Generalized vector potential and Trace Theorem for Lipschitz domains

Abstract

The vector potential is a fundamental concept widely applied across various fields. This paper presents an existence theorem of a vector potential for divergence-free functions in $W^{m,p}({\mathbb{R}}^N,\mathbb{T})$ with general $m,p,N$. Based on this theorem, one can establish the space decomposition theorem for functions in $W_0^{m,p}(\mathrm{curl};\hat{I}\text{c◯},{\mathbb{R}}^N)$ and the trace theorem for functions in $W^{m,p}(\hat{I}\text{c◯})$ within the Lipschitz domain $\hat{I}\text{c◯}\subset {\mathbb{R}}^N$. The methods of proof employed in this paper are straightforward, natural, and consistent.