Abstract
Let
, where A={a1,..., an} and B={b1,...,bm} are systems of distinguished points, and let H be a family of homotopic classes Hi, i=1, ..., j + m, of closed Jordan curves in C, where the classes Hj+ℓ, ℓ=1, ..., m, consist of curves that are homotopic to a point curve in bℓ. Let α={α1,...,αj+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 modulusU=U(α,A,B) of this problem we have the equality\(U = \sum\limits_{i = 1}^{j + m} {\alpha _i^2 \mathcal{U}(\mathcal{D}_i^* )}\), whereD *={D *1 ,...,D * j+m } is a system of domains realizinga maximum for the indicated sum in the family of all systemsD={D 1,...,D j+m } of domains, associated with the family H (byU(D i )) we denote the modulus of the domain Di, associated with the class Hi). In the present paper we investigate the manner in whichU=U(α,A,B) and the moduliU=(D * 1 ) depend on the parameters αi, ak, bℓ; 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=ak (Theorem 4). One gives some applications of the obtained results (Theorem 5).
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Literature cited
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Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 144, pp. 136–148, 1985.
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Solynin, A.Y. The dependence on parameters of the modulus problem for families of several classes of curves. J Math Sci 38, 2131–2139 (1987). https://doi.org/10.1007/BF01474447
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DOI: https://doi.org/10.1007/BF01474447