### Abstract

Let[Figure not available: see fulltext.], where A={a_{1},..., a_{n}} and B={b_{1},...,b_{m}} are systems of distinguished points, and let H be a family of homotopic classes H_{i}, i=1, ..., j + m, of closed Jordan curves in C, where the classes H_{j+ℓ}, ℓ=1, ..., m, consist of curves that are homotopic to a point curve in b_{ℓ}. Let α={α_{1},...,α_{j}+m} be a system of positive numbers. By P=P(α,A,B) we denote the extremal-metric problem for the family H and the numbers α: for the modulus U=U(α,A,B) of this problem we have the equality {Mathematical expression}, where D^{*}={D_{1}^{*},..., D_{j+m}^{*}} is a system of domains realizing a maximum for the indicated sum in the family of all systems D={D_{1},..., D_{j+m}} of domains, associated with the family H (by U(D_{i})) we denote the modulus of the domain D_{i}, associated with the class H_{i}). In the present paper we investigate the manner in which U=U(α,A,B) and the moduli U=(D_{1}^{*}) depend on the parameters α_{i}, a_{k}, b_{ℓ}; moreover, we consider the conditions under which some of the doubly connected domains D_{i}^{*}, i=1,..., j, from the system D^{*} turn out to be degenerate (Theorems 1-3). In particular, one obtains an expression for the gradient of the function M, as function of the parameter a=a_{k} (Theorem 4). One gives some applications of the obtained results (Theorem 5).

Original language | English |
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Pages (from-to) | 2131-2139 |

Number of pages | 9 |

Journal | Journal of Soviet Mathematics |

Volume | 38 |

Issue number | 4 |

DOIs | |

State | Published - Aug 1987 |