带旋度算子方程组中的区域效应

This project will study how the overall geometric feature (convexity), the local geometric property and the topology of the domain affect the regularity, the shape and the number of the solutions of the equations involving the operator curl respectively. The equations involving the operator curl have the special mathematical structures, such as no maximum principle and the degenerate of the system. Compared with that of a single equation, the research of this project will be more complex, difficult and the results will be more abundant.. This project is closely related to the physical background. The local geometric effect of the domain is from the mathematical theory of the superconductivity. The type II superconductors will generate the vortex flux in the transition from the Meissner state to the vortex state. The location of the vortex flux first appeared depends on the local geometric properties of the superconducting materials; the convexity effect is from the theory of the Maxwell equations in electromagnetic field, the geometric characteristics of the ideal conductor affect the smoothness of the electric field and magnetic field in the internal of the conductor.. The study on this project has an important role to perfect the theory of the inequalities involving the operators div and curl in convex domains, to provide the theoretical basis and the research method for establishing the local properties and the boundary behavior of the solutions of the systems associating with the vectors. The research on this project has the important significance to characterize the different properties between the single elliptic equation and the elliptic system.

本项目研究区域的整体几何特征（凸性），局部几何性质以及拓扑结构对带旋度算子方程组解的正则性，解的形状以及解的个数的影响。由于带旋度算子方程组特殊的数学结构，比如没有极大值原理、方程本身的退化性，相比于单个方程，本项目的研究会更复杂和困难，结果也会更丰富。. 本项目与物理背景密切相关。区域的局部几何效应来自于超导中的数学理论，第II类超导体从Meissner态过渡到涡旋态会产生磁通漩涡，它首先发生的位置依赖于超导材料局部的几何性质；凸性效应来自于电磁场中的Maxwell方程组理论，理想导体整体的几何特征对导体内部电场与磁场的光滑性会产生影响。. 本项目的研究对完善凸区域上与向量场散度算子和旋度算子相关的不等式理论有重要作用，为若干与向量场相关的方程组建立解的局部性质和边界行为提供理论基础和研究方法。本项目的研究对刻画单个方程和方程组解的不同性质有重要意义。

本项目研究区域的整体几何特征，局部几何性质对带旋度算子方程组解的正则性、解形状的影响。我们在Lipschitz区域和凸区域建立了向量场散度—旋度不等式，并将这些不等式应用于Maxwell方程组解的估计。区域的局部几何效应来自于超导中的数学理论，第II类超导体从Meissner态过渡到涡旋态会产生磁通漩涡，它首先发生的位置依赖于超导材料局部的几何性质；我们得到了Meissner解的形状，在二维情形证明了磁通漩涡首先发生在曲率最小的地方。. 本项目的研究对完善凸区域上与向量场散度算子和旋度算子相关的不等式理论有重要作用，为若干与向量场相关的方程组建立解的局部性质和边界行为提供理论基础和研究方法。本项目的研究对刻画单个方程和方程组解的不同性质有重要意义。