Global Div-Curl lemma on bounded domains in R3
Global Div-Curl lemma on bounded domains in R3
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DOI：
10.1016/j.jfa.2009.01.010
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发表时间：
2009-06
影响因子：
1.7
通讯作者：
H. Kozono;T. Yanagisawa
中科院分区：
文献类型：
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作者：
H. Kozono;T. Yanagisawa
本文考虑光滑边界为<$Ω的有界区域Ω <$R3中向量场的Div-Curl引理的一个整体形式。假设[公式：见正文]和[公式：见正文]分别在Lr（Ω）和[公式：见正文]中弱收敛于u和v，其中1<r<∞且1/r+1/r′=1。还假设[公式：见正文]在Lq（Ω）中有界，其中q>max{1，3r/（3+r）};[公式：见正文]在Ls（Ω）中有界，其中s>max{1，3r′/（3+r′）}。如果[公式：[见正文]在W1−1/q，q（Ω）中有界，或[公式：见正文]在W1−1/s，s（Ω）中有界（ν：向外垂直于Ω的单位），则成立<$Ωuj <$vjdx→ <$Ωu <$vdx。特别地，如果满足uj <$v =0或vj× v =0，则我们有<$Ωuj <$vjdx→ <$Ωu <$vdx。作为一个直接的后果，我们证明了著名的Div-Curl引理的任何开集在R3。Lr（Ω）的Helmholtz-Weyl分解在证明中起着至关重要的作用。
We consider a global version of the Div-Curl lemma for vector fields in a bounded domain Ω⊂R3with the smooth boundary ∂Ω. Suppose that [Formula: see text] and [Formula: see text] converge to u and v weakly in Lr(Ω) and [Formula: see text] , respectively, where 1<r<∞ with 1/r+1/r′=1. Assume also that [Formula: see text] is bounded in Lq(Ω) for q>max{1,3r/(3+r)} and that [Formula: see text] is bounded in Ls(Ω) for s>max{1,3r′/(3+r′)}, respectively. If either [Formula: see text] is bounded in W1−1/q,q(∂Ω), or [Formula: see text] is bounded in W1−1/s,s(∂Ω) (ν: unit outward normal to ∂Ω), then it holds that ∫Ωuj⋅vjdx→∫Ωu⋅vdx. In particular, if either uj⋅ν=0 or vj×ν=0 on ∂Ω for all j=1,2,… is satisfied, then we have that ∫Ωuj⋅vjdx→∫Ωu⋅vdx. 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.