Regularity Theory for Elliptic Equations and Free Boundaries

椭圆方程和自由边界的正则理论

• 批准号: 1565186
• 负责人: Stefania Patrizi
• 金额: $ 10.36万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-07-15 至 2019-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1565186&HistoricalAwards=false
• 关键词: Regularity; Theory; Elliptic; Equations; Free

The aim of this project is to study the mathematical structures of various differential equations, concentrating on models that may develop moving interfaces. These problems are used to model a number of different phenomena in Physics, Mathematical Finance, Image processing, and Biology. The main mathematical challenge in such problems is to understand the smoothness or regularity of the interfaces. The models combine the difficulty of analyzing the differential equation with the difficulty in locating the moving interface.The nonlocal equations to be investigated in this research project arise naturally in probability, in the study of stochastic processes with jumps. More precisely, free boundary problems involving nonlocal equations appear when considering optimal stopping problems, which are used in the pricing of American options in finance. When the underlying stochastic process is Brownian motion (with continuous paths), the regularity theory for solutions and free boundaries is well understood. However, when considering jump-processes the equations become nonlocal, and the structure of the model is more complicated. One of the main goals of the project is to provide a full understanding of such models, establishing regularity results for solutions and free boundaries in this type of problems.

该项目的目的是研究各种微分方程的数学结构，集中于可能产生移动界面的模型。这些问题被用来模拟物理学、数学金融学、图像处理和生物学中的许多不同现象。这类问题的主要数学挑战是理解界面的光滑性或规律性。该模型结合了联合收割机的困难，分析的微分方程的困难，在定位的移动interfaces.The非局部方程的研究在这个研究项目中自然出现的概率，在研究随机过程的跳跃。更准确地说，涉及非局部方程的自由边界问题出现在考虑最优停止问题时，这是用于金融中美式期权定价。当基本随机过程是布朗运动（具有连续路径）时，解和自由边界的正则性理论是很好理解的。然而，当考虑跳跃过程时，方程变得非局部，模型的结构更加复杂。该项目的主要目标之一是提供对此类模型的全面理解，为此类问题的解决方案和自由边界建立规律性结果。