Let , , be a smooth bounded domain. It is shown that: (a) if and , then the generalized Lebesgue space is smooth; (b) if and , for all , then the generalized Sobolev space is smooth. In both situations, the formulae giving the Gâteaux derivative of the norm, corresponding to each of the above spaces, are given; (c) if and , for all , then is uniformly convex and smooth.
Soit , , un domain borné et régulier. On demontre que : (a) si et , alors l'espace de Lebesgue généralisé est lisse ; (b) si et , pour tout , alors l'espace de Sobolev généralisé est lisse. Dans les deux cas, les formules de la dérivée au sens de Gâteaux de chaque norme des espaces ci-dessus sont données ; (c) si et , pour tout , alors est uniformément convexe et lisse.
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@article{CRMATH_2009__347_15-16_885_0, author = {Dinca, George and Matei, Pavel}, title = {Geometry of {Sobolev} spaces with variable exponent: smoothness and uniform convexity}, journal = {Comptes Rendus. Math\'ematique}, pages = {885--889}, publisher = {Elsevier}, volume = {347}, number = {15-16}, year = {2009}, doi = {10.1016/j.crma.2009.04.028}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.crma.2009.04.028/} }
TY - JOUR AU - Dinca, George AU - Matei, Pavel TI - Geometry of Sobolev spaces with variable exponent: smoothness and uniform convexity JO - Comptes Rendus. Mathématique PY - 2009 SP - 885 EP - 889 VL - 347 IS - 15-16 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2009.04.028/ DO - 10.1016/j.crma.2009.04.028 LA - en ID - CRMATH_2009__347_15-16_885_0 ER -
%0 Journal Article %A Dinca, George %A Matei, Pavel %T Geometry of Sobolev spaces with variable exponent: smoothness and uniform convexity %J Comptes Rendus. Mathématique %D 2009 %P 885-889 %V 347 %N 15-16 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2009.04.028/ %R 10.1016/j.crma.2009.04.028 %G en %F CRMATH_2009__347_15-16_885_0
Dinca, George; Matei, Pavel. Geometry of Sobolev spaces with variable exponent: smoothness and uniform convexity. Comptes Rendus. Mathématique, Volume 347 (2009) no. 15-16, pp. 885-889. doi : 10.1016/j.crma.2009.04.028. http://www.numdam.org/articles/10.1016/j.crma.2009.04.028/
[1] Geometry of Banach Spaces – Selected Topics, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1975
[2] G. Dinca, P. Matei, Geometry of Sobolev spaces with variable exponent and a generalization of the p-Laplacian, Analysis and Applications, in press
[3] Sobolev embeddings with variable exponent, I, Studia Mathematica, Volume 143 (2000), pp. 267-292
[4] Sobolev embeddings with variable exponent, II, Math. Nachr., Volume 246–247 (2002), pp. 53-67
[5] On the spaces and , J. Math. Anal. Appl., Volume 263 (2001), pp. 424-446
[6] Eine Verallgemeinerung der Sobolewschen Räume in unbeschränkten Gebieten, Math. Nachr., Volume 32 (1966), pp. 115-130
[7] On spaces and , Czechoslovak Math. J., Volume 41 (1991), pp. 592-618
[8] Convex Functions and Orlicz Spaces, Gröningen, Noordhoff, 1961
[9] Monotone potential operatoren in theorie und anwendung, Springer Verlag der Wissenschaften, Berlin, 1976
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