Extremal Decomposition of Spatial Domains

Classical results by Lavrent'ev, Alenitsyn, and Kufarev on the product of the conformal radii of planar nonoverlapping domains, as well as a recent result on an extremal decomposition with free poles on the circle are extended to the case of domains in the n-dimensional Euclidean space, n 3, with conformal radii replaced by harmonic ones. The main tool used in the proof is the technique of generalized reduced modules. We also apply the method of extremal metrics and dissymmetrization of families of curves. Bibliography: $5 titles.     

On symmetric configurations in some problems on extremal decomposition. II

Some problems on extremal decomposition in families of nonoverlapping domains containing systems of biangles with free vertices on a circle are considered. Simultaneously, some progress in solving the classical problem on the maximum of a well-known conformal invariant is achieved. This exhibits the role of symmetric configurations in extremal problems under consideration. Bibliography: 11 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 357, 2008, pp. 158–179.     

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. Dedicated to the 100th anniversary of G. M. Goluzin’s birthday __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 337, 2006, pp. 191–211.     

On extremal problems for symmetric disjoint domains

We study two extremal problems for the product of powers of conformal radii of symmetric disjoint domains. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 2, pp. 179–185, February, 1997.     

On symmetric configurations in some problems on extremal decomposition

Some problems on extremal decomposition in families of systems of nonoverlapping domains of various types with free parameters are studied. The functionals considered contain a sum of the reduced modules of biangles. For the problems under consideration, the existence of extremal configurations with appropriate symmetry is investigated. Bibliography: 8 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 350, 2007, pp. 160–172.     

Sharp estimates for inner radii of systems of nonoverlapping domains and open sets

We study extremal problems of the geometric theory of functions of a complex variable. Sharp upper estimates are obtained for the product of inner radii of disjoint domains and open sets with respect to equiradial systems of points. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 12, pp. 1601–1618, December, 2007.     

The maximum of the product of conformal, radii of nonoverlapping domains in the disk

Two problems on the extremal decomposition of the unit disk are considered. It is shown that the associated quadratic differentials in these problems have at least two distinct zeros on the unit circle. Bibliography: 7 titles.Dedicated to the 90th anniversary of G. M. Goluzin's birthTranslated fromZapiski Nauchnykh Seminarov POMI, Vol. 237, 1997, pp. 105–118.     

The maximum of the conformal radius in the families of domains satifying additional conditions

We solve the problems on the maximum of the conformal radius R(D,1) in the family D(R₀) of all simply connected domains D ⊃ ℂ containing the points 0 and 1 and having a fixed value of the conformal radius R(D,0)=R₀, and in the family D(R₀, ρ) of domains from D(R₀) with given hyperbolic distance ρ=ρ_D(0,1) between 0 and 1. Analogs of the mentioned problems for doubly-connected domains with given conformal module are considered. Solution of the above problems is based on results of general character in the theory of problems of extremal decomposition and related module problems. Bibliography: 7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 226, 1996, pp. 93–108.     

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 c_k, 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.     

Connections Between Different Problems on Extremal Decomposition

Connections between various problems on the maximum of the product of conformal radii of nonoverlapping simply connected domains are established. These connections are used for solving two problems of such type. Bibliography: 7 titles.     

On symmetric configurations in some problems on extremal decomposition. III

Some problems on extremal decomposition in families of non overlapping domains containing systems of biangles with distinguished vertices are considered. The present work continues earlier author’s investigations. Bibliography: 8 titles.     

Geometric properties of extremal decompositions and estimates for the modulus of families of curves in an annulus

We study the geometric properties of the boundary for problems on extremal decomposition into two domains of an annulus or a rectangle with a barrier removed. Sharp estimates of moduli are proved for certain families of curves bending around the barrier in a canonical domain. Bibliography: 17 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 204, 1993, pp. 93–114. Translated by A. Yu. Solynin.     

On the product of inner radii of symmetric nonoverlapping domains

Some results concerning extremal problems for nonoverlapping domains with free poles on the unit circle, known for the simply connected case, are generalized to the case of multiply connected domains.     

Inequalities for inner radii of disjoint domains and open sets

In geometric function theory there is a well-known field related to estimates for functionals that depend on the inner radii of nonoverlapping domains. Many authors has contributed to the development of this field. Today, great attention is being paid to the study of extremal problems for nonoverlapping domains with free poles. In this paper, one result of this kind is generalized and strengthened.     

Extremal configurations of certain problems on the capacity and harmonic measure

We study certain extremal problems concerning the capacity of a condenser and the harmonic measure of a compact set. In particular, we answer in the negative Tamrazov's question on the minimum of the capacity of a condenser. We find the solution to Dubinin's problem on the maximum of the harmonic measure of a boundary set in the family of domains containing no “long” segments of given inclination. It is also shown that the segment [1-L, 1] has the maximal harmonic measure at the point z=0 among all curves γ={z=z(t), 0≤t≤1}, z(0)=1, that lie in the unit disk and have given length L, 0<L<1. The proofs are based on Baernstein's method of *-functions, Dubinin's dissymmetrization method, and the method of extremal metrics. Bibliography: 21 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 226, 1996, pp. 170–195.     

Application of a separating transformation to estimates of inner radii of open sets

We obtain solutions of new extremal problems of the geometric theory of functions of a complex variable related to estimates for the inner radii of nonoverlapping domains. Some known results are generalized to the case of open sets. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 10, pp. 1313–1321, October, 2007.     

Representation of extremal functions in the classes Eq in terms of Green's and Neumann's functions

We consider extremal problems in the classes E_q, q≥1, in finitely connected domains and we establish a representation of extremal functions in terms of Green's and Neumann's functions. The equations which we obtain constitute an extension to the multiply connected case of known representation equations for extremal functions in the classes H_q in the unit disk.     

On the complexity of some subgraph problems

We study the complexity of the problem of deciding the existence of a spanning subgraph of a given graph, and of that of finding a maximum (weight) such subgraph. We establish some general relations between these problems, and we use these relations to obtain new NP-completeness results for maximum (weight) spanning subgraph problems from analogous results for existence problems and from results in extremal graph theory. On the positive side, we provide a decomposition method for the maximum (weight) spanning chordal subgraph problem that can be used, e.g., to obtain a linear (or O(nlogn)) time algorithm for such problems in graphs with vertex degree bounded by 3.     

Properties of associated quadratic differentials in certain extremal problems

One investigates properties of extremal configurations in the problem of the maximum of the n-th diameter d_n (E) in the family of continua E of unit capacity and in the problem of the maximum of the corresponding conformal invariant in a family of nonoverlapping domains. It is shown that for these problems the associated quadratic differentials do not have multiple zeros and that the Fekete points of the extremal continuum of the first of the mentioned problems are simple poles of the associated differential. It is also shown that the quadratic differential, associated with a support function of class , does not have zeros of multiplicity >2. The paper continues the previous investigation of the author (MR 88c:30028).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 168, pp. 85–97, 1988.     

Radial projection and the poincaré metric

Some estimates for the Poincaré metric of a planar domain are obtained in terms of the radial projection of the complement of the domain onto the unit circle. These estimates allow us, in particular, to strengthen the well-known Lavrent'ev theorem on the product of conformal radii of nonoverlapping domains. The proofs use the polarization transformation. Bibliography: 18 titles.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 237, 1997, pp. 148–160.This research was partially supported by the Russian Foundation for Basic Research, grant 97-01-00259.