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

Let[Figure not available: see fulltext.], where A={a₁,..., a_n} and B={b₁,...,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={α₁,...,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^*={D₁^*,..., D_j+m^*} is a system of domains realizing a maximum for the indicated sum in the family of all systems D={D₁,..., 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=(D₁^*) 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).