### Abstract

We consider a global version of the Div-Curl lemma for vector fields in a bounded domain Ω ⊂ R^{3} with the smooth boundary ∂Ω. Suppose that {u_{j}}_{j = 1}
^{∞} and {v_{j}}_{j = 1}
^{∞} converge to u and v weakly in L^{r} (Ω) and L^{r′} (Ω), respectively, where 1 < r < ∞ with 1 / r + 1 / r^{′} = 1. Assume also that {div u_{j}}_{j = 1}
^{∞} is bounded in L^{q} (Ω) for q > max {1, 3 r / (3 + r)} and that {rot v_{j}}_{j = 1}
^{∞} is bounded in L^{s} (Ω) for s > max {1, 3 r^{′} / (3 + r^{′})}, respectively. If either {u_{j} ṡ ν |_{∂ Ω}}_{j = 1}
^{∞} is bounded in W^{1 - 1 / q, q} (∂ Ω), or {v_{j} × ν |_{∂ Ω}}_{j = 1}
^{∞} is bounded in W^{1 - 1 / s, s} (∂ Ω) (ν: unit outward normal to ∂Ω), then it holds that ∫_{Ω} u_{j} ṡ v_{j} d x → ∫_{Ω} u ṡ v d x. In particular, if either u_{j} ṡ ν = 0 or v_{j} × ν = 0 on ∂Ω for all j = 1, 2, ... is satisfied, then we have that ∫_{Ω} u_{j} ṡ v_{j} d x → ∫_{Ω} u ṡ v d x. As an immediate consequence, we prove the well-known Div-Curl lemma for any open set in R^{3}. The Helmholtz-Weyl decomposition for L^{r} (Ω) plays an essential role for the proof.

Original language | English |
---|---|

Pages (from-to) | 3847-3859 |

Number of pages | 13 |

Journal | Journal of Functional Analysis |

Volume | 256 |

Issue number | 11 |

DOIs | |

Publication status | Published - 2009 Jun 1 |

Externally published | Yes |

### Fingerprint

### Keywords

- Compact imbedding
- Div-Curl lemma
- Elliptic system of boundary value problem
- Helmholtz-Weyl decomposition

### ASJC Scopus subject areas

- Analysis

### Cite this

^{3}

*Journal of Functional Analysis*,

*256*(11), 3847-3859. https://doi.org/10.1016/j.jfa.2009.01.010

**Global Div-Curl lemma on bounded domains in R ^{3}
.** / Kozono, Hideo; Yanagisawa, Taku.

Research output: Contribution to journal › Article

^{3}',

*Journal of Functional Analysis*, vol. 256, no. 11, pp. 3847-3859. https://doi.org/10.1016/j.jfa.2009.01.010

^{3}Journal of Functional Analysis. 2009 Jun 1;256(11):3847-3859. https://doi.org/10.1016/j.jfa.2009.01.010

}

TY - JOUR

T1 - Global Div-Curl lemma on bounded domains in R3

AU - Kozono, Hideo

AU - Yanagisawa, Taku

PY - 2009/6/1

Y1 - 2009/6/1

N2 - We consider a global version of the Div-Curl lemma for vector fields in a bounded domain Ω ⊂ R3 with the smooth boundary ∂Ω. Suppose that {uj}j = 1 ∞ and {vj}j = 1 ∞ converge to u and v weakly in Lr (Ω) and Lr′ (Ω), respectively, where 1 < r < ∞ with 1 / r + 1 / r′ = 1. Assume also that {div uj}j = 1 ∞ is bounded in Lq (Ω) for q > max {1, 3 r / (3 + r)} and that {rot vj}j = 1 ∞ is bounded in Ls (Ω) for s > max {1, 3 r′ / (3 + r′)}, respectively. If either {uj ṡ ν |∂ Ω}j = 1 ∞ is bounded in W1 - 1 / q, q (∂ Ω), or {vj × ν |∂ Ω}j = 1 ∞ is bounded in W1 - 1 / s, s (∂ Ω) (ν: unit outward normal to ∂Ω), then it holds that ∫Ω uj ṡ vj d x → ∫Ω u ṡ v d x. In particular, if either uj ṡ ν = 0 or vj × ν = 0 on ∂Ω for all j = 1, 2, ... is satisfied, then we have that ∫Ω uj ṡ vj d x → ∫Ω u ṡ v d x. As an immediate consequence, we prove the well-known Div-Curl lemma for any open set in R3. The Helmholtz-Weyl decomposition for Lr (Ω) plays an essential role for the proof.

AB - We consider a global version of the Div-Curl lemma for vector fields in a bounded domain Ω ⊂ R3 with the smooth boundary ∂Ω. Suppose that {uj}j = 1 ∞ and {vj}j = 1 ∞ converge to u and v weakly in Lr (Ω) and Lr′ (Ω), respectively, where 1 < r < ∞ with 1 / r + 1 / r′ = 1. Assume also that {div uj}j = 1 ∞ is bounded in Lq (Ω) for q > max {1, 3 r / (3 + r)} and that {rot vj}j = 1 ∞ is bounded in Ls (Ω) for s > max {1, 3 r′ / (3 + r′)}, respectively. If either {uj ṡ ν |∂ Ω}j = 1 ∞ is bounded in W1 - 1 / q, q (∂ Ω), or {vj × ν |∂ Ω}j = 1 ∞ is bounded in W1 - 1 / s, s (∂ Ω) (ν: unit outward normal to ∂Ω), then it holds that ∫Ω uj ṡ vj d x → ∫Ω u ṡ v d x. In particular, if either uj ṡ ν = 0 or vj × ν = 0 on ∂Ω for all j = 1, 2, ... is satisfied, then we have that ∫Ω uj ṡ vj d x → ∫Ω u ṡ v d x. As an immediate consequence, we prove the well-known Div-Curl lemma for any open set in R3. The Helmholtz-Weyl decomposition for Lr (Ω) plays an essential role for the proof.

KW - Compact imbedding

KW - Div-Curl lemma

KW - Elliptic system of boundary value problem

KW - Helmholtz-Weyl decomposition

UR - http://www.scopus.com/inward/record.url?scp=64549108901&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=64549108901&partnerID=8YFLogxK

U2 - 10.1016/j.jfa.2009.01.010

DO - 10.1016/j.jfa.2009.01.010

M3 - Article

AN - SCOPUS:64549108901

VL - 256

SP - 3847

EP - 3859

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 11

ER -