The dependence on parameters of the modulus problem for families of several classes of curves

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Let[Figure not available: see fulltext.], 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 b. 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 modulus U=U(α,A,B) of this problem we have the equality {Mathematical expression}, where D*={D1*,..., Dj+m*} is a system of domains realizing a maximum for the indicated sum in the family of all systems D={D1,..., Dj+m} of domains, associated with the family H (by U(Di)) we denote the modulus of the domain Di, associated with the class Hi). In the present paper we investigate the manner in which U=U(α,A,B) and the moduli U=(D1*) depend on the parameters αi, ak, b; moreover, we consider the conditions under which some of the doubly connected domains Di*, 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).

Original languageEnglish
Pages (from-to)2131-2139
Number of pages9
JournalJournal of Soviet Mathematics
Issue number4
StatePublished - Aug 1987


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