×

Isoperimetric properties of condenser capacity. (English) Zbl 1462.30048

Summary: For compact subsets \(E\) of the unit disk \(\mathbb{D}\) we study the capacity of the condenser \((\mathbb{D}, E)\) by means of set functionals defined in terms of hyperbolic geometry. In particular, we study experimentally the case of a hyperbolic triangle and arrive at the conjecture that of all triangles with the same hyperbolic area, the equilateral triangle has the least capacity.

MSC:

30C85 Capacity and harmonic measure in the complex plane
30F45 Conformal metrics (hyperbolic, Poincaré, distance functions)

Software:

FMMLIB2D
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Ahlfors, L. V., Conformal Invariants (1973), McGraw-Hill: McGraw-Hill New York · Zbl 0272.30012
[2] Anderson, G. D.; Vamanamurthy, M. K.; Vuorinen, M., Conformal Invariants, Inequalities and Quasiconformal Maps (1997), J. Wiley · Zbl 0885.30012
[3] Baernstein, A., Symmetrization in Analysis. With David Drasin and Richard S. Laugesen, New Mathematical Monographs, vol. 36 (2019), Cambridge University Press: Cambridge University Press Cambridge, xviii+473 pp · Zbl 1509.32001
[4] Bandle, C., Isoperimetric Inequalities and Applications, Monographs and Studies in Mathematics, vol. 7 (1980), Pitman (Advanced Publishing Program): Pitman (Advanced Publishing Program) Boston, Mass.-London, x+228 pp · Zbl 0436.35063
[5] Beardon, A. F., The Geometry of Discrete Groups, Graduate Texts in Math., vol. 91 (1983), Springer-Verlag: Springer-Verlag New York · Zbl 0528.30001
[6] Betsakos, D., Geometric versions of Schwarz’s lemma for quasiregular mappings, Proc. Am. Math. Soc., 139, 4, 1397-1407 (2011) · Zbl 1217.30021
[7] Betsakos, D.; Samuelsson, K.; Vuorinen, M., The computation of capacity of planar condensers, Publ. Inst. Math. (Belgr.) (N.S.), 75, 89, 233-252 (2004) · Zbl 1079.78024
[8] Betsakos, D.; Vuorinen, M., Estimates for conformal capacity, Constr. Approx., 16, 4, 589-602 (2000) · Zbl 0968.30011
[9] Bezrodnykh, S.; Bogatyrev, A.; Goreinov, S.; Grigoriev, O.; Hakula, H.; Vuorinen, M., On capacity computation for symmetric polygonal condensers, J. Comput. Appl. Math., 361, 271-282 (2019) · Zbl 1416.65076
[10] Bianchini, C.; Croce, G.; Henrot, A., On the quantitative isoperimetric inequality in the plane with the barycentric distance
[11] Brock, F.; Solynin, A. Yu., An approach to symmetrization via polarization, Trans. Am. Math. Soc., 352, 4, 1759-1796 (2000), (English summary) · Zbl 0965.49001
[12] Dalichau, H., Conformal Mapping and Elliptic Functions (1993), Munich
[13] De Philippis, G.; Marini, M.; Mukoseeva, E., The sharp quantitative isocapacitary inequality · Zbl 1479.31004
[14] Dubinin, V. N., Condenser Capacities and Symmetrization in Geometric Function Theory (2014), Birkhäuser · Zbl 1305.30002
[15] Garnett, J. B.; Marshall, D. E., Harmonic Measure, New Mathematical Monographs, vol. 2 (2008), Cambridge University Press: Cambridge University Press Cambridge, xvi+571 pp · Zbl 1139.31001
[16] Gehring, F. W., Inequalities for condensers, hyperbolic capacity, and extremal lengths, Mich. Math. J., 18, 1-20 (1971) · Zbl 0228.30014
[17] Greengard, L.; Gimbutas, Z., FMMLIB2D: a MATLAB toolbox for fast multipole method in two dimensions, Version 1.2 (2018)
[18] Hakula, H.; Rasila, A.; Vuorinen, M., On moduli of rings and quadrilaterals: algorithms and experiments, SIAM J. Sci. Comput., 33, 279-302 (2011) · Zbl 1368.65036
