Sergey Klimentov

• Доктор физико-математических наук
• Southern Federal University

This paper is a review on bending of surfaces of positive external curvature of genus p ≥.

We address two-dimensional singular integral equations widely used for constructing and investigating the solutions to the general linear first-order elliptic systems in the 2D domains. Proving the solvability of such integral equations with the use of the Calderón–Zygmund theorem brings us at the statements about the existence of the solution belo...

We prove the next result. If two isometric regular surfaces with regular boundaries, of an arbitrary finite genus, and positive Gaussian curvature in the three-dimensional Euclidean space, consist of two congruent arcs corresponding under the isometry (lying on the boundaries of these surfaces or inside these surfaces) then these surfaces are congr...

In the paper we consider representations of the second kind for solutions to the linear general uniform first order elliptic system in the unit circle D = (z:|z| ≤ 1) written in terms of complex functions: D w ≡ ∂ z w+q 1 (z)∂ z w+q 2 (z)∂ z w+A(z)w+B(z)wR(z), where w = w(z) = u(z)+iv(z) is the sought complex function, q 1 (z) and q 2 (z) are given...

We consider a second kind representation for solutions to a first order general uniformly elliptic linear system in a simply connected plane domain \( G \) with the \( W^{k-\frac{1}{p}}_{p} \)-boundary. We prove that the operator of the system is an isomorphism of Sobolev’s space \( W^{k}_{p}(\overline{G}) \), \( k\geq 1 \), \( p>2 \), under approp...

We construct an example of a bounded solution to a uniformly elliptic Beltrami equation that has no nontangential limit values almost everywhere on the boundary of the unit disk and also an example of a solution to such an equation that is not identically zero and has zero nontangential limit values almost everywhere on the boundary of the unit dis...

We consider the Riemann–Hilbert (Hilbert) problem in classes similar to the Hardy class for general first-order elliptic systems on a plane. We establish basic properties ofHardy classes for solutions of that systems and solvability conditions for boundary-value problems. We construct the example demonstrating that for discontinuous coefficients th...

The aim of this paper is to prove the following version of well-known Kellogg’s theorem. Let , , and is a conformal (one-to-one, onto) mapping. Then, extends to the homeomorphism from to ; moreover, , and the inverse mapping .

We give examples of the absence of representations of the “second kind” for a Beltrami equation. Some sufficient conditions are proposed for the existence of these representations. We specify the regularity “up to the boundary” of a quasiconformal homeomorphism of the unit circle onto itself in the case of a regular complex characteristic.

The Riemann-Hilbert boundary value problem for generalized analytic functions in Smirnov classes is under consideration. The domain is supposed simply connected with Lyapunov or Radon boundary without cusps. In the work the special representation for generalized analytic functions of Smirnov classes is built. This representation has independent int...

In this article the author studies deformations with given infinitesimal change of metric of a closed surface of genus [$p\ge 1$] of positive extrinsic curvature, situated in a three-dimensional Riemannian space. It is established that, in contrast to the case [$p=0$] (investigated by H. Weyl and A. V. Pogorelov), the surface does not admit deforma...

This paper contains a proof of the unbendability of a closed surface of genus p > 1 and positive extrinsic curvature in a 3-dimensional Riemannian space, and the unbendability of a closed surface of genus p = 1 and positive extrinsic curvature in a Riemannian space when one point of the surface is fixed. Bibliography: 8 titles.

It is proved that a regular Riemannian manifold diffeomorphic to a circle and having positive Gaussian curvature bounded from zero is immersible into a three-dimensional Euclidean space in the form of a regular surface if it has smallL p (the norm of the gradient of Gaussian curvature), p > 2, or if it has a sufficiently small area (with any behavi...

We study continuous and infinitesimal bendings of orderk (k-bendings) of a simply connected surface of positive curvature with boundary conditions. We study the question of the extension ofk-bendings satisfying some boundary condition to continuous bendings satisfying an analogous boundary condition.

A new method is proposed for constructing the solutions of boundary-value problems of Riemann-Hilbert type for noncanonical linear and quasilinear first-order elliptic systems in a simply connected bounded region of the plane. For a linear boundary condition we obtain complete results; for a nonlinear boundary condition we study the solvability “in...

This article is a survey of papers connected with obtaining apriori estimates of the norms of the radius-vector of an ovaloid of positive curvature and with related questions of the theory of elliptic differential equations. New results are presented involving estimates of the norms of the radius vector “right up to the boundary” of a locally conve...

~n+2 a curve in the Banach space ~ (~) of (Frechet) class C m, and S o - S. (Here and throughout the note we consider deformations st which preserve the regularity class of the surface.) The deformation S t is called a bending of the surface S of class C TM (with respect to the parameter), if Vt ~ 11 the surface S t is isometric with S; S t is call...

For surfaces of positive Gaussian curvature bounded away from zero the following statement is proved: A piece of a given surface containing a preassigned finite set of points and having a Lyapunov boundary can be deformed with an arbitrary given (as large as we like) bending at these points under the condition that the area of the piece is sufficie...