Lr-Helmholtz-Weyl decomposition for three dimensional exterior domains

In this article the Helmholtz-Weyl decomposition in three dimensional exterior domains is established within the L-r-setting for 1 < r < infinity. In fact, given an L-r-vector field u, there exist h is an element of X-har(r)(Omega), w is an element of (H) over dot(1,r)(Omega)(3) with div w = 0 and p is an element of (H) over dot(1,r)(Omega) such that umay be decomposed uniquely as u = h + rot w + del p. If for the given L-r-vector field u, its harmonic part h is chosen from V-har(r)(Omega), then a decomposition similar to the above one is established, too. However, its uniqueness holds in this case only for the case 1 < r < 3. The proof given relies on an L-r-variational inequality allowing to construct w is an element of (H) over dot(1,r)(Omega)(3) and p is an element of (H) over dot(1,r)(Omega) for given u is an element of L-r(Omega)(3) as weak solutions to certain elliptic boundary value problems. (C) 2021 Elsevier Inc. All rights reserved.

Automated summary

In this article, the Helmholtz-Weyl decomposition in three dimensional exterior domains is established for 1 < r < infinity within the L-r-setting. It has been proven that for a given L-r-vector field u, the decomposition is unique only when 1 < r < 3, with the proof relying on an L-r-variational inequality.