Extremal Decomposition Problems Research Articles

ON the p-Harmonic Robin Radius in the Euclidean Space

For p > 1, the notion of the p-harmonic Robin radius of a domain in the space ℝ n , n ≥ 2, is introduced. In the case where the corresponding part of the boundary degenerates, the Robin–Neumann radius is considered. The monotonicity of the p-harmonic Robin radius under some deformations of a domain is proved. Some extremal decomposition problems in the Euclidean space are solved. The definitions and proofs are based on the technique of moduli of curve families. Bibliography: 23 titles.

The Module Method in a Certain General Extremal Decomposition Problem

An extension of the method of modules of curve families to extremal decomposition problems in which the associated quadratic differentials have poles of arbitrary orders is considered.

Extremal decomposition problems in the Euclidean space

Composition principles for reduced moduli are extended to the case of domains in the $n$-dimensional Euclidean space, $n>2$. As a consequence analogues of extremal decomposition theorems of Kufarev, Dubinin and Kirillova in the planer case are obtained.

Tangency Conditions for Trajectories of Two Quadratic Differentials

The dependence of values of functionals occurring in some extremal decomposition problems on the location of the poles of the associated quadratic differentials is established. The proof is based on an analysis of geometric properties of trajectories of two quadratic differentials.

Quadratic differentials with strip domains in the structure of trajectories in some extremal decomposition problems

Problems on extremal decomposition in which the associated quadratic differentials possess the property indicated in the title are solved. The results obtained supplement the classical results in problems under consideration. Bibliography: 15 titles.

On extremal decomposition problems

The capacity approach and symmetrization method arc adapted to some extremal decomposition problems on the unit disk or an annulus. The problems on the maximum product of the interior radii of pairwise nonoverlapping domains and the maximum product of the Robin radii of such domains are considered. New invariants with respect to the Mobius transformations of the Riemann sphere are introduced. In particular, for these invariants problems on extremal decomposition with free poles on the unit circle are investigated. Bibliography: 19 titles.

An extremal metric approach to some problems on extremal decomposition

The paper presents a simple extremal metric approach to problems on extremal decomposition in families of systems of nonoverlapping domains of different types. The approach is based on using general properties of trajectories of the associated quadratic differentials for simpler extremal decomposition problems. Bibliography: 8 titles.

Capacities of condensers, generalizations of Grötzsch Lemmas, and symmetrization

Principles of symmetry and composition for the conformal capacity of generalized condensers are considered. Interrelations among these principles, the classical Grotzsch lemmas, and certain types of symmetrization are discussed. Applications to estimating the logarithmic and hyperbolic capacities of compacts and to extremal decomposition problems are presented. Bibliography: 27 titles.

The Method of Extremal Metric in Extremal Decomposition Problems with Free Parameters

Let a1,..., an be a system of distinct points on the z-sphere \({\bar C}\), and let \(\mathcal{D}\) be a system of all non-overlapping simply-connected domains D1,..., Dn on \({\bar C}\) such that ak ∈ Dk, k = 1,..., n. Let M (Dk, ak) be the reduced module of the domain Dk with respect to the point ak ∈ Dk. In the present paper, we solve some problems concerning the maximum of weighted sums of the reduced modules M (Dk, ak) in certain families of systems of domains {Dk} described above, where the systems of points {ak} satisfy prescribed symmetry conditions. In each case, the proof is based on an explicit construction of an admissible metric of the module problem, which is equivalent to the extremal problem under consideration, from known extremal metrics of simpler module problems. Bibliography: 7 titles.

On Extremal Decomposition Problems in Families of Systems of Domains of General Type

We prove a theorem extending results of the theory of extremal decomposition problems to families of systems of domains of general type. The considered families of systems of domains contain domains similar in the small to end and strip domains of a quadratic differential having poles of arbitrary orders n k≥ 3 at some marked points ck, k = 1,...,p. In this case, we give a simple definition of reduced modules for the considered systems of domains. Some other definitions for the treated systems of domains are also considered. Some examples are given illustrating the possibilities of applications of the theorem obtained in the problems on extremal decomposition. Bibliography: 13 titles.

Extremal Decomposition Problems in the Space of Riemann Surfaces

An extension of a theorem on extremal decomposition of a Riemann surface is obtained. The problem of extremal decomposition is extended from the case of a Riemann surface $$\Re $$ with a prescribed set $$P \subset \Re $$ of distinguished points to the case of the Teichmuller space $$T_{\Re '} $$ $$\widehat\Re $$ corresponding to $$\Re $$ under quasiconformal homeomorphisms f. For the functional $$\mathcal{M}$$ of our problem on extremal decomposition of a surface $$\widehat\Re $$ , we consider a function $$\mathcal{M}^ * (x)$$ expressing the dependence of the extremal value of $$\mathcal{M}$$ on a point $$x \in T_{\Re '} $$ . Differentiation formulas for the function $$\mathcal{M}^ * (x)$$ are derived. These formulas are different and depend on the genus g of the surface $$\mathcal{M}$$ . The case where the function $$\mathcal{M}^ * (x)$$ is pluriharmonic is considered. Bibliography: 8 titles.