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.
@article{bwmeta1.element.ojs-doi-10_17951_a_2012_66_2_1-24,
author = {Dimitrios Betsakos and Stamatis Pouliasis},
title = {Equality cases for condenser capacity inequalities under symmetrization},
journal = {Annales Universitatis Mariae Curie-Sk\l odowska, sectio A -- Mathematica},
volume = {66},
year = {2012},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.ojs-doi-10_17951_a_2012_66_2_1-24}
}
Dimitrios Betsakos; Stamatis Pouliasis. Equality cases for condenser capacity inequalities under symmetrization. Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica, Tome 66 (2012) . http://gdmltest.u-ga.fr/item/bwmeta1.element.ojs-doi-10_17951_a_2012_66_2_1-24/
Armitage, D. H., Gardiner, S. J., Classical Potential Theory, Springer Monographs in Mathematics, Springer-Verlag, London, 2001.
Bandle, C., Isoperimetric Inequalities and Applications, Monographs and Studies in Mathematics 7, Pitman, London, 1980.
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.
Betsakos, D., Symmetrization and harmonic measure, Illinois J. Math. 52, no. 3 (2008), 919-949.
Blasjo, V., The isoperimetric problem, Amer. Math. Monthly 112, no. 6 (2005), 526-566.
Brelot, M., Etude et extensions du principe de Dirichlet, Ann. Inst. Fourier, Grenoble 5, 371-419 (1954).
Brock, F., Solynin, A. Y., An approach to symmetrization via polarization, Trans. Amer. Math. Soc. 352, no. 4 (2000), 1759-1796.
Cianchi, A., Fusco, N., Steiner symmetric extremals in Pólya-Szego-type inequalities, Adv. Math. 203, no. 2 (2006), 673-728.
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.
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.
Dubinin, V. N., Capacities and geometric transformations of subsets in n-space, Geom. Funct. Anal. 3, no. 4 (1993), 342-369.
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.
Hayman, W. K., Multivalent Functions, Second Edition, Cambridge Tracts in Mathematics, 110, Cambridge University Press, Cambridge, 1994.
Helms, L. L., Potential Theory, Universitext, Springer-Verlag, London, 2009.
Jenkins, J. A., Some uniqueness results in the theory of symmetrization, Ann. of Math. (2) 61 (1955), 106-115.
Jenkins, J. A., Some uniqueness results in the theory of symmetrization II, Ann. of Math. (2) 75 (1962), 223-230.
Kesavan, S., Symmetrization and Applications, Series in Analysis, 3, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006.
Landkof, N. S., Foundations of Modern Potential Theory, Die Grundlehren der mathematischen Wissenschaften, vol. 180, Springer-Verlag, New York-Heidelberg, 1972.
Ohtsuka, M., Dirichlet Problem, Extremal Length and Prime Ends, Van Nostrand Reinhold, 1970.
Pólya, G., Szego, G., Isoperimetric Inequalities in Mathematical Physics, Annals of Mathematics Studies, no. 27, Princeton University Press, Princeton, N. J., 1951.
Sarvas, J., Symmetrization of condensers in n-space, Ann. Acad. Sci. Fenn. Ser. A I no. 522 (1972), 44 pp.
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.
Vaisala, J., Lectures on n-Dimensional Quasiconformal Mappings, Lecture Notes in Mathematics, vol. 229, Springer-Verlag, Berlin-New York, 1971.