Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

 
 

 

Continuous symmetrization via polarization


Author: A. Yu. Solynin
Original publication: Algebra i Analiz, tom 24 (2012), nomer 1.
Journal: St. Petersburg Math. J. 24 (2013), 117-166
MSC (2010): Primary 30C75
DOI: https://doi.org/10.1090/S1061-0022-2012-01234-3
Published electronically: November 15, 2012
MathSciNet review: 3013297
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We discuss a one-parameter family of transformations that changes sets and functions continuously into their $ (k,n)$-Steiner symmetrizations. Our construction consists of two stages. First, we employ a continuous symmetrization introduced by the author in 1990 to transform sets and functions into their one-dimensional Steiner symmetrization. Some of our proofs at this stage rely on a simple rearrangement called polarization. At the second stage, we use an approximation theorem due to Blaschke and Sarvas to give an inductive definition of the continuous $ (k,n)$-Steiner symmetrization for any $ 2\le k \le n$. This transformation provides us with the desired continuous path along which all basic characteristics of sets and functions vary monotonically. In its turn, this leads to continuous versions of several convolution type inequalities and Dirichlet's type inequalities as well as to continuous versions of comparison theorems for solutions of some elliptic and parabolic partial differential equations.


References [Enhancements On Off] (What's this?)

  • 1. S. Abramovich., Monotonicity of eigenvalues under symmetrization, SIAM J. Appl. Math. 28 (1975), 350-361. MR 0382774 (52:3656)
  • 2. L. V. Ahlfors, Conformal invariants: topics in geometric function theory, McGraw-Hill Book Co, New York etc., 1973. MR 0357743 (50:10211)
  • 3. H. W. Alt, Lineare Funktionalanalysis, 2nd ed., Springer-Verlag, 1992.
  • 4. A. Alvino, P.-L. Lions, and G. Trombetti, Comparison results for elliptic and parabolic equations via symmetrization: a new approach, Differential Integral Equations 4 (1991), 25-50. MR 1079609 (91h:35023)
  • 5. -, Comparison results for elliptic and parabolic equations via Schwarz symmetrization, Ann. Inst. H. Poincaré Anal. Non Linéaire 7 (1990), no. 2, 37-65. MR 1051227 (91f:35022)
  • 6. A. Baernstein, II, A unified approach to symmetrization, Partial Differential Equations of Elliptic Type (Cortona, 1992), Sympos. Math., vol. 35, Cambridge Univ. Press, Cambridge, 1994, pp. 47-91. MR 1297773 (96e:26019)
  • 7. A. Baernstein, II and B. A. Taylor, Spherical rearrangements, subharmonic functions, and $ *$-functions in $ n$-space, Duke Math. J. 43 (1976), 245-268. MR 0402083 (53:5906)
  • 8. W. Beckner, Sobolev inequalities, the Poisson semigroup, and analysis on the sphere $ S^n$, Proc. Nat. Acad. Sci. U.S.A. 89 (1992), 4816-4819. MR 1164616 (93d:26018)
  • 9. D. Betsakos, Polarization, conformal invariants, and Brownian motion, Ann. Acad. Sci. Fenn. Math. 23 (1998), no. 1, 59-82. MR 1601843 (99g:31004)
  • 10. -, Polarization, continuous Markov processes, and second order elliptic equations, Indiana Univ. Math. J. 53 (2004), no. 2, 331-345. MR 2056435 (2005c:60094)
  • 11. W. Blaschke, Kreis und Kugel, Chelsea Publ. Co., New York, 1949. MR 0076364 (17:887b)
  • 12. F. Brock, Continuous Steiner-symmetrization, Math. Nachr. 172 (1995), 25-48. MR 1330619 (96c:49004)
  • 13. -, Continuous rearrangement and symmetry of solutions of elliptic problems, Proc. Indian Acad. Sci. Math. Sci. 110 (2000), no. 2, 157-204. MR 1758811 (2001i:35016)
  • 14. 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)
