Mathematical Sciences: Application of the Mulilayer Free Boundary Method to Nonlinear Elliptic Equations in Convex Domains

本课题主要研究一类非线性椭圆型方程的正则性理论和涉及解的重排导数的变分问题，以及与凸对称化有关的问题。主要重点放在可由多层自由边界问题序列逼近的变分问题上。将研究解的水平面的正则性和结构。本项目是在热核聚变研究中的受限环形等离子体准静态平衡模型和旋转流体流动研究的推动下进行的。需要解决的技术问题在聚变能源等领域具有根本性的影响。

This project is concerned with the regularity theory for a class of nonlinear elliptic equations and associated variational problems involving derivatives of the rearrangement of solutions, as well as related questions concerning convex symmetrization. The main emphasis is to be placed on variational problems which are approximable by sequences of multilayer free boundary problems. Regularity and structure of the level surfaces of solutions will be studied. This project has been motivated by a model of quasi static equilibrium of confined toroidal plasmas in thermonuclear fusion research, and the study of rotational fluid flow. The technical questions to be addressed have fundamental implications in areas such as fusion energy.