# Reference for Hodge decomposition

Let $U$ be a bounded open subset of $\mathbb{R}^d$ with Lipschitz boundary, and $g \in L^2(U,\mathbb{R}^d)$ be a solenoidal vector field (i.e. $\nabla \cdot g = 0$). Then $g$ can be written in the form $$g = \nabla \cdot S + \nabla h,$$ where $h$ is harmonic and $S$ takes values in the set of skew-symmetric matrices. Does someone know of a good reference for this result?

• You are describing basically the Hodge decomposition. For Lipschitz domains the result is derived in this AMS memoir (and probably elsewhere too). – Willie Wong Sep 17 '14 at 13:09
• Thanks for this reference. I would be even happier with a reference that only proves the statement for subsets of $\mathbb{R}^d$ (as opposed to Riemannian manifolds), and does not explicitly refer to Hodge theory or differential forms. Is there something like this somewhere? – Elwood Sep 17 '14 at 19:19
• For smooth boundaries, possibly yes, in some advanced calculus textbooks. For Lipschitz boundaries I am slightly doubtful. – Willie Wong Sep 18 '14 at 7:27
• On $\mathbb R^n$ this is also called the Helmholtz decomposition - search also under this name. – Peter Michor Sep 22 '14 at 12:37
• It is written in Girault-Raviart's textbook on Navier Stokes for $d=2,3$. – username Sep 24 '14 at 10:48