  • 15. V. N. Dubinin, Transformation of functions and the Dirichlet principle, Mat. Zametki 38 (1985), no. 1, 49-55; English transl., Math. Notes 38 (1985), no. 1-2, 539-542. MR 0804180 (87j:31005)
  • 16. -, Capacities and geometric transformations of subsets in $ n$-space, Geom. Funct. Anal. 3 (1993), 342-369. MR 1223435 (94f:31008)
  • 17. -, Symmetrization in the geometric theory of functions of a complex variable, Uspekhi Mat. Nauk 49 (1994), no. 1, 3-76; English transl., Russian Math. Surveys 49 (1994), no. 1, 1-79. MR 1307130 (96b:30054)
  • 18. B. Gidas, W. M. Ni, and L.  Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), 209-243. MR 0544879 (80h:35043)
  • 19. J. Kačur, Method of Rothe in evolution equations, Teubner Texts in Math., vol. 80, Teubner-Verlag, Leipzig, 1985. MR 0834176 (87j:35004)
  • 20. B. Kawohl, Rearrangements and convexity of level sets in PDE, Lecture Notes in Math., vol. 1150, Springer-Verlag, Berlin, 1985. MR 0810619 (87a:35001)
  • 21. B. E. Levitskiĭ, $ k$-symmetrization and extremal rings, Kuban. Gos. Univ. Nauchn. Trudy No. 148, Krasnodar, 1971, pp. 35-40. (Russian) MR 0340592 (49:5344)
  • 22. A. McNabb, Partial Steiner symmetrization and some conduction problems, J. Math. Anal. Appl. 17 (1967), 221-227. MR 0203583 (34:3433)
  • 23. M. Marcus, Radial averaging of domains, estimates for Dirichlet integrals and applications, J. Analyse Math. 27 (1974), 47-78. MR 0477029 (57:16573)
  • 24. G. Pólya and G. Szegő, Isoperimetric inequalities in mathematical physics, Ann. of Math. Stud., vol. 27, Princeton Univ. Press, Princeton, NJ, 1951. MR 0043486 (13:270d)
  • 25. J. Sarvas, Symmetrization of condensers in $ n$-space, Ann. Acad. Sci. Fenn. Ser. A1 No. 522 (1972), 44 pp. MR 0348108 (50:606)
  • 26. J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal. 43 (1971), 304-318. MR 0333220 (48:11545)
  • 27. A. Yu. Solynin, Continuous symmetrization of sets, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 185 (1990), 125-139; English transl., J. Soviet Math. 59 (1992), no. 6, 1214-1221. MR 1097593 (92k:28012)
  • 28. -, Functional inequalities via polarization, Algebra i Analiz 8 (1996), no. 6, 148-185; English transl., St. Petersburg Math. J. 8 (1997), no. 6, 1015-1038. MR 1458141 (98e:30001a)
  • 29. -, Ordering of sets, hyperbolic metric and harmonic measure, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 237 (1997), 129-147; English transl., J. Math. Sci. (N. Y.) 95 (1999), no. 3, 2256-2266. MR 1691288 (2000d:30068)
  • 30. J. Steiner, Gesammelte Werke. Bd. 2, Reimer-Verlag, Berlin, 1882.
  • 31. J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal. 43 (1971), 304-318. MR 0333220 (48:11545)
  • 32. G. Talenti, The standard isoperimetric theorem, Handbook of Convex Geometry, Vol. A, North-Holland, Amsterdam, 1993, pp. 73-123. MR 1242977 (94h:49065)
  • 33. V. Wolontis, Properties of conformal invariants, Amer. J. Math. 74 (1952), 587-606. MR 0048585 (14:36c)

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2010): 30C75

Retrieve articles in all journals with MSC (2010): 30C75


Additional Information

A. Yu. Solynin
Affiliation: Department of Mathematics and Statistics, Texas Tech University, Box 41042, Lubbock, Texas 79409
Email: alex.solynin@ttu.edu

DOI: https://doi.org/10.1090/S1061-0022-2012-01234-3
Keywords: Continuous symmetrization, Steiner symmetrization, rearrangement, polarization, integral inequality, boundary-value problem, comparison theorem
Received by editor(s): February 7, 2011
Published electronically: November 15, 2012
Article copyright: © Copyright 2012 American Mathematical Society

American Mathematical Society