L^r-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<∞. In fact, given an L^r-vector field u, there exist h∈X_har^r(Ω), w∈H˙^1,r(Ω)³ with divw=0 and p∈H˙^1,r(Ω) such that u may be decomposed uniquely as u=h+rotw+∇p. If for the given L^r-vector field u, its harmonic part h is chosen from V_har^r(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 L^r-variational inequality allowing to construct w∈H˙^1,r(Ω)³ and p∈H˙^1,r(Ω) for given u∈L^r(Ω)³ as weak solutions to certain elliptic boundary value problems.