Lr-Helmholtz-Weyl decomposition for three dimensional exterior domains

Abstract

In this article the Helmholtz-Weyl decomposition in three dimensional exterior domains is established within the Lr-setting for 1<r<∞. In fact, given an Lr-vector field u, there exist h∈Xharr(Ω), w∈H˙1,r(Ω)3 with divw=0 and p∈H˙1,r(Ω) such that u may be decomposed uniquely asu=h+rotw+∇p. If for the given Lr-vector field u, its harmonic part h is chosen from Vharr(Ω), 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 Lr-variational inequality allowing to construct w∈H˙1,r(Ω)3 and p∈H˙1,r(Ω) for given u∈Lr(Ω)3 as weak solutions to certain elliptic boundary value problems.