Mathematical Sciences: Regularity Theory for Certain Nonlinear Elliptic Equations Involving Derivatives of Rearrangements of Solutions

工作将集中在一类非线性椭圆型偏微分方程正则理论和涉及解的重排的导数的相关变分问题。这些方程被称为Grad微分方程，是核聚变界广泛使用的模型。此外，还将讨论有关凸对称化和插值化的问题以及某些自由边界问题。这些问题源于热核聚变中受限环形等离子体的准静态平衡模型和旋转流体流动。从纯数学的观点来看，它们的分析涉及到函数重排、凸对称化、自由边界问题和非线性非局部变分问题的正则性理论等研究中有趣而重要的问题。主要重点放在可用多相自由边界问题序列逼近的变分问题上。最近的工作已经建立了这类近似格式的收敛准则以及某些解的各种正则性结果。规划了一种建立凸域弱解的解析性的新方法。这项工作的一部分包括研究Schwarz对称化到凸域的推广，以及其他减少泛函的方法，如Dirichlet范数。//

Work will focus on the regularity theory for a class of nonlinear elliptic partial differential equations and associated variational problems involving derivatives of the rearrangement of solutions. The equations, known as Grad differential equations, are a widely used model in the nuclear fusion community. In addition, questions concerning convex symmetrization and interpolation and certain free boundary problems will be treated. These questions arise from models of quasi-static equilibrium of confined toroidal plasmas in thermonuclear fusion and from rotational fluid flow. From a purely mathematical standpoint their analysis involves interesting and important problems in the study of rearrangements of a function, convex symmetrization, free boundary problems and regularity theory for nonlinear nonlocal variational problems. The main emphasis will be on variational problems which can be approximated by sequences of multiphase free boundary problems. Recent work has established convergence criteria for such approximation schemes as well as various regularity results for certain solutions. A new approach for establishing analyticity of weak solutions for convex domains is planned. Part of this effort involves the study of generalizations of Schwarz symmetrization to convex domains as well as other methods for decreasing functionals like the Dirichlet norm. //