General Magneto-Static Model

Abstract

In this paper we study a nonlinear magneto-static model on a general domain which is multiply-connected and has $m$ holes, and under a nonlinear relation between magnetic induction $\mathbf{B}$ and magnetic field $\mathbf{H}$. The equation contains a Neumann field ${\mathbf{h}}_1\in {\mathbb{H}}_1(\Omega )$ and a Dirichlet field ${\mathbf{h}}_2\in {\mathbb{H}}_2(\Omega )$, which represent the effects of domain topology. For a general electric current, the equation contains an unknown gradient $\nabla p(x)$, which represents the electric field. Existence results of solutions of boundary value problems of this model under various types of boundary conditions are proved, which exhibit the effects of domain topology.