两类含旋度算子的拟线性方程组

The effects of lower order term, boundary condition and the topology of domain on existence and regularity theory of quasilinear elliptic system involving operator curl will be studied in this project, which consists of five problems related to two types of system..The first type is the extended magnetostatic Born-Infeld model (degenerate elliptic, the main term of the corresponding functional is degree-one growth) which plays an important role in the nonlinear electrodynamics. We have obtained the existence and Hölder regularity of the solutions when the topology of the domain is trivial, the lower order term is a convex function or a concave function, and the boundary is small enough. Naturally we would like to study further the effects of three essential factors on the existence and regularity theory: nontrivial domain topology, lower order term which is neither convex nor concave, and large boundary value..The second type is the q-curlcurl system. Due to its typical quasilinear structure, most difficulties dealing with the general systems involving operator curl will appear. Although we have proved the existence and the interior regularity of solutions when q>1, the boundary regularity is not clear and cannot derived directly from the interior estimates. Here we would like to deduce the boundary estimates for two types of boundary value problem, which will help us understand the effects of different types of boundary conditions on the regularity theory. Especially, when q=1, the 1-curlcurl systems have some commons with the other two types of systems introduced above, but there are material differences between them and few results on 1-curlcurl systems. Hence we plan to start from 1-curlcurl operator, define the suitable admissible vector space and study the least eigenvalue problem on 1-curlcurl operator.

本项目拟研究方程组低阶项、边值条件、区域拓扑对含旋度算子的拟线性椭圆方程组的存在性和正则性理论的影响，共两类方程组。.第一类是非线性电动力学中占据重要地位的扩展的静磁场Born-Infeld模型（退化椭圆，对应泛函主项一阶增长）。区域单连通无洞、低阶项为凸函数或凹函数、边值足够小时已得解的存在性和Hölder连续性，我们希望进一步考察非平凡的区域拓扑、非凸非凹低阶项以及边值度量对解的存在性和正则性的影响。 .第二类是具有典型拟线性数学结构的q-curlcurl方程组，一般含旋度算子的拟线性方程组的主要难点大都在这类方程组中有所体现。q>1时已得解的存在性和内部正则性，但边界正则性却无法从内估计直接得到，我们希望借边界估计的研究来理解边值条件类型对解的正则性的影响。特别地，q=1时这类方程组的研究尚未开展，我们拟从1-curlcurl算子的最小特征值问题着手，这需要建立适当的允许向量场空间。