×

Condenser energy under holomorphic motions. (English) Zbl 1273.31001

A condenser in the extended complex plane \(\widehat{\mathbb C}\) is a set \(D\setminus K\), where \(D\) is a domain and \(K\) is a compact subset of \(D\) such that \(D\setminus K\) is connected. The compact sets \(\widehat{\mathbb C}\setminus D\) and \(K\) are the plates of the condenser. Let \(u\) be the solution of the generalized Dirichlet problem on \(D\setminus K\) with boundary values \(0\) on \(\partial D\) and \(1\) on \(\partial K\). The equilibrium energy of the condenser is given by \[ \text{md}(D\setminus K)=2\pi\left (\int_{D\setminus K}|\nabla u|^2\right )^{-1}. \] The author studies the behavior of the equilibrium energy when one of the plates moves under a holomorphic motion \(f_\lambda(z)\) (namely a function holomorphic in \(\lambda\) and injective in \(z\)). The main result is that the function \(\lambda\mapsto \text{md}(f_\lambda(D)\setminus K)\) is superharmonic. Moreover, a characterization of the case of harmonicity is given. In the case where the holomorphic motion is a dilation (\(f_\lambda(z)=\lambda z\)), the author proves that harmonicity occurs if and only if the condenser is essentially an annulus centered at the origin.

MSC:

31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions
31A05 Harmonic, subharmonic, superharmonic functions in two dimensions
30C85 Capacity and harmonic measure in the complex plane
PDFBibTeX XMLCite
Full Text: Euclid

References:

[1] D. H. Armitage and S. J. Gardiner, Classical potential theory , Springer Monographs in Mathematics, Springer, London, 2001. · Zbl 0972.31001 · doi:10.1007/978-1-4471-0233-5
[2] K. Astala and G. J. Martin, Holomorphic motions , Papers on analysis, Rep. Univ. Jyväskylä Dep. Math. Stat., vol. 83, Univ. Jyväskylä, Jyväskylä, 2001, pp. 27-40.
[3] S. Axler, P. Bourdon and W. Ramey, Harmonic function theory , 2nd ed., Springer-Verlag, New York, 2001. · Zbl 0959.31001
[4] T. Bagby, The modulus of a plane condenser , J. Math. Mech. 17 (1967), 315-329. · Zbl 0163.35204
[5] D. Betsakos, On the equilibrium measure and the capacity of certain condensers , Illinois J. Math. 44 (2000), no. 3, 681-689. · Zbl 0959.31002
[6] D. Betsakos, Elliptic, hyperbolic, and condenser capacity; geometric estimates for elliptic capacity , J. Anal. Math. 96 (2005), 37-55. · Zbl 1091.30021 · doi:10.1007/BF02787824
[7] C. J. Earle and S. Mitra, Variation of moduli under holomorphic motions , Contemp. Math., vol. 256, Amer. Math. Soc., Providence, RI, 2000, pp. 39-67. · Zbl 0972.30005 · doi:10.1090/conm/256/03996
[8] V. N. Dubinin, 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 (1994), no. 1, 1-79. · doi:10.1070/RM1994v049n01ABEH002002
[9] V. N. Dubinin, Generalized condensers and the asymptotics of their capacities under a degeneration of some plates (Russian), Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 302 (2003), Anal. Teor. Chisel i Teor. Funkts. 19 , 38-51, 198-199; translation in J. Math. Sci. (N. Y.) 129 (2005), no. 3, 3835-3842. · Zbl 1241.31003 · doi:10.1007/s10958-005-0319-4
[10] V. N. Dubinin and D. Karp, Capacities of certain plane condensers and sets under simple geometric transformations , Complex Var. Elliptic Equ. 53 (2008), no. 6, 607-622. · Zbl 1151.31002 · doi:10.1080/17476930701734292
[11] G. B. Folland, Real analysis; modern techniques and their applications , 2nd ed., Wiley, New York, 1999. · Zbl 0924.28001
[12] L. L. Helms, Introduction to potential theory , Wiley-Interscience, New York, 1969. · Zbl 0188.17203
[13] S. G. Krantz and H. R. Parks, A primer of real analytic functions , 2nd ed., Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Boston, Boston, MA, 2002. · Zbl 1015.26030 · doi:10.1007/978-0-8176-8134-0
[14] R. Kühnau, Randeffekte beim elektrostatischen Kondensator [ Boundary effects in the electrostatic condenser ] (German), Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 254 (1998), Anal. Teor. Chisel i Teor. Funkts. 15 , 132-144, 246-247; translation in J. Math. Sci. (N. Y.) 105 (2001), no. 4, 2210-2219. · Zbl 0984.30015 · doi:10.1023/A:1011385226334
[15] N. S. Landkof, Foundations of modern potential theory , Springer-Verlag, New York, 1972. · Zbl 0253.31001
[16] R. Laugesen, Extremal problems involving logarithmic and Green capacity , Duke Math. J. 70 (1993), no. 2, 445-480. · Zbl 0788.31002 · doi:10.1215/S0012-7094-93-07009-3
[17] S. Pouliasis, Invariance of Green equilibrium measure on the domain , to appear in Filomat. · Zbl 1404.31014
[18] T. Ransford, Potential theory in the complex plane , Cambridge University Press, Cambridge, 1995. · Zbl 0828.31001 · doi:10.1017/CBO9780511623776
[19] Z. Slodkowski, Holomorphic motions and polynomial hulls , Proc. Amer. Math. Soc. 111 (1991), 347-355. · Zbl 0741.32009 · doi:10.2307/2048323
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.