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

工作将集中在一类正则性理论， 非线性椭圆型偏微分方程及其相关 涉及重排导数的变分问题 解决方案。 这些方程被称为格拉德微分方程 方程，是核聚变中广泛使用的模型 社区此外，关于凸 对称化、插值与某些自由边界 问题将得到处理。 这些问题源于 约束环形等离子体的准静态平衡 热核聚变和旋转流体流。 从 从纯数学的角度来看，他们的分析涉及到 重排研究中有趣而重要的问题 凸对称化，自由边界问题， 非线性非局部变分问题的正则性理论 主要重点将放在变分问题， 用多相自由边界序列近似 问题最近的工作建立了收敛标准， 这种近似方案以及各种正则性结果 对于某些解决方案。 一种新的建立 规划了凸域弱解的解析性。 这项工作的一部分涉及研究的一般化， 凸域的施瓦茨对称化及其它方法 对于像狄利克雷范数这样的递减泛函。 //

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. //