Global Div-Curl lemma on bounded domains in R3

Hideo Kozono, Taku Yanagisawa

Research output: Contribution to journalArticle

6 Citations (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)3847-3859
Number of pages13
JournalJournal of Functional Analysis
Volume256
Issue number11
DOIs
Publication statusPublished - 2009 Jun 1
Externally publishedYes

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Keywords

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

ASJC Scopus subject areas

  • Analysis

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