TY - JOUR
T1 - Global Div-Curl lemma on bounded domains in R3
AU - Kozono, Hideo
AU - Yanagisawa, Taku
N1 - Copyright:
Copyright 2009 Elsevier B.V., All rights reserved.
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
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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 -