On se propose d'établir quelques propriétés des petits espaces de Lebesgue introduits par Fiorenza [7], notamment la convergence monotone de Lévi et des propriétés d'équivalence de normes. En combinant ces propriétés avec les inégalités de Poincaré–Sobolev pour le réarrangement relatif [11], nous donnons quelques estimations précises concernant les espaces de Sobolev associés à ces espaces et les régularités des solutions d'équations quasilinéaires lorsque les données sont dans ces espaces.
We prove some new properties of the small Lebesgue spaces introduced by Fiorenza [7]. Combining these properties with the Poincaré–Sobolev inequalities for the relative rearrangement (see [11]), we derive some new and precises estimates either for small Lebesgue–Sobolev spaces or for quasilinear equations with data in the small Lebesgue spaces.
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Alberto Fiorenza 1 ; Jean-Michel Rakotoson 2
@article{CRMATH_2002__334_1_23_0, author = {Alberto Fiorenza and Jean-Michel Rakotoson}, title = {Petits espaces de {Lebesgue} et quelques applications}, journal = {Comptes Rendus. Math\'ematique}, pages = {23--26}, publisher = {Elsevier}, volume = {334}, number = {1}, year = {2002}, doi = {10.1016/S1631-073X(02)02199-4}, language = {fr}, }
Alberto Fiorenza; Jean-Michel Rakotoson. Petits espaces de Lebesgue et quelques applications. Comptes Rendus. Mathématique, Volume 334 (2002) no. 1, pp. 23-26. doi : 10.1016/S1631-073X(02)02199-4. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02199-4/
[1] Sulle migliori costanti di maggiorazione per una classe di equazioni ellitiche degeneri, Ricerche Mat., Volume 27 (1978), pp. 413-428
[2] Sobolev imbedding into BMO, VMO, L∞, Ark. Mat., Volume 36 (1998), pp. 317-340
[3] Symmetrization techniques on unbounded domains: Application to a chemotaxis system on , J. Differential Equations, Volume 145 (1998) no. 1, pp. 156-183
[4] On the differentiability of functions in , Proc. Amer. Math. Soc., Volume 91 (1984) no. 2, pp. 326-328
[5] L∞ estimates for nonlinear elliptic problems with p-growth in the gradient, J. Inequal. Appl., Volume 3 (1999), pp. 109-125
[6] Some relations between pseudo-rearrangement and relative rearrangement, Nonlinear Anal., Volume 41 (2000) no. 7–8, pp. 855-869
[7] Duality and reflexivity in grand Lebesgue spaces, Collect. Math., Volume 51 (2000) no. 2, pp. 131-148
[8] Regularity and comparison results in Grand Sobolev spaces for parabolic equations with measure data, Appl. Math. Lett., Volume 14 (2001), pp. 979-981
[9] On the integrability of the Jacobian under minimal hypotheses, Arch. Rational Mech. Anal., Volume 119 (1992), pp. 129-143
[10] Directional derivate of the increasing rearrangement mapping and application to a queer differential equation in plasma physics, Duke Math. J., Volume 48 (1981), pp. 475-495
[11] Rakotoson J.M., General pointwise relations for the relative rearrangement and applications (to appear)
[12] Rakotoson J.M., Some new applications of the pointwise relations for the relative rearrangement, Differential Integral Equations (to appear)
[13] A co-area formula with applications to monotone rearrangement and to regularity, Arch. Rational Mech. Anal., Volume 109 (1990) no. 3, pp. 213-238
[14] Editor's note, On the differentiability of functions in , Ann. Math., Volume 113 (1981) no. 2, pp. 381-385
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