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| On a Vector Version of a Fundamental Lemma of J. L. Lions |
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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 |
| Page view： 2310
Net amount： 1863 |
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.9041738-CityU 100612). |
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| Abstract： |
Let $\Omega$ be a bounded and connected open subset of
$\mathbb{R}^N$ with a Lipschitz-continuous 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 well-known inequality due to J.Ne\v{c}as, weak versions of the classical Donati and Saint-Venant
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, Saint-Venant compatibility conditions |
Classification： |
46F05, 47A05, 74B05 |
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