|
Universal approximation of symmetrizations by polarizations
Author:
Jean van Schaftingen
Journal:
Proc. Amer. Math. Soc. 134 (2006), 177-186
MSC (2000):
Primary 26D10; Secondary 28D05, 46E30
Posted:
August 11, 2005
MathSciNet review:
2170557
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: Any symmetrization (Schwarz, Steiner, cap or increasing rearrangement) can be approximated by a universal sequence of polarizations which converges in  norm for any admissible function in  for  and uniformly for admissible continuous functions. A new Pólya-Szegö inequality is proved for the increasing rearrangement.
References
- 1.
G. Alberti, Some remarks about a notion of rearrangement, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 29 (2000), no. 2, 457-472. MR 1784182 (2001k:46050)
- 2.
M. Badiale, Monotonicity of solutions for elliptic systems on unbounded domains, Boll. Un. Mat. Ital. A (7) 6 (1992), no. 1, 59-69. MR 1164736 (93g:35037)
- 3.
A. Baernstein, II, A unified approach to symmetrization, Partial Differential Equations of Elliptic Type (A. Alvino et al., eds.), Sympos. Math., no. 35, Cambridge Univ. Press, 1995, pp. 47-49. MR 1297773 (96e:26019)
- 4.
H. J. Brascamp, E. H. Lieb, and J. M. Luttinger, A general rearrangement inequality for multiple integrals, J. Funct. Anal. 17 (1974), 227-237. MR 0346109 (49:10835)
- 5.
F. Brock and A. Yu. Solynin, An approach to symmetrization via polarization, Trans. Amer. Math. Soc. 352 (2000), no. 4, 1759-1796. MR 1695019 (2001a:26014)
- 6.
G. Carbou, Unicité et minimalité des solutions d'une équation de Ginzburg-Landau, Ann. Inst. H. Poincaré Anal. Non Linéaire 12 (1995), no. 3, 305-318. MR 1340266 (96m:35078)
- 7.
J. A. Crowe, J. A. Zweibel, and P. C. Rosenbloom, Rearrangements of functions, J. Funct. Anal. 66 (1986), 432-438. MR 0839110 (87j:28003)
- 8.
E. H. Lieb and M. Loss, Analysis, second ed., Grad. Stud. Math., vol. 14, American Mathematical Society, Providence, RI, 2001. MR 1817225 (2001i:00001)
- 9.
P. Mani-Levitska, Random Steiner symmetrizations, Studia Sci. Math. Hungar. 21 (1986), no. 3-4, 373-378. MR 0919382 (89a:52012)
- 10.
G. Pólya and G. Szegö, Isoperimetric inequalities in mathematical physics, Princeton Univ. Press, 1951. MR 0043486 (13:270d)
- 11.
J. Sarvas, Symmetrization of condensers in
 -space., Ann. Acad. Sci. Fenn., Ser. A I no. 522 (1972), 44 pp. MR 0348108 (50:606)
- 12.
D. Smets and M. Willem, Partial symmetry and asymptotic behavior for some elliptic variational problems, Calc. Var. Partial Differential Equations 18 (2003), no. 1, 57-75. MR 2001882 (2004m:35092)
- 13.
M. Willem, Analyse fonctionnelle élémentaire, Cassini, Paris, 2003.
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2000):
26D10,
28D05,
46E30
Retrieve articles in all journals
with MSC (2000):
26D10,
28D05,
46E30
Additional Information
Jean van Schaftingen
Affiliation:
Département de Mathématiques, Université Catholique de Louvain, Chemin du cyclotron 2, B-1348 Louvain-la-Neuve, Belgium
Email:
vanschaftingen@math.ucl.ac.be
DOI:
http://dx.doi.org/10.1090/S0002-9939-05-08325-5
PII:
S 0002-9939(05)08325-5
Received by editor(s):
May 13, 2003
Received by editor(s) in revised form:
July 15, 2004
Posted:
August 11, 2005
Additional Notes:
The author is a research fellow of the Fonds National de la Recherche Scientifique (Belgium).
Communicated by:
David S. Tartakoff
Article copyright:
© Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain after
28 years from publication.
|
