TY - JOUR
T1 - On a Vector Version of a Fundamental Lemma of J. L. Lions
AU - CIARLET, Philippe G.
AU - MALIN, Maria
AU - MARDARE, Cristinel
PY - 2018/1
Y1 - 2018/1
N2 - Let Ω be a bounded and connected open subset of ℝN with a Lipschitz-continuous boundary, the set Ω being locally on the same side of ∂Ω. 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 v = (vi) ∈ (D'(Ω))N, such that all the components 1/2(∂jvi + ∂ivj), 1 ≤ i, j ≤ N, of its symmetrized gradient matrix field are in the space H−1(Ω), is in effect in the space (L2(Ω))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. Nečas, 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.
AB - Let Ω be a bounded and connected open subset of ℝN with a Lipschitz-continuous boundary, the set Ω being locally on the same side of ∂Ω. 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 v = (vi) ∈ (D'(Ω))N, such that all the components 1/2(∂jvi + ∂ivj), 1 ≤ i, j ≤ N, of its symmetrized gradient matrix field are in the space H−1(Ω), is in effect in the space (L2(Ω))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. Nečas, 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.
KW - Donati compatibility conditions
KW - J. L. Lions lemma
KW - Nečas inequality
KW - Saint-Venant compatibility conditions
UR - http://www.scopus.com/inward/record.url?scp=85040104912&partnerID=8YFLogxK
UR - https://www.scopus.com/record/pubmetrics.uri?eid=2-s2.0-85040104912&origin=recordpage
U2 - 10.1007/s11401-018-1049-5
DO - 10.1007/s11401-018-1049-5
M3 - 21_Publication in refereed journal
VL - 39
SP - 33
EP - 46
JO - Chinese Annals of Mathematics. Series B
JF - Chinese Annals of Mathematics. Series B
SN - 0252-9599
IS - 1
ER -