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Generalized vector potential and Trace Theorem for Lipschitz domains (2405.05228v1)

Published 8 May 2024 in math-ph, math.AP, math.FA, and math.MP

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 $W{m,p}_0(\operatorname{curl};\Omega,\mathbb{R}N)$ and the trace theorem for functions in $W{m,p}(\Omega)$ within the Lipschitz domain $\Omega \subset \mathbb{R}N$. The methods of proof employed in this paper are straightforward, natural, and consistent.

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