Minimal rearrangements of Sobolev functions: A new proof

Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire

Volumn 20, Issue 2, 2003, Pages 333-339

  Ferone, Adele ^a   Volpicelli, Roberta ^b  

^a SECOND UNIVERSITY OF NAPLES   (Italy)

^b UNIVERSITY OF NAPLES FEDERICO II   (Italy)

Author keywords

Polar factorization; P lya Szeg type inequalities; Rearrangements

Indexed keywords

BOUNDARY CONDITIONS; ERROR ANALYSIS; INTEGRATION; NONLINEAR SYSTEMS; PROBLEM SOLVING; SET THEORY; THEOREM PROVING;

DISTRIBUTION FUNCTIONS; FACTORIZATION; ISOPERIMETRIC THEOREM;

FUNCTION EVALUATION;

EID: 0345529248     PISSN: 02941449     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0294-1449(02)00012-4     Document Type: Article

References (27)

1. Alvino A., Lions P.L., Trombetti G. A remark on comparison result via Schwartz symmetrization. Proc. Roy. Soc. Edinburgh. 102-A:(1-2):1986;37-48.
2. Alvino A., Ferone V., Lions P.L., Trombetti G. Convex symmetrization and applications. Ann. Inst. H. Poincaré, Anal. Non Linéaire. 14:(2):1997;275-293.
3. Aronsson G., Talenti G. Estimating the integral of a function in terms of a distribution function of its gradient. Boll. Un. Mat. Ital. (5). 18-B:(3):1981;885-894.
4. Aubin T. Problèmes isopérimétriques et espaces de Sobolev. C. R. Acad. Sci. Paris. 280:1975;279-281.
5. Betta M.F., Brock F., Mercaldo A., Posteraro M.R. A weighted isoperimetric inequality and applications to symmetrization. J. Inequal. Appl. 4:(3):1999;215-240.
6. Brock F. Weighted Dirichlet-type inequalities for Steiner symmetrization. Calc. Var. Partial Differential Equations. 8:1999;15-25.
7. Brothers J.E., Ziemer W.P. Minimal rearrangements of Sobolev functions. J. Reine Angew. Math. 384:1988;153-179.
8. ∞ . Ark. Mat. 36:(2):1998;317-340.
9. De Giorgi E. Su una teoria generale della misura (r-1) -dimensionale in uno spazio ad r dimensioni. Ann. Mat. Pura Appl. 36:(4):1954;191-213.
10. Duff G.F.D. A general integral inequality for the derivative of an equimeasurable rearrangement. Canad. J. Math. 28:(4):1976;793-804.
11. Federer H. Geometric Measure Theory. 1969;Springer, Berlin.
12. Fleming W., Rishel R. An integral formula for total gradient variation. Arch. Math. 11:1960;218-222.
13. Friedman A., McLeod R. Strict inequalities for integrals of decreasingly rearranged functions. Proc. Roy. Soc. Edinburgh. 102-A:(3-4):1986;277-289.
14. Hilden K. Symmetrization of functions in Sobolev spaces and the isoperimetric inequality. Manuscripta Math. 18:(3):1976;215-235.
15. Kawohl B. Rearrangements and Convexity of Level Sets in P.D.E. Lecture Notes in Math. 1150:1985;Springer, Berlin.
16. Klimov V.S. Imbedding theorems and geometric inequalities. Izv. Akad. Nauk USSR Ser. Mat. 40:(3):1976;645-671.
17. Maz'ja V.M. Sobolev Spaces. 1985;Springer, Berlin.
18. Mossino J. Inégalités Isopérimétriques et Applications en Physic. Collection Travaux en Cours. 1984;Hermann Paris.
19. Morrey C.B. Multiple Integrals in the Calculus of Variations. 1966;Springer, Berlin.
20. Pólya G., Szegö G. Isoperimetric Inequalities in Mathematical Physics. Ann. of Math. Studies. 27:1951;Princeton University Press, Princeton.
21. Rakotoson J.M., Temam R. A co-area formula with applications to monotone rearrangement and to regularity. Arch. Rational Mech. Anal. 109:(3):1990;213-238.
22. Ryff J.V. Measure preserving transformations and rearrangements. J. Math. Anal. Appl. 31:(3):1970;449-458.
23. Sperner E. Zur Symmetrisierung von Funktionen auf Sphären. Math. Z. 134:1973;317-327.
24. Sperner E. Symmetrisierung für Funktionen mehrerer reeler Variablen. Manuscripta Math. 11:1974;159-170.
25. Talenti G. Best constant in Sobolev inequality. Ann. Mat. Pura Appl. Cl. Sci. (4). 110:1976;353-372.
26. Talenti G. A weighted version of a rearrangement inequality. Ann. Univ. Ferrara. 43:1997;121-133.
27. A. Uribe, Minima of Dirichlet norm and Topeliz operators, Preprint, 1985.