Simmetrizzazione e disuguaglianze di tipo Pòlya-Szegö
Fusco, Nicola
Bollettino dell'Unione Matematica Italiana, Tome 8-A (2005), p. 529-548 / Harvested from Biblioteca Digitale Italiana di Matematica

Si presentano alcuni risultati recenti riguardanti la disuguaglianza di Pòlya- Szegö e la caratterizzazione dei casi in cui essa si riduce ad un'uguaglianza. Particolare attenzione viene rivolta alla simmetrizzazione di Steiner di insiemi di perimetro finito e di funzioni di Sobolev.

We present some recent results concerning the Pòlya-Szegö inequality and the characterization of the equality cases. In particular we will be concerned with the Steiner symmetrization of sets of finite perimeter and Sobolev functions.

Publié le : 2005-10-01
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     author = {Nicola Fusco},
     title = {Simmetrizzazione e disuguaglianze di tipo P\`olya-Szeg\"o},
     journal = {Bollettino dell'Unione Matematica Italiana},
     volume = {8-A},
     year = {2005},
     pages = {529-548},
     zbl = {1182.26050},
     mrnumber = {2182414},
     language = {it},
     url = {http://dml.mathdoc.fr/item/BUMI_2005_8_8B_3_529_0}
}
Fusco, Nicola. Simmetrizzazione e disuguaglianze di tipo Pòlya-Szegö. Bollettino dell'Unione Matematica Italiana, Tome 8-A (2005) pp. 529-548. http://gdmltest.u-ga.fr/item/BUMI_2005_8_8B_3_529_0/

[1] Alvino, A. - Ferone, V. - Lions, P. L. - Trombetti, G., Convex symmetrization and rearrangements, Ann. Inst. H. Poincaré, Anal. Non Linèaire, 14 (1997), 275-293. | MR 1441395 | Zbl 0877.35040

[2] Ambrosio, L. - Fusco, N. - Pallara, D., Functions of bounded variation and free discontinuity problems, Oxford University Press, Oxford, 2000. | MR 1857292 | Zbl 0957.49001

[3] Baernstein Ii, A., A unified approach to symmetrization, in Partial differential equations of ellyptic type, A.Alvino, Fabes, E. & Talenti, G. eds., Symposia Math. 35, Cambridge Univ. Press, 1994. | MR 1297773 | Zbl 0830.35005

[4] Betta, F. - Brock, F. - Mercaldo, A. - Posteraro, M., A weighted isoperimetric inequality and applications to symmetrization, J. Ineq. Appl., 4 (1999), 215-240. | MR 1734159 | Zbl 1029.26018

[5] Brock, F., Weighted Dirichlet-type inequalities for Steiner Symmetrization, Calc. Var., 8 (1999), 15-25. | MR 1666874 | Zbl 0947.35056

[6] Brothers, J. - Ziemer, W., Minimal rearrangements of Sobolev functions, J. Reine. Angew. Math., 384 (1988), 153-179. | MR 929981 | Zbl 0633.46030

[7] Burchard, A., Steiner symmetrization is continuous in $W^{1, p}$, Geom. Funct. Anal., 7 (1997), 823-860. | MR 1475547 | Zbl 0912.46034

[8] Chlebík, M. - Cianchi, A. - Fusco, N., Perimeter inequalities for Steiner symmetrization: cases of equalities, in corso di stampa su Annals of Mathematics.

[9] Cianchi, A., Second order derivatives and rearrangements, Duke Math. J., 105 (2000), 355-385. | MR 1801766 | Zbl 1017.46023

[10] Cianchi, A., Rearrangements of functions in Besov spaces, Math. Nachr., 230 (2001), 19-35. | MR 1854875 | Zbl 1022.46021

[11] Cianchi, A. - Fusco, N., Functions of bounded variation and rearrangements, Arch. Rat. Mech. Anal., 165 (2002), 1-40. | MR 1947097 | Zbl 1028.49035

[12] Cianchi, A. - Fusco, N., Steiner symmetric extremals in Pólya-Szegö type inequalities, preprint del Dip. Mat. e Appl. Univ. Napoli, n. 17 (2003). | MR 2228056 | Zbl 1110.46021

[13] De Giorgi, E., Nuovi teoremi relativi alle misure (r-1)-dimensionali in uno spazio a r dimensioni, Ricerche Mat., 4 (1955), 95-113. | MR 74499 | Zbl 0066.29903

[14] De Giorgi, E., Sulla proprietà isoperimetrica dell’ipersfera, nella classe degli insiemi aventi frontiera orientata di misura finita, Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Nat. Sez. I, 5 (1958), 33-44. | MR 98331 | Zbl 0116.07901

[15] Ehrhard, A., Inegalités isoperimetriques et integrales de Dirichlet gaussiennes, Ann. Sci. École Norm. Sup., 17 (1984), 317-332. | MR 760680 | Zbl 0546.49020

[16] Esposito, L. - Trombetti, C., Convex symmetrization and Pólya-Szegö inequality, Nonlinear Anal., 56 (2004), 43-62. | MR 2031435 | Zbl 1038.26014

[17] Ferone, A. - Volpicelli, R., Minimal rearrangements of Sobolev functions: a new proof, Ann. Inst. H.Poincaré, Anal. Nonlinéaire, 20 (2003), 333-339 | MR 1961519 | Zbl 1038.49039

[18] Ferone, A. - Volpicelli, R., Convex symmetrization: the equality case in the Pólya-Szegö inequality, in corso di stampa su Calc. Var. and PDE's.

[19] Kawohl, B., Rearrangements and convexity of level sets in PDE, Lecture Notes in Math.1150, Springer-Verlag, Berlin (1985). | MR 810619 | Zbl 0593.35002

[20] Kawohl, B., On the isoperimetric nature of a rearrangement inequality and its consequences for some variational problems, Arch. Rat. Mech. Anal., 94 (1986), 227-243. | MR 846062 | Zbl 0603.49030

[21] Pòlya, G. - Szegö, G., Isoperimetric inequalities in Mathematical Physics, Ann. of Math. Studies, 27, Princeton University Press, Princeton, 1951. | MR 43486 | Zbl 0044.38301

[22] Steiner, J., Gesammelte Werke, 2, Reimer, Berlin, 1882.

[23] Talenti, G., Best constant in Sobolev inequality, Ann. Mat. Pura Appl., 110 (1976), 353-372. | MR 463908 | Zbl 0353.46018

[24] Talenti, G., The standard isoperimetric theorem, in Handbook of convex geometry, P. M.Gruber and J. M.Wills eds., North-Holland, Amsterdam, 1993. | MR 1242977 | Zbl 0799.51015

[25] Talenti, G., A weighted version of a rearrangement inequality, Ann. Univ. Ferrara, 43 (1997), 121-133. | MR 1686750 | Zbl 0936.26007

[26] Tonelli, L., Sulla proprietà di minimo della sfera, Rend. Circ. Mat. Palermo, 39 (1915), 109-138.

[27] Uribe, A., Minima of the Dirichlet norm and Toeplitz operators, preprint (1985).

[28] Vol’Pert, A. I., Spaces BV and quasi-linear equations, Math. USSR Sb., 17 (1967), 225-267. | Zbl 0168.07402