Certain Free Boundary Problems

若干自由边界问题。Arshak petrosyan本课题研究某些椭圆型和抛物型问题的自由边界的性质，特别是正则性。我们将研究许多不同的自由边界问题。本课题研究的大多数问题都是关于非局部约束的变分问题的解。所提出的一些问题是由亚椭圆算子控制的，它们缺乏一致的椭圆性，但具有足够丰富的几何结构（例如在卡诺群中），允许发展一个令人满意的自由边界理论。其他问题类似于传统的变分不等式，但缺乏不等式约束。对于这些问题，主要的方法是基于相当深的单调性公式（有时不止一个）的应用，它允许证明类似于与障碍问题相关的变分不等式的结果。此外，自由边界通常具有显著的性质，有时类似于最小曲面的性质。自由边界问题涉及在未知先验区域内满足的方程（或方程组）。方程所在区域的边界称为自由边界，通常以附加的边条件为特征。该问题的求解既要确定方程的解，又要确定具有自由边界的区域。自由边界问题出现在从势理论和几何到最优控制、机器人和超导的应用中。

Certain Free Boundary Problems.Abstract of Proposed Research Arshak PetrosyanThis project is to study the properties, especially the regularity, of the free boundary in certain elliptic and parabolic problems. A number of different free boundary problems will be investigated. Most of the problems to be studied in this project arise as the solution of variational problems with nonlocal constraints. Some of the proposed problems are governed by subelliptic operators, which lack the uniform ellipticity, but have a sufficiently rich geometric structure (e.g. in Carnot groups) that allows the development of a satisfactory theory of free boundaries. Other problems resemble traditional variational inequalities, but lack an inequality constraint. For those problems, the main approach is based on application of rather deep monotonicity formulas (sometimes more than one) that allow the proof of results similar to those for variational inequalities associated with obstacle problems. Moreover the free boundaries often have remarkable properties, sometimes akin to those of minimal surfaces. Free boundary problems involve equations (or systems of equations) that are satisfied in a region that is unknown apriori. The boundary of the region where the equation holds is called the free boundary and is generally characterized by extra side conditions. The solution of the problem involves identifying both the solution of the equation and the region with the free boundary. Free boundary problems arise in applications ranging from potential theory and geometry to optimal control, robotics, and superconductivity.