Continuous and Discrete Free Boundary Problems for Partial Differential Equations

自由边界问题是偏微分方程(PDE)的问题，它是在一个区域中定义的，其边界事先是未知的(即“自由”)。然后必须在自由边界上提供另一个定量条件，以排除不确定性。这类问题出现在许多应用和工业领域。典型的例子是描述冰的融化和凝固的经典斯特凡问题：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多孔介质的过滤中，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，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)、生物学(不同物种占据的区域)等。由于在各种科学和实际问题中的广泛应用，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、生物学家和其他科学家提供了合作的机会。第一部分涉及自然地呈现余维2的自由边界的问题，这通常被称为薄自由边界。这类问题在弹性力学、数学金融、边界相变等领域都有广泛的应用。它们还涉及到非局部积分-微分算子的问题，例如通过扩张算子的分数拉普拉斯算子。第二部分讨论了各种生长/聚集模型中引人入胜的极限组态，例如阿贝尔沙堆生长模型，它可以被视为位势理论中庞加莱平衡问题的离散版本，并且与经典的障碍问题密切相关。虽然近年来在这类问题上取得了重大进展，但仍有许多重要问题有待回答，目前的项目旨在向这一方向作出贡献。要研究的特殊问题包括各种薄的自由边界问题：对于变系数抛物型偏微分方程组，对于几乎极小点(椭圆情形)，恰当地表述的两相和多相问题，以及系统地研究离散值最小超调和优项的极限(所谓的离散值Perron方法)。这可能会导致更好地理解阿贝尔沙堆和其他增长/聚集模型和相关偏微分方程中多边形和分段二次极限的形成。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Free boundary problems are problems for partial differential equations (PDEs) which are defined in a domain with a boundary that is not known beforehand (i.e. "free"). A further quantitative condition must be then provided at the free boundary to exclude indeterminacy. Problems of this sort arise in a large number of areas of applied and industrial interest. The paradigm example is the classical Stefan problem describing the melting and solidification of the 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, 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 on. Because of the abundance of applications in various sciences and real world problems, free boundary problems are considered today as one of the most important directions in the mainstream of the analysis of partial differential equations and offer opportunities of collaboration among mathematicians, physicists, engineers, material scientists, finance practitioners and other industrial researchers, biologists, and other scientists.This project consists of two main parts. The first part concerns problems that naturally exhibit free boundaries of codimension two, which are often called thin free boundaries. Such problems appear in many applications, such as elasticity, math finance, boundary phase transitions. They are also related to problems for nonlocal integro-differential operators such as the fractional Laplacian through extension operators. The second part concerns the fascinating limiting configurations in various growth/aggregation models such as the Abelian sandpile growth model, which can be viewed as a discrete version of Poincare's balayage problem in potential theory and is closely related to the classical obstacle problem. While there has been a significant progress in such problems in recent years, there are still many important questions that remain to be answered and the current project aims to contribute in that directions. Particular questions to be studied include various thin free boundary problems: for parabolic PDEs with variable coefficients, for almost minimizers (elliptic case), properly stated two- and multi-phase problems, as well as the systematic study of the limits of discrete-valued least superharmonic majorants (so-called discrete-valued Perron method). This may lead to better understanding of the formation of polygonal and piecewise-quadratic limits in Abelian sandpile and other growth/aggregation models and related PDEs.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.