# General Magneto-Static Model

## TL;DR

This paper develops a comprehensive nonlinear magneto-static model for complex multiply-connected domains, analyzing the influence of topology and boundary conditions on the existence of solutions.

## Contribution

It introduces a general magneto-static model incorporating domain topology effects and proves existence results for solutions under various boundary conditions.

## Key findings

- Existence of solutions depends on domain topology.
- The model accounts for nonlinear B-H relations.
- Boundary conditions significantly influence solution existence.

## 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 $\bold B$ and magnetic field $\bold H$. The equation contains a Neumann field $\bold h_1\in\Bbb H_1(\Omega)$ and a Dirichlet field $\bold h_2\in\Bbb H_2(\Omega)$, 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.