Partial Differential Equations and Fourier Analysis

PI：David S.Jerison，MITDMS-0244991摘要：这个项目的主要目标是了解非线性椭圆型和抛物型偏微分方程解的水平曲面的光滑性(或非光滑性)和其他定量性质。PI将考虑模拟火焰锋面的半线性方程，作为半线性方程的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建议将这种正则性结果扩展到三维空间中的一类广泛的方程，包括尽可能多的物理激励的例子。鉴于自由边界的存在和正则性与相应的极小曲面问题之间的强烈相似之处，人们怀疑正则性将在更高的维度上崩溃，在那里人们期望找到奇异的能量最小化解，类似于著名的西蒙斯锥和Bernstein问题的反例。最后，PI将检查水平集的全局行为。例如，考虑与凸平面区域中最小的非零本征值相对应的Neumann本征函数。J·劳赫猜想，它所有的水平曲线都与边界接触。这个项目专注于边界未知且必须确定的非线性微分方程中的问题：所谓的自由边界。冰川融化的经典斯特凡问题就是一个例子。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”)作为时间函数的位置。本建议的方法适用的具体问题还包括火焰前锋、流动中的油和水之间的界面以及船的尾迹轮廓。PI既处理均衡问题，又处理进化问题。最近，皮耶里森证实，三维中的某些均衡问题有良好的解，而以前人们只理解二维的情况。因为三是物理空间的维度，这一发现为其他有物理意义的数学模型打开了大门，例如可压缩流体和毛细管的模型。另一种自由边界问题是决策理论(Pi Stroock)中出现的那种，它根本不是物理问题，而是数学自由边界理论所适用的问题。例如，人们想知道继续进行医学试验什么时候可能弊大于利。同样，在金融背景下，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，也就是说，这种期权可以在到期前的任何时候行使，而不是在固定的时间行使。

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.