### Abstract

We show that every L^{r}-vector field on Ω can be uniquely decomposed into two spaces with scalar and vector potentials, and the harmonic vector space via operators rot and div, where Ω is a bounded domain in ℝ^{3} with the smooth boundary ∂Ω. Our decomposition consists of two kinds of boundary conditions such as u-v _{∂Ω} = 0 and u × _{∂Ω} = 0, where v denotes the unit outward normal to ∂Ω. Our results may be regarded as an extension of the well-known de Rham-Hodge-Kodaira decomposition of C∞-forms on compact Riemannian manifolds into L^{r}-vector fields on Ω. As an application, the generalized Biot-Savart law for the incompressible fluids in Ω is obtained. Furthermore, various bounds of u in L^{r} for higher derivatives are given by means of rot u and div u.

Original language | English |
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Pages (from-to) | 1853-1920 |

Number of pages | 68 |

Journal | Indiana University Mathematics Journal |

Volume | 58 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2009 |

Externally published | Yes |

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### Keywords

- Betti number
- Div-curl lemma
- Harmonic vector fields
- L-vector fields

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

^{r}-variational inequality for vector fields and the helmholtz-weyl decomposition in bounded domains.

*Indiana University Mathematics Journal*,

*58*(4), 1853-1920. https://doi.org/10.1512/iumj.2009.58.3605