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?

  • 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 Wong Sep 17 '14 at 13:09
  • $\begingroup$ 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? $\endgroup$ – Elwood Sep 17 '14 at 19:19
  • $\begingroup$ For smooth boundaries, possibly yes, in some advanced calculus textbooks. For Lipschitz boundaries I am slightly doubtful. $\endgroup$ – Willie Wong Sep 18 '14 at 7:27
  • 1
    $\begingroup$ On $\mathbb R^n$ this is also called the Helmholtz decomposition - search also under this name. $\endgroup$ – Peter Michor Sep 22 '14 at 12:37
  • 1
    $\begingroup$ It is written in Girault-Raviart's textbook on Navier Stokes for $d=2,3$. $\endgroup$ – username Sep 24 '14 at 10:48

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.