Helmholtz (-Hodge) decomposition commonly used in physics includes decomposition of a (sufficiently smooth) vector field $F = -\mathrm{grad}(U) + \mathrm{curl}(W)$ on bounded simply connected domain $\Omega \subseteq \mathbb{R}^3$ (with smooth boundary), with scalar $U$ and vector field $W$ which are explicitly given by integrals on Wiki page https://en.wikipedia.org/wiki/Helmholtz_decomposition

The proof presented on the same Wiki page (an early instance of which appeared in Aris: "Vectors, Tensors, and the Basic Equations of Fluid Mechanics"; see also Griffiths: "Introduction to Electrodynamics", Appendix C) silently assumes, among other things, that vector field $F$ is test (smooth and compactly supported), whereas one would like to have decomposition for vector fields which are merely $C^2(\Omega) \cap C^0(\overline{\Omega})$.

Is there some standard reference where the decomposition, with explicit expressions for $U$ and $W$, is proven in this modest setting?

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