The Helmholtz decomposition of vector fields for two-dimensional exterior Lipschitz domains

Abstract

Let $ \Omega $ be an exterior Lipschitz domain in $ \mathbb{R}^2 $. It is proved that the Helmholtz decomposition of the vector fields in $ L_p (\Omega; \mathbb{R}^2) $ exists if $ p $ satisfies $ \lvert1/ p - 1/ 2 \rvert < 1/ 4+ \varepsilon $ with some constant $ \varepsilon = \varepsilon (\Omega) \in (0, 1/ 4] $, where it is allowed to take $ \varepsilon = 1/ 4 $ if $ \partial \Omega \in C^1 $.

Keywords

• exterior lipschitz domains
• helmholtz decomposition