Continuous symmetrization via polarization

Solynin, A.

doi:10.1090/S1061-0022-2012-01234-3

Continuous symmetrization via polarization
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by A. Yu. Solynin

St. Petersburg Math. J. 24 (2013), 117-166

DOI: https://doi.org/10.1090/S1061-0022-2012-01234-3

Published electronically: November 15, 2012

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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

Shoshana Abramovich, Monotonicity of eigenvalues under symmetrization, SIAM J. Appl. Math. 28 (1975), 350–361. MR 382774, DOI 10.1137/0128030
Lars V. Ahlfors, Conformal invariants: topics in geometric function theory, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. MR 0357743
H. W. Alt, Lineare Funktionalanalysis, 2nd ed., Springer-Verlag, 1992.
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), no. 1, 25–50. MR 1079609
A. Alvino, G. Trombetti, and P.-L. Lions, Comparison results for elliptic and parabolic equations via Schwarz symmetrization, Ann. Inst. H. Poincaré C Anal. Non Linéaire 7 (1990), no. 2, 37–65 (English, with French summary). MR 1051227, DOI 10.1016/S0294-1449(16)30303-1
Albert Baernstein II, A unified approach to symmetrization, Partial differential equations of elliptic type (Cortona, 1992) Sympos. Math., XXXV, Cambridge Univ. Press, Cambridge, 1994, pp. 47–91. MR 1297773
Albert Baernstein II and B. A. Taylor, Spherical rearrangements, subharmonic functions, and $^*$-functions in $n$-space, Duke Math. J. 43 (1976), no. 2, 245–268. MR 402083
William Beckner, Sobolev inequalities, the Poisson semigroup, and analysis on the sphere $S^n$, Proc. Nat. Acad. Sci. U.S.A. 89 (1992), no. 11, 4816–4819. MR 1164616, DOI 10.1073/pnas.89.11.4816
Dimitrios Betsakos, Polarization, conformal invariants, and Brownian motion, Ann. Acad. Sci. Fenn. Math. 23 (1998), no. 1, 59–82. MR 1601843
Dimitrios Betsakos, Polarization, continuous Markov processes, and second order elliptic equations, Indiana Univ. Math. J. 53 (2004), no. 2, 331–345. MR 2056435, DOI 10.1512/iumj.2004.53.2380
Wilhelm Blaschke, Kreis und Kugel, Chelsea Publishing Co., New York, 1949 (German). MR 0076364
Friedemann Brock, Continuous Steiner-symmetrization, Math. Nachr. 172 (1995), 25–48. MR 1330619, DOI 10.1002/mana.19951720104
Friedemann Brock, Continuous rearrangement and symmetry of solutions of elliptic problems, Proc. Indian Acad. Sci. Math. Sci. 110 (2000), no. 2, 157–204. MR 1758811, DOI 10.1007/BF02829490
Friedemann Brock and Alexander Yu. Solynin, An approach to symmetrization via polarization, Trans. Amer. Math. Soc. 352 (2000), no. 4, 1759–1796. MR 1695019, DOI 10.1090/S0002-9947-99-02558-1
V. N. Dubinin, Transformation of functions and the Dirichlet principle, Mat. Zametki 38 (1985), no. 1, 49–55, 169 (Russian). MR 804180
V. N. Dubinin, Capacities and geometric transformations of subsets in $n$-space, Geom. Funct. Anal. 3 (1993), no. 4, 342–369. MR 1223435, DOI 10.1007/BF01896260
V. N. Dubinin, Symmetrization in the geometric theory of functions of a complex variable, Uspekhi Mat. Nauk 49 (1994), no. 1(295), 3–76 (Russian); English transl., Russian Math. Surveys 49 (1994), no. 1, 1–79. MR 1307130, DOI 10.1070/RM1994v049n01ABEH002002
B. Gidas, Wei Ming Ni, and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), no. 3, 209–243. MR 544879
Jozef Kačur, Method of Rothe in evolution equations, Teubner-Texte zur Mathematik [Teubner Texts in Mathematics], vol. 80, BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1985. With German, French and Russian summaries. MR 834176
Bernhard Kawohl, Rearrangements and convexity of level sets in PDE, Lecture Notes in Mathematics, vol. 1150, Springer-Verlag, Berlin, 1985. MR 810619, DOI 10.1007/BFb0075060
B. E. Levickiĭ, $k$-symmetrization and extremal rings, Kuban. Gos. Univ. Naučn. Trudy 148 Mat. Anal. (1971), 35–40 (Russian). MR 0340592
A. McNabb, Partial Steiner symmetrization and some conduction problems, J. Math. Anal. Appl. 17 (1967), 221–227. MR 203583, DOI 10.1016/0022-247X(67)90147-3
Moshe Marcus, Radial averaging of domains, estimates for Dirichlet integrals and applications, J. Analyse Math. 27 (1974), 47–78. MR 477029, DOI 10.1007/BF02788642
G. Pólya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics, Annals of Mathematics Studies, No. 27, Princeton University Press, Princeton, N. J., 1951. MR 0043486
Jukka Sarvas, Symmetrization of condensers in $n$-space, Ann. Acad. Sci. Fenn. Ser. A. I. 522 (1972), 44. MR 348108
James Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal. 43 (1971), 304–318. MR 333220, DOI 10.1007/BF00250468
A. Yu. Solynin, Continuous symmetrization of sets, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 185 (1990), no. Anal. Teor. Chisel i Teor. Funktsiĭ. 10, 125–139, 186 (Russian); English transl., J. Soviet Math. 59 (1992), no. 6, 1214–1221. MR 1097593, DOI 10.1007/BF01374083
A. Yu. Solynin, Polarization and functional inequalities, Algebra i Analiz 8 (1996), no. 6, 148–185 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 8 (1997), no. 6, 1015–1038. MR 1458141
A. Yu. Solynin, Ordering of sets, hyperbolic metric, and harmonic measure, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 237 (1997), no. Anal. Teor. Chisel i Teor. Funkts. 14, 129–147, 230 (Russian, with Russian summary); English transl., J. Math. Sci. (New York) 95 (1999), no. 3, 2256–2266. MR 1691288, DOI 10.1007/BF02172470
J. Steiner, Gesammelte Werke. Bd. 2, Reimer-Verlag, Berlin, 1882.
James Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal. 43 (1971), 304–318. MR 333220, DOI 10.1007/BF00250468
Giorgio Talenti, The standard isoperimetric theorem, Handbook of convex geometry, Vol. A, B, North-Holland, Amsterdam, 1993, pp. 73–123. MR 1242977, DOI 10.1016/B978-0-444-89596-7.50008-0
Vidar Wolontis, Properties of conformal invariants, Amer. J. Math. 74 (1952), 587–606. MR 48585, DOI 10.2307/2372264