Nonlinear Diffusion Equations and Free-Boundary Problems

题目：非线性扩散方程和自由边界问题PI：Panagiota Daskalopoulos，哥伦比亚大学摘要本项目 关注非线性椭圆和抛物方程和自由边界问题的研究，与更复杂的微分几何问题有关，包括高斯曲率流，Ricci流和Weyl问题与非负高斯曲率和物理应用，如薄液膜动力学和火焰传播。该项目的第一部分将研究的几何形状和规律性的自由，由拟线性和完全非线性几何流的退化引起的边界问题，如具有平坦边的高斯曲率流或包括调和流在内的更一般的曲率流。边界问题可能具有重要的几何甚至拓扑应用。另一条新的研究路线将研究边界问题的规律性。退化Monge-Ampere方程及相关椭圆自由边界问题解它的主要目标是发展新的技术来建立完全非线性退化椭圆型方程的最优正则性。 第三部分研究Stefan型自由边界问题，包括Hele-Shaw流动和火焰传播中的自由边界问题，研究几何与正则性之间的联系以及奇异性的形成。使用的几何方面的问题是至关重要的，在proposedapproach。拟议活动的最后一部分 将研究各种奇异扩散模型解的渐近行为。 特别是，它将处理第二类炸毁行为， 二维Ricci流的最大解。这些解对应于非紧曲面上的完备黎曼共形度量.该项目将广泛的数学活跃领域联系起来，特别是非线性偏微分方程、几何和经典分析.所提出的关于退化非线性抛物方程和自由边界问题的几何和正则性的研究活动可能会导致重要的几何甚至拓扑应用. Stefan型自由边界问题的研究活动与各种重要的物理模型密切相关，包括高活化能极限下预混等扩散火焰的传播，本项目将研究的奇异扩散模型 在各种物理应用中出现，如种群动力学，气体动力学理论和薄液膜动力学。学生和博士后将作为该项目的一部分进行培训。 将特别强调鼓励有才华的女本科生、研究生和博士后在数学或相关科学领域取得成功。 将为研究生设计和实施连接偏微分方程和几何分析的新课程。

Title: Nonlinear Diffusion Equations and Free boundary problemsPI: Panagiota Daskalopoulos, Columbia UniversityABSTRACTThis project concerns with the study of nonlinear elliptic and parabolic equations and free-boundary problems, in connection with more complex problems of differential geometry,includingthe Gauss curvature flow, the Ricci flow and the WeylProblem withnonnegative Gaussian curvature and with physical applications such as thin liquidfilm dynamics and flame propagation.The first part of the project will study thegeometry and regularity of free-boundary problems arising from the degeneracy ofquasilinear and fully-nonlinear geometric flows,such as the Gauss curvature flow withflat sides or more general curvature flows including the Harmonic flow.The understanding of such models of equations and free-boundary problemsmay have significant geometric and even topological applications.A different new line of research will study the regularity of solutionsofdegenerate Monge-Ampere equations and related ellipticfree-boundary problems. Its main goal is to develop new techniques toestablish the optimal regularity in fully-nonlineardegenerate elliptic equations. The proposed work is also motivated bythe well known Weyl problem with nonnegative Gaussian curvature.The aim of the third part of the project is to study the connectionbetween the geometry and the regularity as well as the formation ofsingularities in Stefan type free-boundary problems including also theHele-Shaw flow and free-boundary problem in flame propagation. The useof the geometric aspects of the problems is crucial in the proposedapproach. The last part of the proposed activity will study the asymptoticbehavior of solutions of variousmodels of singular diffusion. In particular, it will deal with the type II blow up behavior of maximal solutions of the two dimensionalRicci Flow. These solutions correspond to complete Riemannian conformalmetrics on a non-compact surface.This project links a wide range of active fields of mathematics, inparticular nonlinear partial differential equations, geometry andclassical analysis.The proposed research activity on the geometry and regularity ofdegenerate nonlinear parabolic equations and free-boundary problems mayresult to significant geometric and even topological applications. Theproposed research activity on Stefan type free-boundary problems isclosely relatedto various important physical models, including the propagation of thepremixed equi-diffusional flames in the limit of high activation energy.The models of singular diffusion which will be studiedin this project arise in variousphysical applications such as population dynamics, the kinetic theory ofgases and thin liquid film dynamics.Students and postdocs will be trained as part of this project. Special emphasis will be given to theencouragement of talented femaleundergraduate students, graduate students and postdocsto pursue a successfulcareer in mathematics or related sciences. New courses linking PartialDifferential Equations and Geometric Analysis for graduate students will be designedand implemented.