Necessary and sufficient conditions for the existence of Helmholtz decompositions in general domains

Consider a general domain \(\varOmega \subseteq {\mathbb {R}}^n, n\ge 2\) , and let \(1 < q <\infty \) . Our first result is based on the estimate for the gradient \(\nabla p \in G^q(\varOmega )\) in the form \(\Vert \nabla p\Vert _q \le C \,\sup |\langle \nabla p,\nabla v\rangle _{\varOmega }|/\Vert \nabla v\Vert _{q'}\) , \(\nabla v \in G^{q'}(\varOmega ), q' = \frac{q}{q-1}\) , with some constant \(C=C(\varOmega ,q)>0\) . This estimate was introduced by Simader and Sohr (Mathematical Problems Relating to the Navier–Stokes Equations. Series on Advances in Mathematics for Applied Sciences, vol. 11, pp. 1–35. World Scientific, Singapore, 1992) for smooth bounded and exterior domains. We show for general domains that the validity of this gradient estimate in \(G^q(\varOmega )\) and in \(G^{q'}(\varOmega )\) is necessary and sufficient for the validity of the Helmholtz decomposition in \(L^q(\varOmega )\) and in \(L^{q'}(\varOmega )\) . A new aspect concerns the estimate for divergence free functions \(f_0 \in L^q_{\sigma }(\varOmega )\) in the form \(\Vert f_0\Vert _q \le C \sup |\langle f_0,w\rangle _{\varOmega }|/ \Vert w\Vert _{q'}, w\in L^{q'}_{\sigma }(\varOmega )\) , for the second part of the Helmholtz decomposition. We show again for general domains that the validity of this estimate in \(L^q_{\sigma }(\varOmega )\) and in \(L^{q'}_{\sigma }(\varOmega )\) is necessary and sufficient for the validity of the Helmholtz decomposition in \(L^q(\varOmega )\) and in \(L^{q'}(\varOmega )\) .