Steiner symmetrization is continuous in W1,p

Geometric and Functional Analysis

Volumn 7, Issue 5, 1997, Pages 823-860

  Burchard, A ^a  

^a PRINCETON UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0031285553     PISSN: 1016443X     EISSN: None     Source Type: Journal    
DOI: 10.1007/s000390050027     Document Type: Article

References (49)

1. L.V. AHLFORS, Conformal Invariants: Topics in Geometric Function Theory, McGraw-Hill Series in Higher Mathematics, 1973.
2. F. ALMGREN, E.H. LIEB, Continuity and discontinuity of the spherically decreasing rearrangement, Journal of the AMS 2 (1989), 683-773.
3. A. ALVINO, G. TROMBETTI, P.-L. LIONS, On optimization problems with prescribed rearrangements, Nonlinear Analysis 13 (1989), 185-220.
4. A. BAERNSTEIN, A unified approach to symmetrization, Symposia Matematica 35 (1995), 47-91.
5. A. BAERNSTEIN, B.A. TAYLOR, Spherical rearrangements, subharmonic functions, and *-functions in n-space, Duke Math. J. 43 (1976), 245-268.
6. C. BANDLE, Isoperimetric Inequalities and Applications, Pitman, Boston, 1986.
7. W. BECKNER, Inequalities of Fourier analysis, Ann. Math. (2) 102 (1975), 159-182.
8. n, Proc. Natl. Acad. Sci. USA 89 (1992), 4816-4819.
9. Y. BENYAMINI, Two-point symmetrization, the isoperimetric inequality on the sphere and some applications, in "Texas Functional Analysis Seminar 1983/84, Austin, Texas", Longhorn Notes, University of Texas Press, Austin, TX (1984), 53-76.
10. W. BLASCHKE, Eine isoperimetrische Eigenschaft des Kreises, Math. Z. 3 (1918), 52-57.
11. B. BOLLOBÁS, I.B. LEADER, Compressions and isoperimetric inequalities, J. Combin. Theory Ser. A 56 (1991), 47-62.
12. H.J. BRASCAMP, E.H. LIEB, Best constants in Young's inequality, its converse, and its generalization to more than three functions, Advances in Mathematics 20 (1976), 151-173.
13. H.J. BRASCAMP, E.H. LIEB, J.M. LUTTINGER, A general rearrangement inequality for multiple integrals, J. Funct. Anal. 17 (1974), 227-237.
14. F. BROCK, A.Yu. SOLYNIN, An approach to symmetrization via polarization, Technical report, Universität zu Köln, 1996.
15. J. BROTHERS, W.P. ZIEMER, Minimal rearrangements of Sobolev functions, Crelle J. Reine Angew. Math. 384 (1988), 153-179.
16. Yu.D. BURAGO, V.A. ZALGALLER, Geometric Inequalities (English translation of the Russian original), Springer Grundlehren 285, 1980.
17. C. CARATHEODORY, E. STUDY, Zwei Beweise des Satzes, daß der Kreis unter allen Figuren gleichen Umfangs den größten Inhalt hat, Math. Annalen 68 (1909), 133-144.
18. I. CHAVEL, Eigenvalues in Riemannian Geometry, Academic Press, Orlando, Fla, 1984.
19. 1,p(R), Ann. Scuola Norm. Sup. Pisa, Cl. Sci (4) 11 (1984), 57-85.
20. P.S. CROOKE, An isoperimetric bound for a Sobolev constant, Colloq. Math. 38 (1977/78), 263-267.
21. D.L. DONOHO, P.B. STARK, A note on rearrangements, spectral concentration, and the zero-order prolate spheroidal wavefunction, IEEE Trans. Inform. Theory 39 (1993), 257-260.
22. G.F.D. DUFF, A general inequality for the derivative of an equimeasurable rearrangement, Canad. J. Math. 28 (1976), 793-804.
23. G. FEDERER, Geometric Measure Theory, Springer Grundlehren 153, 1969.
24. H. FEDERER, W.H. FLEMING, Normal and integral currents, Ann. of Math. (2) 72 (1960), 458-520.
