Certain Problems with Lower Dimensional Free Boundaries

这个项目的主要目标是研究自然地表现出余维2的自由边界的问题，这些边界存在于先验给定的流形中。这种类型的自由边界问题自然而然地出现在许多应用中，从弹性理论(斯诺里尼问题，或薄障碍问题)，数学金融(美式期权)，燃烧，边界热控制，以及更一般的，在边界相变问题中。最近出现的这类问题的一个重要来源是研究由非局部积分-微分算子管理的自由边界问题，例如分数次拉普拉斯算子。近年来，关于低维自由边界的一些问题已经有了显著的进展。然而，仍有一些根本问题尚未解决，当前项目的目标就是做到这一点。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，以及定常和时变情况下的部分速度图-勒让德变换。自由边界问题是指定义在边界事先未知的区域上的偏微分方程组的问题。然后，必须在自由边界处提供另一个定量条件，以防止不确定性。这类问题出现在许多应用和工业领域。典型的例子是经典的Stefan问题，它用来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。其他重要的例子出现在通过多孔介质(例如，油田)的过滤中，其中自由边界作为饱和区域和非饱和区域之间的前沿出现，而其他的例子来自燃烧(火焰前沿的传播)、数学金融(执行期权的最佳时间)、生物(由不同物种占据的区域)等等。由于在各种科学和现实问题中的广泛应用，自由边界问题被认为是当今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和生物学家之间的合作提供了机会。

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.