Certain Free Boundary Problems

某些自由边界问题。建议研究摘要Arshak Petrosyan该项目是研究某些椭圆和抛物问题中自由边界的性质，特别是正则性。将研究一些不同的自由边界问题。在这个项目中要研究的大部分问题都是作为具有非局部约束的变分问题的解决方案而出现的。一些提出的问题是由亚椭圆算子，缺乏统一的椭圆性，但有足够丰富的几何结构（如卡诺集团），允许发展一个令人满意的理论自由边界。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，主要的方法是基于应用相当深的单调性公式（有时不止一个），允许证明的结果类似于与障碍问题相关的变分不等式。此外，自由边界往往有显着的性质，有时类似于极小曲面。 自由边界问题涉及在一个先验未知的区域中满足的方程（或方程组）。方程成立的区域的边界称为自由边界，通常以附加边条件为特征。 该问题的解决方案涉及到识别方程的解决方案和自由边界的区域。自由边界问题出现在从位势理论和几何到最优控制、机器人学和超导性的应用中。

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.