Certain Aspects of Free Boundary Problems

提案DMS-0401179 PI：Arshak Petrosyan，普渡大学标题：自由边界问题的某些方面摘要本提案的目的是研究某些缺乏“经典”性质的自由边界问题，如均匀椭圆性（抛物性）或解的非负性。更具体地说，PI建议研究以下问题，这将有助于理解新困难的本质和解决这些困难的可能途径：1）Heisenberg群中次Laplacian障碍问题中导数的连续性; 2）平面上p-Laplacian障碍问题的齐次整体解的分类; 3）退化相变模型中水平集的一致收敛性; 4）平面上退化Bernoulli型问题自由边界的光滑性，5）解无符号限制的单相障碍和Stefan型问题。有些问题可以通过使用最近开发的方法来解决，而一些旧方法的复兴可能在其他情况下证明是成功的。部分结果以及经验证据（在某些情况下）支持预期和假设的结果。自由边界是先验未知集，出现在偏微分方程和变分问题的解决方案。典型的例子是相变和流体力学问题中的界面和运动边界。感兴趣的主要问题是自由边界及其结构的规则性（光滑性）。由于许多数学家的贡献，自由边界理论在过去的几十年里已经发展成为数学中一个非常深刻和美丽的部分.然而，在一些自由边界问题中出现的应用，从几何和最优控制机器人和超导，某些传统的假设可能会打破。这个建议的目的是了解哪些结果可以从自由边界的经典理论结转到这些问题的情况。

Proposal DMS-0401179PI: Arshak Petrosyan, Purdue UniversityTitle: Certain aspects of free boundary problemsABSTRACTThe aim of this proposal is to study certain free boundary problemswith the lack of "classical" properties such as the uniformellipticity (parabolicity) or the nonnegativity ofthe solutions. More specifically, the PI proposes to study thefollowing questions that will help to understand the essence of thenew difficulties and the possible ways of their resolution:1) the continuity of the derivatives in the obstacle problem for thesub-Laplacian in the Heisenberg group; 2) the classification ofhomogeneous global solutions of the obstacle problem for thep-Laplacian in the plane; 3) uniform convergence of the level setsin a degenerate phase-transition model; 4) smoothness of the freeboundary in a degenerate Bernoulli-type problem in the plane; and 5)one-phase obstacle and Stefan type problems with no sign restrictionson solutions. Some of the problems can be attacked by using recentlydeveloped methods, while a revitalization of some older methods mightprove successful in other cases. Partial results as well asempirical evidence (in some cases) are provided to support the expectedand conjectured results.Free boundaries are apriori unknown sets, coming up in solutions ofpartial differential equations and variational problems. Typicalexamples are the interfaces and moving boundaries in problems on phasetransitions and fluid mechanics. Main questions of interest are theregularity (smoothness) of free boundaries and their structure. Thanksto the contributions by many mathematicians the theory of freeboundaries has developed over the last decades to a very deep andbeautiful part of mathematics. However, in a number of freeboundary problems that arise in applications, ranging from geometryand optimal control to robotics and superconductivity, certaintraditional assumptions may break down. The aim of thisproposal is to understand which results could be carried over from theclassical theory of free boundaries to the case of those problems.