关于电磁场方程组解的W^(1,p)正则性研究

On W^(1,p) Regularity of A System Arising from Electromagnetic Fields

本文研究了R^3中有界区域Ω上的电磁场方程组弱解的W^(1,p)估计.该方程组来自于磁场所满足的稳态麦克斯韦方程组.在假定系数矩阵的逆属于VMO空间的条件下,利用R^3中向量场的旋度和散度的性质,将该方程组转化为标量椭圆型方程组,从而根据椭圆型方程组的正则性理论,得到解的W^(1,p)估计,其中1 <p <∞.

We establish the fundamental W^1,p estimate for the weak solution of a system in a bounded domain Ω in R^3. The system is related to the steady-state of Maxwell’s equations for the magnetic field. The inverse of the principle coefficient matrix is assumed to be in the VMO space. We transform the system to scalar elliptic equations by using the properties of curl and divergence of vector fields in R3. By the regularity theory of elliptic equations, we get the W1,p estimate for 1 < p <∞.