Volume 14, Issue 3

The paper deals with the Robin function and eigenvalue problems generated by the Robin operator. First we show that Green's function of an 𝑛-fold connected domain is the Robin function of an appropriate simply connected domain. The main part of the paper deals with eigenvalue problems for the Robin operator: the mixed Stekloff eigenvalue problem and the membrane problem with mixed boundary conditions. Isoperimetric inequalities are proved for the sum of reciprocal eigenvalues.

The Virasoro algebra is a central extension of the Witt algebra, the complexified Lie algebra of the sense preserving diffeomorphism group of the circle Diff 𝑆 1 . It appears in Quantum Field Theories as an infinite dimensional algebra generated by the coefficients of the Laurent expansion of the analytic component of the momentum-energy tensor, Virasoro generators. The background for the construction of the theory of unitary representations of Diff 𝑆 1 is found in the study of Kirillov's manifold Diff 𝑆 1 =𝑆 1 . It possesses a natural Kählerian embedding into the universal Teichmüller space with the projection into the moduli space realized as an infinite-dimensional body of the coefficients of univalent quasiconformally extendable functions. The differential of this embedding leads to an analytic representation of the Virasoro algebra based on Kirillov's operators. In this paper we overview several interesting connections between the Virasoro algebra, Teichmüller theory, Löwner representation of univalent functions, and propose a finite-dimensional grading of the Virasoro algebra such that the grades form a hierarchy of finite dimensional algebras which, in their turn, are the first integrals of Liouville partially integrable systems for coefficients of univalent functions.

It is shown that any uncountable commutative group (𝐺, +) admits a representation 𝐺 = ∪{𝐺 𝑗 : 𝑗 ∈ 𝐽}, where 𝐽 is a countable set, 𝐺 𝑗 is a subgroup of 𝐺 for each 𝑗 ∈ 𝐽, and 𝐺/𝐺 𝑗 is uncountable. A related fact for uncountable solvable groups is established in terms of invariant (quasiinvariant) measures and an application of the above-mentioned results to the measure extension problem is given.

Systems of higher order nonlinear functional-differential inequalities arising in the theory of boundary value problems as well as in the stability theory are considered on a finite and an infinite interval. On a finite interval, a priori estimates are obtained for solutions of these systems satisfying boundary conditions of periodic type, and on an infinite interval, a priori estimates are established for the derivatives of bounded solutions.

Unimprovable conditions of the existence and uniqueness of doubly periodic solutions are established for nonlinear hyperbolic equations of higher order with two independent variables.

We study the widest possible generalizations of the complex analytic function theory agreeing with the Riemann and Weierstrass approaches.

We give an alternate and simpler proof of the important theorem stating that all invariant distances on the universal Teichmüller space 𝐓 coincide, and solve for 𝐓 the problem of Kra on isometric embeddings of a disk into Teichmüller spaces.

We describe the dual space of the boundary trace space for functions with a finite Dirichlet integral for a domain with a vertex of an isolated cusp at the boundary. This leads to conditions of solvability of the Neumann problem for elliptic equations of second order. In particular, we give an explicit necessary and sufficient condition for 𝑞 such that the Neumann problem is solvable if the boundary function is in 𝐿 𝑞 over the boundary of a domain with an outer peak.

We investigate stagnation zones of solutions of partial differential elliptic equations. With the domain width being much less than its length and special boundary conditions, these solutions can be almost constant over large subdomains. Such domains are called stagnation zones (𝑠-zones). We estimate the size, the location of these 𝑠-zones and study the behavior of solutions on 𝑠-zones.

The unimprovable sufficient conditions are established for the unique solvability of the periodic problem where ℓ 𝑖 : 𝐶([0, 𝑤]) → 𝐿([0, 𝑤]) are linear bounded operators, 𝑞 ∈ 𝐿([0, 𝑤]) and 𝑐 𝑗 ∈ 𝑅

In the space , 𝑛-dimensional surfaces are considered having the parametrizations which are functions of the Sobolev class with 𝑝 > 𝑛. The first and the second fundamental tensor are defined. The Peterson–Codazzi equations for such functions are understood in some generalized sense. It is proved that if the first and the second fundamental tensor of one surface are close to the first and, respectively, to the second fundamental tensor of the other surface, then these surfaces will be close up to the motion of the space . A difference between the fundamental tensors and the nearness of the surfaces are measured with the help of suitable 𝑊-norms. The proofs are based on a generalization of Frobenius' theorem about completely integrable systems of the differential equations which was proved by Yu. E. Borovskiĭ. The integral representations of functions by differential operators with complete integrability condition are used, which were elaborated by the author in his other works.

For a higher order differential equation with the polyharmonic operator, the Dirichlet and Riquier boundary value problems are studied in some polyhedral angles. Uniqueness theorems for solutions with a bounded “energy integral” of the corresponding BVPs are proved. Recurrent formulas are constructed for representation of fundamental solutions and Green's functions. The asymptotic behavior of solutions at infinity is studied.

Originally I. N. Vekua's theory of generalized analytic functions dealt only with linear systems of partial differential equations in the plane. The present paper shows why I. N. Vekua's ideas are also fruitful for the solution of linear and non-linear partial differential equations in higher dimensions. One of the highlights of the theory of generalized analytic functions in the plane is the reduction of boundary value problems for general (linear or nonlinear) equations to boundary value problems for holomorphic functions using the well-known weakly singular and strongly singular 𝑇- and П-operators, respectively. The present paper is mainly aimed at reducing boundary value problems in higher dimensions to boundary value problems for monogenic functions.