Equality cases for condenser capacity inequalities under symmetrization
Dimitrios Betsakos; Stamatis Pouliasis
Annales UMCS, Mathematica (2012)
- Volume: 66, Issue: 2, page 1-24
- ISSN: 2083-7402
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topDimitrios Betsakos, and Stamatis Pouliasis. "Equality cases for condenser capacity inequalities under symmetrization." Annales UMCS, Mathematica 66.2 (2012): 1-24. <http://eudml.org/doc/268272>.
@article{DimitriosBetsakos2012,
abstract = {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.},
author = {Dimitrios Betsakos, Stamatis Pouliasis},
journal = {Annales UMCS, Mathematica},
keywords = {Steiner symmetrization; Schwarz symmetrization; polarization; condenser; capacity; Green function.; Green function},
language = {eng},
number = {2},
pages = {1-24},
title = {Equality cases for condenser capacity inequalities under symmetrization},
url = {http://eudml.org/doc/268272},
volume = {66},
year = {2012},
}
TY - JOUR
AU - Dimitrios Betsakos
AU - Stamatis Pouliasis
TI - Equality cases for condenser capacity inequalities under symmetrization
JO - Annales UMCS, Mathematica
PY - 2012
VL - 66
IS - 2
SP - 1
EP - 24
AB - 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.
LA - eng
KW - Steiner symmetrization; Schwarz symmetrization; polarization; condenser; capacity; Green function.; Green function
UR - http://eudml.org/doc/268272
ER -
References
top- [1] Armitage, D. H., Gardiner, S. J., Classical Potential Theory, Springer Monographs in Mathematics, Springer-Verlag, London, 2001. Zbl0972.31001
- [2] Bandle, C., Isoperimetric Inequalities and Applications, Monographs and Studies in Mathematics 7, Pitman, London, 1980. Zbl0436.35063
- [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. Zbl1160.35308
- [4] Betsakos, D., Symmetrization and harmonic measure, Illinois J. Math. 52, no. 3 (2008), 919-949. Zbl1180.31009
- [5] Blåsjö, 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). Zbl0067.33002
- [7] Brock, F., Solynin, A. Y., An approach to symmetrization via polarization, Trans. Amer. Math. Soc. 352, no. 4 (2000), 1759-1796. Zbl0965.49001
- [8] Cianchi, A., Fusco, N., Steiner symmetric extremals in Pólya-Szegö-type inequalities, Adv. Math. 203, no. 2 (2006), 673-728. Zbl1110.46021
- [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.[Crossref] Zbl0787.31010
- [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. Zbl0830.30014
- [13] Hayman, W. K., Multivalent Functions, Second Edition, Cambridge Tracts in Mathematics, 110, Cambridge University Press, Cambridge, 1994. Zbl0904.30001
- [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. Zbl0064.07501
- [16] Jenkins, J. A., Some uniqueness results in the theory of symmetrization II, Ann. of Math. (2) 75 (1962), 223-230. Zbl0106.04902
- [17] Kesavan, S., Symmetrization and Applications, Series in Analysis, 3, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006. Zbl1110.35002
- [18] Landkof, N. S., Foundations of Modern Potential Theory, Die Grundlehren der mathematischen Wissenschaften, vol. 180, Springer-Verlag, New York-Heidelberg, 1972. Zbl0253.31001
- [19] Ohtsuka, M., Dirichlet Problem, Extremal Length and Prime Ends, Van Nostrand Reinhold, 1970. Zbl0197.08404
- [20] Pólya, G., Szegö, G., Isoperimetric Inequalities in Mathematical Physics, Annals of Mathematics Studies, no. 27, Princeton University Press, Princeton, N. J., 1951. Zbl0044.38301
- [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. Zbl0548.31003
- [23] Väisälä, J., Lectures on n-Dimensional Quasiconformal Mappings, Lecture Notes in Mathematics, vol. 229, Springer-Verlag, Berlin-New York, 1971. Zbl0221.30031
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