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Exercises on the Theme of Continuous Symmetrization

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Abstract

In this paper, we discuss the Pólya–Szegő continuous symmetrization and its applications to isoperimetric inequalities. In particular, we survey results concerning monotonicity properties of certain characteristics, including torsional rigidity of cylindrical beams and principal frequency of a uniformly stretched elastic membrane of a drum, of triangles and other domains. Several remaining open problems on continuous symmetrization and relevant properties of domains and functions are also discussed.

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References

  1. Acker, A., Payne, L.E., Philippin, G.: On the convexity of level lines of the fundamental mode in the clamped membrane problem, and the existence of convex solutions in a related free boundary proble. Z. Angew. Math. Phys. 32(6), 683–694 (1981)

    Article  MathSciNet  Google Scholar 

  2. Anderson, J.M., Barth, K.F., Brannan, D.A.: Research problems in complex analysis. Bull. Lond. Math. Soc. 9(2), 129–162 (1977)

    Article  MathSciNet  Google Scholar 

  3. Baernstein, A.: II, Symmetrization in analysis. With David Drasin and Richard S. Laugesen. With a foreword by Walter Hayman. New Mathematical Monographs, vol. 36. Cambridge University Press, Cambridge (2019)

  4. Barnard, R.W., Hadjicostas, P., Solynin, AYu.: The Poincaré metric and isoperimetric inequalities for hyperbolic polygons. Trans. Am. Math. Soc. 357(10), 3905–3932 (2005)

    Article  Google Scholar 

  5. Brascamp, H.J., Lieb, E.H.: Some inequalities for Gaussian measures and the long-range order of the one-dimensional plasma. In: Arthurs, A. (ed.) Funct. Integr. Appl. Proc. Int. Conf. London 1974, Oxford (1975), pp. 1–14

  6. Brock, F.: Continuous Steiner-symmetrization. Math. Nachr. 172, 25–48 (1995)

    Article  MathSciNet  Google Scholar 

  7. Brock, F.: Continuous rearrangement and symmetry of solutions of elliptic problems. Proc. Indian Acad. Sci. (Math. Sci.) 110(2), 157–204 (2000)

    Article  MathSciNet  Google Scholar 

  8. Brock, F., Solynin, AYu.: An approach to symmetrization via polarization. Trans. Am. Math. Soc. 352(4), 1759–1796 (2000)

    Article  MathSciNet  Google Scholar 

  9. Dubinin, V.N.: Transformation of functions and the Dirichlet principle. Mat. Zametki 38(1), 49–55, 169 (1985)

  10. Dubinin, V.N.: Condenser capacities and symmetrization in geometric function theory. Translated from the Russian by Nikolai G. Kruzhilin. Springer, Basel, xii+344 pp (2014)

  11. Fleeman, M., Simanek, B.: Torsional rigidity and Bergman analytic content of simply connected regions. Comput. Methods Funct. Theory 19(1), 37–63 (2019)

    Article  MathSciNet  Google Scholar 

  12. Freitas, P., Siudeja, B.: Bounds for the first Dirichlet eigenvalue of triangles and quadrilaterals. ESAIM Control Optim. Calc. Var. 16(3), 648–676 (2010)

    Article  MathSciNet  Google Scholar 

  13. Goluzin, G. M.: Geometric theory of functions of a complex variable. Translations of Mathematical Monographs, vol. 26. American Mathematical Society, Providence, vi+676 pp (1969)

  14. Goodman, A.W.: Univalent Functions. Mariner Publishing Co., Inc., Tampa, xvii+246 pp (1983)

  15. Goodman, A.W., Saff, E.B.: On univalent functions convex in one direction. Proc. Am. Math. Soc. 73(2), 183–187 (1979)

    Article  MathSciNet  Google Scholar 

  16. Haegi, H.R.: Extremalprobleme und Ungleichungen konformer Gebietsgrössen. Compos. Math. 8, 81–111 (1950)

    MathSciNet  MATH  Google Scholar 

  17. Hamel, F., Nadirashvili, N., Sire, Y.: Convexity of level sets for elliptic problems in convex domains or convex rings: Two counterexamples. Am. J. Math. 138, 499–527 (2016)

    Article  MathSciNet  Google Scholar 

  18. Hayman, W.K.: Multivalent Functions, 2nd edn. Cambridge Tracts in Mathematics and Mathematical Physics, vol. 48. Cambridge University Press, Cambridge (1994)

