## 抄録

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.

本文言語 | English |
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ページ（範囲） | 1853-1920 |

ページ数 | 68 |

ジャーナル | Indiana University Mathematics Journal |

巻 | 58 |

号 | 4 |

DOI | |

出版ステータス | Published - 2009 |

外部発表 | はい |

## ASJC Scopus subject areas

- 数学 (全般)

## フィンガープリント

「L^{r}-variational inequality for vector fields and the helmholtz-weyl decomposition in bounded domains」の研究トピックを掘り下げます。これらがまとまってユニークなフィンガープリントを構成します。