determined: a so-called free boundary. The classical Stefan problem of melting ice is an example. In the Stefan problem, the question of interest is the location, as a function of time, of the interface (``free boundary'') between water and ice. The particular problems to which the methods of the present proposal apply also include flame fronts, the interface between oil and water in a flow and the profile of the wake of a boat. The PI's treat both equilibrium and evolution problems. Recently PI Jerison established that certain equilibrium problems in three dimensions have well-behaved solutions, where previously only the two-dimensional case was understood. Because three is the dimension of physical space, this discovery opens the door to other physically meaningful mathematical models, such as models of compressible fluids and capillarity. Another kind of free boundary problem, one which is not at all physical in origin but to which mathematical free boundary theory applies, is the sort that arises in decision theory (PI Stroock). For example, one wants to know when continuing a medical trial is likely to cause more harm than good. Similarly, in a financial context, one wants to know when interest rates and stock prices indicate that it would be wise to buy or sell a stock option. Such questions arise when one is trying to price an American option, that is, an option that can be exercised at any time before it expires instead of at a fixed time.

PI：大卫S. Jerison，MITDMS-0244991摘要：本项目的主要目标是了解非线性椭圆和抛物型偏微分方程解的光滑性（或非光滑性）和水平面的其他定量性质。 PI将考虑模拟火焰前沿的半线性方程，作为半线性方程奇异极限的自由边界问题，以及下一段中提到的其他模型。 三维空间中二维自由边界的正则性是最近才建立起来的。 PI的建议表明，这种规律性的结果扩展到广泛的一类方程在三维空间，包括尽可能多的物理动机的例子。 鉴于自由边界的存在性和正则性与相应的极小曲面问题之间的强烈相似性，人们怀疑正则性将在高维中被打破，在高维中人们期望找到奇异的能量最小化解，类似于西蒙斯锥和伯恩斯坦问题的反例的著名例子。最后，PI将检查水平集的全局行为。例如，考虑对应于凸平面域中的最小非零本征值的诺依曼本征函数。 J. Rauch指出其所有的水平曲线都与边界接触。本项目主要研究非线性微分方程中的问题，其中边界是未知的，必须确定：所谓的自由边界。 经典的Stefan融冰问题就是一个例子。 在斯特凡问题中，感兴趣的问题是水和冰之间的界面（“自由边界”）的位置，作为时间的函数。应用本发明的方法的具体问题还包括火焰前缘、流中油和水之间的界面以及船尾流的轮廓。PI处理平衡和演化问题。 最近PI Jerison建立了某些平衡问题在三维有良好的表现解决方案，以前只有二维的情况下被理解。 由于三维是物理空间的维度，这一发现为其他物理意义上的数学模型打开了大门，例如可压缩流体和毛细作用的模型。 另一种自由边界问题，一个不是在所有物理的起源，但数学自由边界理论适用，是那种出现在决策理论（PI Stroock）。 例如，人们想知道什么时候继续进行医学试验可能弊大于利。 同样，在金融环境中，人们想知道什么时候利率和股票价格表明买入或卖出股票期权是明智的。当人们试图为美式期权定价时，就会出现这样的问题，美式期权是指可以在到期前的任何时间而不是在固定时间行使的期权。