## Abstract

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.

Original language | English |
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Article number | 109144 |

Journal | Journal of Functional Analysis |

Volume | 281 |

Issue number | 8 |

DOIs | |

Publication status | Published - 2021 Oct 15 |

## Keywords

- Exterior domains
- Harmonic vector fields
- Helmholtz-Weyl decomposition
- Vector and scalar potentials

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