  19. Jenkins, J.A.: On certain geometrical problems associated with capacity. Math. Nachr. 39, 349–356 (1969)

    Article  MathSciNet  Google Scholar 

  20. Kuz’mina, G.V.: Moduli of families of curves and quadratic differentials. A translation of Trudy Mat. Inst. Steklov. 139 (1980). Proc. Steklov Inst. Math. no. 1, vii+231 pp (1982)

  21. Laugesen, R.S., Siudeja, B.A.: Dirichlet eigenvalue sums on triangles are minimal for equilaterals. Commun. Anal. Geom. 19(5), 855–885 (2011)

    Article  MathSciNet  Google Scholar 

  22. Laugesen, R.S, Siudeja, B.A.: Triangles and Other Special Domains. Shape Optimization and Spectral Theory, pp. 149–200. De Gruyter Open, Warsaw (2017)

  23. Lions, P.-L.: Two geometric properties of solutions of semilinear problems. Appl. Anal. 12(4), 499–527 (1981)

    Google Scholar 

  24. Makar-Limanov, L.G.: The solution of the Dirichlet problem for the equation \(\Delta u=-1\) in a convex region. Mat. Zametki 9, 89–92 (1971)

    MathSciNet  MATH  Google Scholar 

  25. McNabb, A.: A partial Steiner symmetrization and some conduction problems. J. Math. Anal. Appl. 17, 221–227 (1967)

    Article  MathSciNet  Google Scholar 

  26. Pólya, G., Szegő, G.: Isoperimetric Inequalities in Mathematical Physics. Ann. Math. Studies, vol. 27. Princeton University Press, Princeton (1951)

  27. Pólya, G., Szegő, G.: Problems and Theorems in Analysis. I. Series, Integral Calculus, Theory of Functions. Classics in Mathematics. Springer, Berlin, xx+389 pp (1998)

  28. Prokhorov, D.V.: Level lines of functions that are convex in the direction of an axis. Mat. Zametki 44(4), 523–527, 558 (1988) [Translation in Math. Notes 44(3–4), 767–769 (1989)]

  29. Ruscheweyh, S., Salinas, L.C.: On the preservation of direction-convexity and the Goodman–Saff conjecture. Ann. Acad. Sci. Fenn. Ser. A I Math. 14(1), 63–73 (1989)

    Article  MathSciNet  Google Scholar 

  30. Siudeja, B.: Isoperimetric inequalities for eigenvalues of triangles. Indiana Univ. Math. J. 59(3), 1097–1120 (2010)

    Article  MathSciNet  Google Scholar 

  31. Solynin, A.Yu.: Extremal decompositions of the plane or disk into two nonoverlapping domains. Preprint of the Kuban State University, Krasnodar, Deponirovano in VINITI, no. 7800 (1984)

  32. Solynin, A.Yu.: Continuous symmetrization of sets. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 185 (1990) [Anal. Teor. Chisel i Teor. Funktsii. 10, 125–139; translation in J. Soviet Math. 59(6), 1214–1221 (1990)]

  33. Solynin, A.Yu.: Polarization and functional inequalities. Algebra i Analiz 8(6), 148–185 (1996) [English transl., St. Petersburg Math. J. 8(6), 1015–1038 (1997)]

  34. Solynin, A.Yu.: Continuous symmetrization via polarization. Algebra i Analiz 24(1), 157–222 (2012) [translation in St. Petersburg Math. J. 24(1), 117–166 (2012)]

  35. Solynin, AYu., Zalgaller, V.A.: An isoperimetric inequality for logarithmic capacity of polygons. Ann. Math. (2) 159(1), 277–303 (2004)

    Article  MathSciNet  Google Scholar 

  36. Solynin, AYu., Zalgaller, V.A.: The inradius, the first eigenvalue, and the torsional rigidity of curvilinear polygons. Bull. Lond. Math. Soc. 42(5), 765–783 (2010)

    Article  MathSciNet  Google Scholar 

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Correspondence to Alexander Yu. Solynin.

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Communicated by Vladimir V. Andrievskii.

In the memory of Stephan Ruscheweyh, a wonderful person and an excellent mathematician whose work for the mathematical society inspired so many of us.

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Solynin, A.Y. Exercises on the Theme of Continuous Symmetrization. Comput. Methods Funct. Theory 20, 465–509 (2020). https://doi.org/10.1007/s40315-020-00331-y

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