The Modulus of a Plane Condenser

Bagby, Thomas

doi:10.1512/iumj.1968.17.17017

Cited by 47 publications

(35 citation statements)

References 4 publications

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“…The following facts summarize Theorems 1,2,3 of [3]; see also [1]. Let (Ω, A, B) be a condenser such that ∞ ∈ Ω and cap (Ω, A, B) > 0.…”

Section: Some Facts About Condensersmentioning

confidence: 74%

“…The inequality (1.4) is proved by using only properties of extremal length. The proof of the equality statement makes use of the potential theory of condensers; in particular, the results of T.Bagby [3] play an essential role. In addition, an important tool is Brelot's generalized Dirichlet principle [5] with its equality statement.…”

Section: = −Amentioning

confidence: 99%

“…By a preliminary assumption we made at the beginning of this section, d e (K 1 ) < 1. Therefore the function u is not constant, it is continuous everywhere (see [3,Theorem 2]) and attains its maximum value V only on K 1 . This implies that u is superharmonic in a neighborhood of K 1 .…”

Section: A Shove Theorem For Elliptic Capacitymentioning

confidence: 99%

“…is the Dirichlet integral of u, dm is the 2-dimensional Lebesgue measure, and the infimum is taken over all C 1 functions in Ω with boundary values 1 at A and 0 at B; the capacity of an arbitrary condenser is defined by means of an auxiliary Möbius transformation. In Section 2 we review some facts about the capacity of condensers; we refer to [1], [3], [6], [11] for more information.…”

Section: Introductionmentioning

confidence: 99%

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Elliptic, hyperbolic, and condenser capacity; geometric estimates for elliptic capacity

Betsakos

2005

J. Anal. Math.

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Abstract. The elliptic capacity of a compact set E in the plane is defined by R.Kühnau by least energy considerations analogous to the definition of logarithmic capacity but using the elliptic distance in place of the euclidian distance. We prove an identity connecting elliptic capacity, extremal length, and the capacity of the condenser with plates E and E * , where E * is antipodal to E in the Riemann sphere. An analogous identity for hyperbolic capacity, conjectured by P.Duren and J.Pfaltzgraff, is also established. We show that elliptic capacity is always smaller than hyperbolic capacity, thus answering a question of P.Duren and R.Kühnau. We prove some geometric estimates for elliptic capacity. These estimates involve geometric transformations such as polarization, circular symmetrization, projection, and shoving.

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“…The following facts summarize Theorems 1,2,3 of [3]; see also [1]. Let (Ω, A, B) be a condenser such that ∞ ∈ Ω and cap (Ω, A, B) > 0.…”

Section: Some Facts About Condensersmentioning

confidence: 74%

Section: = −Amentioning

confidence: 99%

Section: A Shove Theorem For Elliptic Capacitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Elliptic, hyperbolic, and condenser capacity; geometric estimates for elliptic capacity

Betsakos

2005

J. Anal. Math.

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“…LEMMA 3.1 ( [Ba,Theorem 5] Thus it follows that ( 3 7 mod(A n ,g n ;C) < 2 mod(A n ,g n ;C) mod(A n , B n ;Q) ~ mod(A n , B' n ;Q'…”

Section: Mod(a N B N ; Q T ) < (H + E) Mod(a' N B' N \ < 2(h + €)mentioning

confidence: 99%

On Symmetric Quasicircles

Wu¹,

Yang²

2000

J Aust Math Soc A

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We study an important subclass of quasicircles, namely, symmetric quasicircles. Several characterizations for quasicircles, such as the reverse triangle inequality, the M -condition and the quasiconformal extension property, have been extended to symmetric quasicircles by Becker and Pommerenke and by Gardiner and Sullivan. In this paper we establish several relations among various domain constants such as quasiextremal distance constants, (local) reflection constants and (local) extension constants for this class. We also give several characterizations for symmetric quasicircles such as the strong quadrilateral inequality and the strong extremal distance property. They correspond to the quadrilateral inequality and the extremal distance property for quasicircles.1991 Mathematics subject classification (Amer. Math. Soc): primary 30C62, 3 lBxx.

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Balayage and Convergence of Rational Interpolants

Ambroladze

Wallin

1999

Journal of Approximation Theory

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We investigate the following problem: For which open simply connected domains do there exist interpolation schemes (a set of interpolation points) such that for any analytic function defined in the domain the corresponding interpolating polynomials converge to the function when the degree of the polynomials tends to infinity? We also study similar problems for rational interpolants. These problems are connected to the balayage (sweeping out) problems of measures. Academic Press

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The Modulus of a Plane Condenser

Cited by 47 publications

References 4 publications

Elliptic, hyperbolic, and condenser capacity; geometric estimates for elliptic capacity

Elliptic, hyperbolic, and condenser capacity; geometric estimates for elliptic capacity

On Symmetric Quasicircles

Balayage and Convergence of Rational Interpolants

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