Let Ω be a bounded and connected open subset of with a Lipschitz-continuous boundary ∂Ω, the set Ω being locally on one side of ∂Ω. It is shown in this Note that a fundamental characterization of the space due to Jacques-Louis Lions is in effect equivalent to a variety of other properties. One of the keys for establishing these equivalences is a specific “approximation lemma”, itself one of these equivalent properties.
Soit Ω un ouvert borné et connesce de de frontière ∂Ω lipschitzienne, l'ensemble Ω étant localement du même côté de ∂Ω. On montre dans cette Note qu'une caractérisation fondamentale de l'espace due à Jacques-Louis Lions est en fait équivalente à un certain nombre d'autres propriétés. L'une des clés pour établir ces équivalences est un « lemme d'approximation » spécifique, qui constitue lui-même l'une de ces propriétés équivalentes.
Published online:
@article{CRMATH_2014__352_9_691_0, author = {Amrouche, Ch\'erif and Ciarlet, Philippe G. and Mardare, Cristinel}, title = {Remarks on a lemma by {Jacques-Louis} {Lions}}, journal = {Comptes Rendus. Math\'ematique}, pages = {691--695}, publisher = {Elsevier}, volume = {352}, number = {9}, year = {2014}, doi = {10.1016/j.crma.2014.08.003}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.crma.2014.08.003/} }
TY - JOUR AU - Amrouche, Chérif AU - Ciarlet, Philippe G. AU - Mardare, Cristinel TI - Remarks on a lemma by Jacques-Louis Lions JO - Comptes Rendus. Mathématique PY - 2014 SP - 691 EP - 695 VL - 352 IS - 9 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2014.08.003/ DO - 10.1016/j.crma.2014.08.003 LA - en ID - CRMATH_2014__352_9_691_0 ER -
%0 Journal Article %A Amrouche, Chérif %A Ciarlet, Philippe G. %A Mardare, Cristinel %T Remarks on a lemma by Jacques-Louis Lions %J Comptes Rendus. Mathématique %D 2014 %P 691-695 %V 352 %N 9 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2014.08.003/ %R 10.1016/j.crma.2014.08.003 %G en %F CRMATH_2014__352_9_691_0
Amrouche, Chérif; Ciarlet, Philippe G.; Mardare, Cristinel. Remarks on a lemma by Jacques-Louis Lions. Comptes Rendus. Mathématique, Volume 352 (2014) no. 9, pp. 691-695. doi : 10.1016/j.crma.2014.08.003. http://www.numdam.org/articles/10.1016/j.crma.2014.08.003/
[1] C. Amrouche, P.G. Ciarlet, C. Mardare, On a lemma due to Jacques-Louis Lions and its relation to other fundamental results, in preparation.
[2] Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension, Czechoslov. Math. J., Volume 44 (1994), pp. 109-140
[3] Solution of the first boundary value problem for the equation of continuity of an incompressible medium, Sov. Math. Dokl., Volume 20 (1979), pp. 1094-1098
[4] On the equations and with zero boundary conditions, Hokkaido Math. J., Volume 19 (1990), pp. 67-87
[5] Linear and Nonlinear Functional Analysis with Applications, SIAM, Philadelphia, 2013
[6] Another approach to linearized elasticity and a new proof of Korn's inequality, Math. Models Methods Appl. Sci., Volume 15 (2005), pp. 259-271
[7] Les équations en mécanique et en physique, Dunod, Paris, 1972
[8] Functional spaces for Norton–Hoff materials, Math. Methods Appl. Sci., Volume 8 (1986), pp. 206-222
[9] Finite Element Methods for Navier–Stokes Equations, Springer, Berlin, 1986
[10] On Poincaré's and J.-L. Lions' lemmas, C. R. Acad. Sci. Paris, Ser. I, Volume 340 (2005), pp. 27-30
[11] Équations aux dérivées partielles, Presses de l'Université de Montréal, 1965
[12] Topics in Nonlinear Analysis, Publications mathématiques d'Orsay No. 78.13, Université de Paris-Sud, Département de mathématique, Orsay, 1978
Cited by Sources: