Analytical and Geometrical Aspects of Non Linear Partial Differential Equations

点击翻译按钮获取中文摘要

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.