Equality cases for condenser capacity inequalities under symmetrization

Betsakos, Dimitrios ; Pouliasis, Stamatis

Annales Universitatis Mariae Curie-Skłodowska. Mathematica , Tome 66 (2012) no. 2.

Résumé

It is well known that certain transformations decrease the capacity of a condenser. We prove equality statements for the condenser capacity inequalities under symmetrization and polarization without connectivity restrictions on the condenser and without regularity assumptions on the boundary of the condenser.

Détail

Keywords: Steiner symmetrization, Schwarz symmetrization, polarization, condenser, capacity, Green function

Bibliographie

[1] Armitage, D. H., Gardiner, S. J., Classical Potential Theory, Springer Monographs in Mathematics, Springer-Verlag, London, 2001.

[2] Bandle, C., Isoperimetric Inequalities and Applications, Monographs and Studies in Mathematics 7, Pitman, London, 1980.

[3] Betsakos, D., Equality cases in the symmetrization inequalities for Brownian transition functions and Dirichlet heat kernels, Ann. Acad. Sci. Fenn. Math. 33, no. 2 (2008), 413-427.

[4] Betsakos, D., Symmetrization and harmonic measure, Illinois J. Math. 52, no. 3 (2008), 919-949.

[5] Blasjo, V., The isoperimetric problem, Amer. Math. Monthly 112, no. 6 (2005), 526-566.

[6] Brelot, M., Etude et extensions du principe de Dirichlet, Ann. Inst. Fourier, Grenoble 5, 371-419 (1954).

[7] Brock, F., Solynin, A. Y., An approach to symmetrization via polarization, Trans. Amer. Math. Soc. 352, no. 4 (2000), 1759-1796.

[8] Cianchi, A., Fusco, N., Steiner symmetric extremals in Pólya-Szego-type inequalities, Adv. Math. 203, no. 2 (2006), 673-728.

[9] Dubinin, V. N., Transformation of functions and the Dirichlet principle, (Russian) Mat. Zametki 38, no. 1 (1985), 49-55; translation in Math. Notes 38 (1985), 539-542.

[10] Dubinin, V. N., Transformation of condensers in space, (Russian) Dokl. Akad. Nauk SSSR 296, no. 1 (1987), 18-20; translation in Soviet Math. Dokl. 36 (1988), no. 2, 217-219.

[11] Dubinin, V. N., Capacities and geometric transformations of subsets in n-space, Geom. Funct. Anal. 3, no. 4 (1993), 342-369.

[12] Dubinin, V. N., Symmetrization in the geometric theory of functions of a complex variable, (Russian), Uspekhi Mat. Nauk 49 (1994), no. 1(295), 3-76; translation in Russian Math. Surveys 49, no. 1 (1994), 1-79.

[13] Hayman, W. K., Multivalent Functions, Second Edition, Cambridge Tracts in Mathematics, 110, Cambridge University Press, Cambridge, 1994.

[14] Helms, L. L., Potential Theory, Universitext, Springer-Verlag, London, 2009.

[15] Jenkins, J. A., Some uniqueness results in the theory of symmetrization, Ann. of Math. (2) 61 (1955), 106-115.

[16] Jenkins, J. A., Some uniqueness results in the theory of symmetrization II, Ann. of Math. (2) 75 (1962), 223-230.

[17] Kesavan, S., Symmetrization and Applications, Series in Analysis, 3, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006.

[18] Landkof, N. S., Foundations of Modern Potential Theory, Die Grundlehren der mathematischen Wissenschaften, vol. 180, Springer-Verlag, New York-Heidelberg, 1972.

[19] Ohtsuka, M., Dirichlet Problem, Extremal Length and Prime Ends, Van Nostrand Reinhold, 1970.

[20] Pólya, G., Szego, G., Isoperimetric Inequalities in Mathematical Physics, Annals of Mathematics Studies, no. 27, Princeton University Press, Princeton, N. J., 1951.

[21] Sarvas, J., Symmetrization of condensers in n-space, Ann. Acad. Sci. Fenn. Ser. A I no. 522 (1972), 44 pp.

[22] Shlyk, V. A., A uniqueness theorem for the symmetrization of arbitrary condensers, (Russian) Sibirsk. Mat. Zh. 23, no. 2 (1982), 165-175. Siberian Math. J. 23 (1982), 267-276.

[23] Vaisala, J., Lectures on n-Dimensional Quasiconformal Mappings, Lecture Notes in Mathematics, vol. 229, Springer-Verlag, Berlin-New York, 1971.