The perimeter inequality under Steiner symmetrization: Cases of equality

Annals of Mathematics

Volumn 162, Issue 1, 2005, Pages 525-555

  Chlebík, Miroslav ^a   Cianchi, Andrea ^b   Fusco, Nicola ^c  

^a MAX PLANCK INSTITUTE FOR MATHEMATICS IN THE SCIENCES   (Germany)

^b UNIVERSITY OF FLORENCE   (Italy)

^c UNIVERSITY OF NAPLES FEDERICO II   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 29844442856     PISSN: 0003486X     EISSN: 0003486X     Source Type: Journal    
DOI: 10.4007/annals.2005.162.525     Document Type: Article

References (23)

1. L. AMBROSIO, N. Fusco, and D. PALLARA, Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press, Oxford, 2000.
2. A. BAERNSTEIN II, Integral means, univalent functions and circular symmetrization, Acta Math. 133 (1974), 139-169.
3. A. BAERNSTEIN II, A unified approach to symmetrization, in Partial Differential Equations of Elliptic Type (A. Alvino, E. Fabes, and G. Talenti, eds.) (Cortona, 1992)
4. Symposia Math. 35, 47-91, Cambridge Univ. Press, Cambridge, 1994.
5. W. BECKNER, Inequalites in Fourier analysis, Ann. of Math. 102 (1975), 159-182.
6. F. BROCK and A. Yu. SOLYNIN, An approach to symmetrization via polarization, Trans. Amer. Math. Soc. 352 (2000), 1759-1796.
7. J. BROTHERS and W. ZIEMER, Minimal rearrangements of Sobolev functions, J. reine angew. Math. 384 (1988), 153-179.
8. A. BURCHARD, Cases of equality in the Riesz rearrangement inequality, Ann. of Math. 143 (1996), 499-527.
9. L. A. CAFFARELLI, D. JERISON, and E. LIEB, On the case of equality in the Brunn-Minkowski inequality for capacity, Adv. Math. 117 (1996), 193-207.
10. E. CARLEN and M. Loss, Extremals of functional with competing symmetries, J. Funct. Anal. 88 (1990), 437-456.
11. A. CIANCHI and N. FUSCO, Functions of bounded variation and rearrangements, Arch. Rational Mech. Anal. 165 (2002), 1-40.
12. E. DE GIORGI, Su una teoria generale della, misura (r - 1)-dimensionale in uno spazio ad r dimensioni, Ann. Mat. Pura Appl. 36 (1954), 191-213.
13. V. N. DUBININ, Capacities and geometric transformations of subsets in n-space, Geom. Funct. Anal. 3 (1993), 342-369.
14. L. C. EVANS and R. F. GARIEPY, Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992.
15. H. FEDERER, Geometric Measure Theory, Springer-Verlag, New York, 1969.
16. M. GIAQUINTA, G. MODICA, and J. SOUČEK, Cartesian Currents in the Calculus of Variations, Part I: Cartesian Currents, Part II: Variational Integrals, Springer-Verlag, New York, 1998.
17. B. KAWOHL, On the isoperimetric nature of a rearrangement inequality and its consequences for some variational problems, Arch. Rational Mech. Anal. 94 (1986), 227-243.
18. E. LIEB, Sharp constants in the Hardy-Littlewood-Sobolev inequality and related inequalities, Ann. of Math. 118 (1983), 349-374.
19. J. STEINER, Einfacher Beweis der isoperimetrischen Hauptsätze, J. reine angew Math. 18 (1838), 281-296
20. and Gesammelte Werke Vol. 2, 77-91, Reimer, Berlin, 1882.
21. G. TALENTI, The standard isoperimetric theorem, in Handbook of Convex Geometry, Vol. A, B, 73-123 (P. M. Gruber and J. M. Wills eds.), North-Holland, Amsterdam, 1993.
22. A. I. VOL'PERT, Spaces BV and quasilinear equations, Math. USSR. Sb. 17 (1967), 225-267.
23. W. P. ZIEMER, Weakly Differentiate Functions, Springer-Verlag, New York, 1989.