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, ) 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 $$L^{q'}(\varOmega )$$ . [ABSTRACT FROM AUTHOR]