25. P. FRANKL, Z. FÜREDI, A short proof for a theorem of Harper about Hamming spheres, Discrete Math. 34 (1981), 311-313.
26. H. HADWIGER, Einfache Herleitung der isoperimetrischen Ungleichung für abgeschlossene Punktmengen, Math. Ann. 124 (1952), 158-160.
27. H. HADWIGER, D. OHMANN, Brunn-Minkowski Satz und Isoperimetrie, Math. Z. 66 (1956), 1-8.
28. G.E. HARDY, J.E. LITTLEWOOD, G. PÓLYA, Inequalities, Cambridge University Press, London and New York, 1952.
29. K. HILDÉN, Symmetrization of functions in Sobolev spaces and the isoperimetric inequality, Manuscripta Math. 18 (1976), 215-235.
30. B. KAWOHL, Rearrangements and convexity of level sets in PDE, Lecture Notes in Mathematics 1150, Springer Verlag, Berlin-New York, 1985.
31. T. LACHAND-ROBERT, Letter to E.H. Lieb, May 1995. Noted as "open problem" at the meeting on qualitative aspects of PDE, organized by H. Berestycki, B. Kawohl, and G. Talenti, Oberwolfach, Germany February 12-18, 1995.
32. E.H. LIEB, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Studies in Applied Mathematics 57 (1977), 93-105.
33. E.H. LIEB, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. Math. 118 (1983), 349-374.
34. E.H. LIEB, M. Loss, Analysis, AMS Graduate Studies in Mathematics, 1996.
35. M. Loss, Personal communication, July 1995.
36. L. LUSTERNIK, Die Brunn-Minkowskische Ungleichung für beliebige meßbare Mengen, Dokl. Acad. Sci. URSS 3 (1935), 55-58.
37. P. MANI-LEVITSKA, Random Steiner symmetrizations, Studia Sci. Math. Hungar. 21 (1986), 373-378.
38. W. MAZJA, Einbettungssätze für Sobolewsche Räume. Teil 1 (German translation of the Russian original), Teubner Verlagsgesellschaft, Leipzig, 1979.
39. F. PÓLYA AND G. SZEGO, Isoperimetric Inequalities, Princeton University Press, Princeton, 1951.
40. F. RIESZ, Sur une inégalité intégrale, J. London Math. Soc. 5 (1930), 162-168.
41. S.L. SOBOLEV, On a theorem of functional analysis, Mat. Sb. (N.S.) 4 (1938), 471-497 (Russian original); AMS Transl. (2), 34 (1963), 39-68 (English translation).
42. S.L. SOBOLEV, On a theorem of functional analysis, Mat. Sb. (N.S.) 4 (1938), 471-497 (Russian original); AMS Transl. (2), 34 (1963), 39-68 (English translation).
43. A.YU. SOLYNIN, Application of the polarization for proofs of functional inequalities (in Russian), Russian Academy of Sciences, St. Petersburg, PDMI preprint 10, 1995.
44. E. SPERNER, Symmetrisierung für Funktionen mehrerer reeller Variablen, Manuscripta Math. 11 (1974), 159-170.
45. J. STEINER, Einfacher Beweis der isoperimetrischen Hauptsätze, Crelle J. Reine Angew. Math. 18 (1838) 281-296; Ges. Werke 2 (Berlin, 1882), 77-91 (in German).
46. J. STEINER, Einfacher Beweis der isoperimetrischen Hauptsätze, Crelle J. Reine Angew. Math. 18 (1838) 281-296; Ges. Werke 2 (Berlin, 1882), 77-91 (in German).
47. G. TALENTI, The standard isoperimetric theorem, Handbook of Convex Geometry, Vol. A, 73-123, North Holland, Amsterdam, 1993.
48. G. TALENTI, Best constant in Sobolev inequality, Ann. Mat. Pura Appl (4) 110 (1976), 353-372.
49. W. ZIEMER, Weakly differentiate functions. Sobolev spaces and functions of bounded variation, Graduate Texts in Mathematics 120. Springer-Verlag, New York, 1989.