The div-curl lemma for sequences whose divergence and curl are compact in W^{-1,1}

Details The div-curl lemma for sequences whose divergence and curl are compact in W^{-1,1} The div-curl lemma for sequences whose divergence and curl are compact in W^{-1,1}

2009-07-02

arxiv.org/abs/0907.0397v1

Mathematics

Analysis of PDEs

Scientific paper

It is shown that $u_k \cdot v_k$ converges weakly to $u\cdot v$ if $u_k\weakto u$ weakly in $L^p$ and $v_k\weakly v$ weakly in $L^q$ with $p, q\in (1,\infty)$, $1/p+1/q=1$, under the additional assumptions that the sequences $\Div u_k$ and $\curl v_k$ are compact in the dual space of $W^{1,\infty}_0$ and that $u_k\cdot v_k$ is equi-integrable. The main point is that we only require equi-integrability of the scalar product $u_k\cdot v_k$ and not of the individual sequences.

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