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?
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4$\begingroup$ You are describing basically the Hodge decomposition. For Lipschitz domains the result is derived in this AMS memoir (and probably elsewhere too). $\endgroup$– Willie WongCommented Sep 17, 2014 at 13:09
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$\begingroup$ 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? $\endgroup$– ElwoodCommented Sep 17, 2014 at 19:19
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$\begingroup$ For smooth boundaries, possibly yes, in some advanced calculus textbooks. For Lipschitz boundaries I am slightly doubtful. $\endgroup$– Willie WongCommented Sep 18, 2014 at 7:27
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1$\begingroup$ On $\mathbb R^n$ this is also called the Helmholtz decomposition - search also under this name. $\endgroup$– Peter MichorCommented Sep 22, 2014 at 12:37
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1$\begingroup$ It is written in Girault-Raviart's textbook on Navier Stokes for $d=2,3$. $\endgroup$– usernameCommented Sep 24, 2014 at 10:48
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