
 
On a Vector Version of a Fundamental Lemma of J. L. Lions 
 
Citation： 
Philippe G. CIARLET,Maria MALIN,Cristinel MARDARE.On a Vector Version of a Fundamental Lemma of J. L. Lions[J].Chinese Annals of Mathematics B,2018,39(1):33~46 
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Net amount： 666 
Authors： 
Philippe G. CIARLET; Maria MALIN;Cristinel MARDARE 
Foundation： 
This work was supported by a grant from the Research
Grants Council of the Hong Kong Special Administrative Region, China
(No.9041738CityU 100612). 


Abstract： 
Let $\Omega$ be a bounded and connected open subset of
$\mathbb{R}^N$ with a Lipschitzcontinuous boundary, the set
$\Omega$ being locally on the same side of $\partial \Omega$. A
vector version of a fundamental lemma of J. L. Lions, due to C.Amrouche, the first author, L.\ Gratie and S. Kesavan, asserts that
any vector field $\bm{v} = (v_i) \in (\mathcal{D}'(\Omega))^N$, such
that all the components $\frac12 (\partial_j v_i + \partial_i v_j )
, \, 1 \leq i, \, j \leq N$, of its symmetrized gradient matrix
field are in the space $H^{1} (\Omega)$, is in effect in the space
$(L^2(\Omega))^N$. The objective of this paper is to show that this
vector version of J. L. Lions lemma is equivalent to a certain
number of other properties of interest by themselves. These include
in particular a vector version of a wellknown inequality due to J.Ne\v{c}as, weak versions of the classical Donati and SaintVenant
compatibility conditions for a matrix field to be the symmetrized
gradient matrix field of a vector field, or a natural vector version
of a fundamental surjectivity property of the divergence operator. 
Keywords： 
J. L. Lions lemma, Nev{c}as inequality, Donati compatibilityconditions, SaintVenant compatibility conditions 
Classification： 
46F05, 47A05, 74B05 

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