Analytical and Geometrical Aspects of Non Linear Partial Differential Equations

2002年3月11日。PI: Luis Caffarelli [caffarel@math.utexas.edu]，德克萨斯大学，austin - ms -0140338摘要：我们的研究重点是将非线性PDE的不同领域结合在一起的一系列现象。例如，在分层介质中传播的火焰锋面汇集了火焰边缘快速相变的相互作用，并将大尺度（前速度）和小尺度（可能是随机的，薄层）联系在一起。锋面形成的最新模型，汇集了最优分配的相互作用，为锋面形成（蒙日安培方程）提供了“变分框架”，解决了涡度如何在锋面内部传输并影响锋面的问题。最后，金融工程中的问题，将完全非线性方程（例如，当只有波动率和相关性的粗略界限可用时，作为“极值”或“信任”区间）与相变问题结合在一起，当交易策略在某些参数值上不连续变化时，或者当潜在空间随机变化时，对交易范围和同质化存在约束时。我们计划研究来自不同科学的几个模型，这些模型汇集了互补非线性现象的相互作用。一个很好的例子是火焰在薄层材料中的传播：在近距离观察，火焰锋面会“摆动”，在某些层中移动得更快，这取决于它们的组成。从远处看，火焰将呈现为均匀的锋面。然而，它的速度将以一种非常微妙的方式取决于火焰边缘点火过程的性质（自由边界问题）和非常精细的性质（可能是随机组织的，因为它通常发生在自然界中）薄层。另一个例子与天气锋面的形成有关：根据一些模型，锋面在全球范围内组织自己，试图以最佳方式“传播它的能量”，在内部，涡度（风的“旋转成分”）以一种与锋面组织互动的方式传输。第三个例子来自金融工程，当寻求最优（或最安全）的交易策略时，但是，在大多数情况下，对于投资组合的不同组成部分或参数（利率、汇率等）相互作用的不同方式，只有粗略的了解。这种粗糙或不完整的知识产生了非线性策略，它与交易范围的约束、不连续交易策略等相结合。

March 11, 2002.PI: Luis Caffarelli [caffarel@math.utexas.edu], University of Texas, AustinDMS-0140338Abstract:Our research focus in a series of phenomena that bring togetherdifferent areas of non linear PDE. For instance, a flame front propagating in a layered medium brings together the interaction of the fast phase transition at the edge of the flame, with the linking of large scales (front speeds) and small scales (possibly random , thin layering).Recent models of front formation, bring together the interaction ofoptimal allocation,that provides the " variational framework" for frontformation (the Monge Ampere equation ) with the issue of how vorticity istransported within, and affects the front. Finally, problems in financial engineering, bring together fully non linear equations ( for instance as "extremals" or intervals of"trust" when only rough bounds on volatility and correlations areavailable) with issues of phase transition , when trading strategychanges discontinuously at certain values of the parameters,or there areconstraints on the range of trading, and homogeneisation, when theunderlying space changes randomly.We plan to study several models from different sciences that bringtogether interactions of complementary non linear phenomena.A good example would be flame propagation in a thinly layeredmaterial: Observed at short range, the flame front will "wiggle", movingfaster in some of the layers, depending on their composition.From far away, the flame will appear as a homogeneous front.Its speed , though, will depend in a very subtle way from both the natureof the ignition process at the edge of the flame ( free boundaryproblem) and the properties of the very fine, ( possibly randomlyorganized, as it usually occurs in nature) thin layering.Another example has to do with the formation of weather fronts: accordingto some models, globally the front organizes itself trying to " spread itsenergy" in an optimal way, and within, vorticity ( the "rotationalcomponent" of wind) is transported in an interactive way with the frontorganization. A third example comes from financial engineering when seeking optimal ( orsafest) trading strategies, but , as in most cases, only a rough knowledgeis available of the way different ways in which different components ofthe portfolio, or parameters ( interest rates, exchange rates, etc) interact.This rough or incomplete knowledge, gives rise to non linearstrategies, that couple with constraints in the trading range,discontinuous trading strategies, etc.