Comptes Rendus
Partial Differential Equations/Functional Analysis
Geometry of Sobolev spaces with variable exponent: smoothness and uniform convexity
[Géométrie des espaces de Sobolev à coefficients variables : lissitude et convexité uniforme]
Comptes Rendus. Mathématique, Volume 347 (2009) no. 15-16, pp. 885-889.

Soit ΩRN, N2, un domain borné et régulier. On demontre que : (a) si pL(Ω) et essinfxΩp(x)>1, alors l'espace de Lebesgue généralisé (Lp()(Ω),p()) est lisse ; (b) si pC(Ω¯) et p(x)>1, pour tout xΩ¯, alors l'espace de Sobolev généralisé (W01,p()(Ω),1,p()) 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 pC(Ω¯) et p(x)2, pour tout xΩ¯, alors (W01,p()(Ω),1,p()) est uniformément convexe et lisse.

Let ΩRN, N2, be a smooth bounded domain. It is shown that: (a) if pL(Ω) and essinfxΩp(x)>1, then the generalized Lebesgue space (Lp()(Ω),p()) is smooth; (b) if pC(Ω¯) and p(x)>1, for all xΩ¯, then the generalized Sobolev space (W01,p()(Ω),1,p()) 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 pC(Ω¯) and p(x)2, for all xΩ¯, then (W01,p()(Ω),1,p()) is uniformly convex and smooth.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2009.04.028

George Dinca 1 ; Pavel Matei 2

1 Faculty of Mathematics and Computer Science, 14, Academiei St, 010014 Bucharest, Romania
2 Department of Mathematics, Technical University of Civil Engineering, 124, Lacul Tei Blvd., 020396 Bucharest, Romania
@article{CRMATH_2009__347_15-16_885_0,
     author = {George Dinca and Pavel Matei},
     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},
}
TY  - JOUR
AU  - George Dinca
AU  - Pavel Matei
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
DO  - 10.1016/j.crma.2009.04.028
LA  - en
ID  - CRMATH_2009__347_15-16_885_0
ER  - 
%0 Journal Article
%A George Dinca
%A Pavel Matei
%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
%R 10.1016/j.crma.2009.04.028
%G en
%F CRMATH_2009__347_15-16_885_0
George Dinca; Pavel Matei. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.04.028/

[1] J. Diestel 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] D.E. Edmunds; J. Rákosník Sobolev embeddings with variable exponent, I, Studia Mathematica, Volume 143 (2000), pp. 267-292

[4] D.E. Edmunds; J. Rákosník Sobolev embeddings with variable exponent, II, Math. Nachr., Volume 246–247 (2002), pp. 53-67

[5] X.L. Fan; D. Zhao On the spaces Lp(x)(Ω) and Wm,p(x)(Ω), J. Math. Anal. Appl., Volume 263 (2001), pp. 424-446

[6] K. Gröger Eine Verallgemeinerung der Sobolewschen Räume in unbeschränkten Gebieten, Math. Nachr., Volume 32 (1966), pp. 115-130

[7] O. Kováčik; J. Rákosník On spaces Lp(x) and Wk,p(x), Czechoslovak Math. J., Volume 41 (1991), pp. 592-618

[8] M.A. Krasnosel'skij; Ya.B. Rutickij Convex Functions and Orlicz Spaces, Gröningen, Noordhoff, 1961

[9] A. Langenbach Monotone potential operatoren in theorie und anwendung, Springer Verlag der Wissenschaften, Berlin, 1976

  • Alexander J. Zaslavski Special Issue: Fixed-Point Theory and Its Applications, Dedicated to the Memory of Professor William Arthur Kirk, Symmetry, Volume 16 (2024) no. 11, p. 1408 | DOI:10.3390/sym16111408
  • Brigida Bonino; Claudio Estatico; Marta Lazzaretti Dual descent regularization algorithms in variable exponent Lebesgue spaces for imaging, Numerical Algorithms, Volume 92 (2023) no. 1, p. 149 | DOI:10.1007/s11075-022-01458-w
  • Mostafa Bachar; Mohamed A. Khamsi; Osvaldo Méndez Uniform Convexity in Variable Exponent Sobolev Spaces, Symmetry, Volume 15 (2023) no. 11, p. 1988 | DOI:10.3390/sym15111988
  • Marta Lazzaretti; Luca Calatroni; Claudio Estatico Modular-Proximal Gradient Algorithms in Variable Exponent Lebesgue Spaces, SIAM Journal on Scientific Computing, Volume 44 (2022) no. 6, p. A3463 | DOI:10.1137/21m1464336
  • Alessandro Fedeli; Claudio Estatico; Andrea Randazzo; Matteo Pastorino, 2020 XXXIIIrd General Assembly and Scientific Symposium of the International Union of Radio Science (2020), p. 1 | DOI:10.23919/ursigass49373.2020.9232449
  • Claudio Estatico; Alessandro Fedeli; Matteo Pastorino; Andrea Randazzo Microwave Imaging by Means of Lebesgue-Space Inversion: An Overview, Electronics, Volume 8 (2019) no. 9, p. 945 | DOI:10.3390/electronics8090945
  • Claudio Estatico; Alessandro Fedeli; Matteo Pastorino; Andrea Randazzo Quantitative Microwave Imaging Method in Lebesgue Spaces With Nonconstant Exponents, IEEE Transactions on Antennas and Propagation, Volume 66 (2018) no. 12, p. 7282 | DOI:10.1109/tap.2018.2869201
  • Jan Lang; Osvaldo Méndez Γ-convergence of the energy functionals for the variable exponent p(·)-Laplacian and stability of the minimizers with respect to integrability, Journal d'Analyse Mathématique, Volume 134 (2018) no. 2, p. 575 | DOI:10.1007/s11854-018-0018-y
  • Jan Lang; Osvaldo Méndez Stability of the Norm-Eigenfunctions of the p() p ( · ) -Laplacian, Integral Equations and Operator Theory, Volume 85 (2016) no. 2, p. 245 | DOI:10.1007/s00020-015-2275-9
  • Jan Lang; Osvaldo Méndez Convergence properties of modular eigenfunctions for the -Laplacian, Nonlinear Analysis: Theory, Methods Applications, Volume 104 (2014), p. 156 | DOI:10.1016/j.na.2014.03.008
  • PHILIPPE G. CIARLET; GEORGE DINCA; PAVEL MATEI FRÉCHET DIFFERENTIABILITY OF THE NORM IN A SOBOLEV SPACE WITH A VARIABLE EXPONENT, Analysis and Applications, Volume 11 (2013) no. 04, p. 1350012 | DOI:10.1142/s0219530513500127
  • David V. Cruz-Uribe; Alberto Fiorenza Structure of Variable Lebesgue Spaces, Variable Lebesgue Spaces (2013), p. 13 | DOI:10.1007/978-3-0348-0548-3_2

Cité par 12 documents. Sources : Crossref

Commentaires - Politique