[19] Hariri, P.; Klén, R.; Vuorinen, M., Conformally Invariant Metrics and Quasiconformal Mappings, Springer Monographs in Mathematics (2020), Springer · Zbl 1450.30003
[20] Jenkins, J. A., Univalent Functions and Conformal Mapping (1958), Springer-Verlag Berlin Heidelberg · Zbl 0083.29606
[21] Keen, L.; Lakic, N., Hyperbolic Geometry from a Local Viewpoint, London Mathematical Society Student Texts, vol. 68 (2007), Cambridge University Press: Cambridge University Press Cambridge, x+271 pp · Zbl 1190.30001
[22] Kesavan, S., Symmetrization & Applications, Series in Analysis, vol. 3 (2006), World Scientific Publishing Co. Pte. Ltd.: World Scientific Publishing Co. Pte. Ltd. Hackensack, NJ, xii+148 pp · Zbl 1110.35002
[23] Lehto, O.; Virtanen, K. I., Quasiconformal Mappings in the Plane (1973), Springer: Springer Berlin · Zbl 0267.30016
[24] Liesen, J.; Séte, O.; Nasser, M. M.S., Fast and accurate computation of the logarithmic capacity of compact sets, Comput. Methods Funct. Theory, 17, 689-713 (2017) · Zbl 1381.65026
[25] Maz’ya, V., Lectures on isoperimetric and isocapacitary inequalities in the theory of Sobolev spaces, Contemp. Math., 338, 307-340 (2003) · Zbl 1062.31009
[26] Nasser, M. M.S., Numerical conformal mapping via a boundary integral equation with the generalized Neumann kernel, SIAM J. Sci. Comput., 31, 1695-1715 (2009) · Zbl 1198.30009
[27] Nasser, M. M.S., Fast solution of boundary integral equations with the generalized Neumann kernel, Electron. Trans. Numer. Anal., 44, 189-229 (2015) · Zbl 1330.65185
[28] Nasser, M. M.S.; Vuorinen, M., Computation of conformal invariants, Appl. Math. Comput., 389, Article 125617 pp. (2021) · Zbl 1474.65060
[29] Nasser, M. M.S.; Vuorinen, M., Conformal invariants in simply connected domains, Comput. Methods Funct. Theory, 20, 747-775 (2020) · Zbl 1459.65035
[30] Pólya, G.; Szegö, G., Isoperimetric Inequalities in Mathematical Physics, Annals of Mathematics Studies, vol. 27 (1951), Princeton University Press: Princeton University Press Princeton, N. J., xvi+279 pp · Zbl 0044.38301
[31] Sarvas, J., Symmetrization of condensers in n-space, Ann. Acad. Sci. Fenn., Ser. A 1 Math., I, 522 (1972), 44 pp · Zbl 0245.30013
[32] Schinzinger, R.; Laura, P. A.A., Conformal Mapping. Methods and Applications (2003), Dover Publications, Inc.: Dover Publications, Inc. Mineola, NY, xxiv+583 pp · Zbl 1063.30007
[33] Solynin, A. Yu., Solution of the Pólya-Szegö isoperimetric problem, Zap. Nauč. Semin. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI). Zap. Nauč. Semin. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), J. Sov. Math., 53, 3, 311-320 (1991), Anal. Teor. Chisel i Teor. Funktsii. 9, 140-153, 190; translation in · Zbl 0717.30020
[34] Solynin, A. Yu., Some extremal problems on the hyperbolic polygons, Complex Var. Theory Appl., 36, 3, 207-231 (1998) · Zbl 1054.30507
[35] Solynin, A. Yu.; Zalgaller, V. A., An isoperimetric inequality for logarithmic capacity of polygons, Ann. Math. (2), 159, 1, 277-303 (2004) · Zbl 1060.31001
[36] Trefethen, L. N., Analysis and design of polygonal resistors by conformal mapping, Z. Angew. Math. Phys., 35, 692-704 (1984)
[37] Vasil’ev, A., Moduli of Families of Curves for Conformal and Quasiconformal Mappings (2002), Springer-Verlag: Springer-Verlag Berlin · Zbl 0999.30001
[38] Vuorinen, M., Conformal Geometry and Quasiregular Mappings, Lecture Notes in Mathematics, vol. 1319 (1988), Springer-Verlag: Springer-Verlag Berlin, xx+209 pp · Zbl 0646.30025
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.