We consider energy minimization problems in classes of real signed measures consistent with a finite or infinite collection of signed sets that are allowed to intersect each other, and with nonzero real numbers attached to these sets.

We investigate pairwise products of moduli of families of curves on a Riemannian Möbius strip and obtain estimates for these products. As one of the factors, we consider the modulus of a family of arcs from a broad class of families of this sort (for each of these families, we determine the modulus and extremal metric).

We present results on the moduli and extremal metrics of families of curves on certain nonorientable or twisted Riemannian manifolds. We also introduce and investigate weighted l-moduli of families of curves and the corresponding extremal metrics.

The paper gives a survey of results completely solving the differential contour-solid problem of analytic functions in an open subset G of the complex plane, which was discussed as an open problem at the informal seminar held in 1994 in Zurich by participants of the International Congress of Mathematicians. This problem has a long prehistory and includes questions (unsolved at that time) concerning conditions of validity of differential contour-solid statements on the continuous extendability of a derivative to boundary points and on the differentiability of an analytic function at boundary points of the set G. In June, 1995, the author established that these statements are always true for arbitrary open sets G and any boundary points. These and more general theorems are given in this paper. We also present some other results, among which contour-solid theorems and a representation formula for the generalized solution of the Dirichlet problem for the derivative of a function should be mentioned.

The paper contains a generalization of the results obtained earlier concerning the subharmonic extension of functions and the extension of functions in the Hardy-Orlicz classes. We give the unified proofs of these results.