Analytical and Geometrical Aspects of Non Linear Partial Differential Equations

摘要：我们的研究集中在一系列将非线性偏微分方程的不同领域结合在一起的现象上。caffarel@math.utexas.edu例如，在层状介质中传播的火焰前锋将火焰边缘处的快速相变的相互作用与大尺度的链接结合在一起，（前端速度）和小规模（可能是随机的，薄层）。最近的锋面形成模型，汇集了最佳分配的相互作用，这为锋面的形成提供了“变分框架”（Monge Ampere方程），并提出了涡度如何在锋面内传递和影响锋面的问题。最后，金融工程中的问题，将完全非线性方程组合在一起，（例如，当只有波动性和相关性的粗略界限时，作为“极值”或“信任”区间），当交易策略在某些参数值处不连续变化时，或者交易范围受到限制时，当底层空间随机变化时。我们计划研究来自不同科学的几个模型，这些模型将互补的非线性现象的相互作用结合在一起。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，火焰前锋会“摆动”，在某些层中移动得更快，这取决于它们的成分。从远处看，火焰将看起来像一个均匀的前锋。尽管它的速度，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的形成有关：根据一些模型，锋面在全球范围内组织起来，试图以最佳方式“传播能量”，在锋面内部，涡度（风的“旋转分量”）以与锋面组织相互作用的方式传输。第三个例子来自金融工程，（或最安全的）交易策略，但在大多数情况下，只有一个粗略的知识是可用的方式不同的方式，不同的组成部分的投资组合，或参数，（利率，汇率等）相互作用。这种粗略或不完整的知识，引起非线性策略，加上交易范围的限制，不连续交易策略等。

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.