Certain Problems with Lower Dimensional Free Boundaries

低维自由边界的某些问题

• 批准号: 1101139
• 负责人: Arshak Petrosyan
• 金额: $ 22.55万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-08-01 至 2015-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1101139&HistoricalAwards=false
• 关键词: Certain; Problems; Lower; Dimensional; Free

The main objective of this project is to study problems that naturally exhibit free boundaries of codimension two that live in an apriori given manifold. Free boundary problems of this type appear naturally in many applications, ranging from the theory of elasticity (the Signorini problem, or the thin obstacle problem), mathematical finance (American options), combustion, boundary heat control, and more generally, in problems with boundary phase transitions. An important source of such problems that has emerged recently is the study of free boundary problems governed by nonlocal integro-differential operators, such as the fractional Laplacian. There has been a significant progress in recent years in some problems with lower-dimensional free boundaries. However, there are still a number of fundamental questions that have yet to be addressed and the current project aims to do just that. Particular questions include monotonicity formulas for nonlocal operators, appropriate generalizations of the boundary Harnack principle, and the partial hodograph-Legendre transform, both in stationary and time-dependent situations.Free boundary problems are problems for partial differential equations that are defined in domains whose boundaries are not known beforehand (i.e., are "free"). A further quantitative condition must be then provided at the free boundary to prevent indeterminacy. Problems of this sort arise in a large number of areas of applied and industrial interest. The paradigmatic example is the classical Stefan problem, which is to model the melting and solidification of ice: the free boundary here is the moving interface between the regions occupied by the water and the ice. Other important examples occur in filtration through porous media (e.g., an oil field), where free boundaries occur as fronts between saturated and unsaturated regions, and others come from combustion (propagation of the flame front), mathematical finance (optimal time for exercising an option), biology (regions occupied by different species), and so forth. Because of the abundance of applications in various sciences and real world problems, free boundary problems are considered today to be one of the most important directions in the mainstream of the analysis of partial differential equations and offer opportunities for collaboration between mathematicians, physicists, engineers, materials scientists, financial practitioners and other industrial researchers, and biologists.

该项目的主要目标是研究自然表现出存在于先验给定流形中的余维二自由边界的问题。这种类型的自由边界问题自然出现在许多应用中，包括弹性理论（西格诺里尼问题或薄障碍问题）、数学金融（美式期权）、燃烧、边界热控制，以及更普遍的边界相变问题。最近出现的此类问题的一个重要来源是对由非局部积分微分算子（例如分数拉普拉斯算子）控制的自由边界问题的研究。近年来，在一些低维自由边界问题上取得了重大进展。然而，仍有许多基本问题有待解决，当前项目的目的就是解决这个问题。具体问题包括非局部算子的单调性公式、边界 Harnack 原理的适当推广，以及在稳态和时间相关情况下的部分记录图-勒让德变换。自由边界问题是在事先未知边界（即“自由”）的域中定义的偏微分方程问题。然后必须在自由边界处提供进一步的定量条件以防止不确定性。此类问题出现在许多应用和工业领域。典型的例子是经典的斯特凡问题，它模拟冰的融化和凝固：这里的自由边界是水和冰占据的区域之间的移动界面。其他重要的例子发生在多孔介质（例如油田）的过滤中，其中自由边界出现在饱和区域和不饱和区域之间的前沿，其他例子来自燃烧（火焰前沿的传播）、数学金融（行使选项的最佳时间）、生物学（不同物种占据的区域）等等。由于在各种科学和现实世界问题中的丰富应用，自由边界问题今天被认为是偏微分方程分析主流中最重要的方向之一，并为数学家、物理学家、工程师、材料科学家、金融从业者和其他工业研究人员以及生物学家之间的合作提供了机会。