On a method of constructing the solutions of boundary-value problems of the theory of bendings of surfaces of positive curvature

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 a neighborhood of zero. Applications are given to the problem of isometric transformations of a surface diffeomorphic to the disk and having positive curvature all the way to the boundary under prescribed boundary conditions.Translated from Ukrainskií Geometricheskií Sbornik, Issue 29, 1986, pp. 56–82.     

Deformability of surfaces of positive curvature

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 sufficiently small.Translated from Matematicheskie Zametki, Vol. 19, No. 5, pp. 815–823, May, 1976.I thank V. T. Fomenko for guidance.     

Unique Determination of Locally Convex Surfaces with Boundary and Positive Curvature of Genus p ≥ 0

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 congruent.

     

Semiconductor disk laser in the nanosecond lasing mode

Pulse generation (pulse duration 370 to 34 ns) is obtained for the semiconductor disk laser optically pumped by the Cr:LiCAF laser beam. Dynamics of its lasing spectrum during the pump pulse is investigated. Generation pulse variations under the mismatch of the pump spot and the region of the main disk lasing mode are studied.     

Generation of long-living charged nanoparticles at ablation in air and their role in pulsed microdrilling

The ablation of steel in air by short laser pulses was shown to form a long-living cloud of electrically charged submicron particles. These particles, being resident in the atmosphere of deep laser-produced channels within tens of seconds and carrying an electric charge during portions of a second, are able to initiate low-threshold gas breakdown resulting in the significant screening of the following pulses with a duration of ～100 ps. The clouds contained mostly positively charged particles shaped as ideal spheres. The statistics of their diameters nearly followed the Poisson law with the peak at 400 nm. The total volume of the charged particles was nearly equal to the volume of the ablatively removed material. A new approach was proposed to eliminate the screening, which implied the use of an external electric field. This enabled the enhancement of drilling rates by up to 50 times. The electric charge, mobility, and discharge rates of the particles were measured using a conductivity technique.     

The Riemann–Hilbert problem in Hardy classes for general first-order elliptic systems

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 the solvability features differ from the pictures of solvability of analogous problems for holomorphic and generalized analytic functions. In particular, the problem with positive index can be unsolvable.     

Effect of submillisecond radiation of the erbium laser on absorbing liquid

Pressure pulses occurring in water under the effect of submillisecond radiation of the Er³⁺:YAG laser on the free and covered liquid surface were measured. The behavior of pressure pulses is caused by the photoacoustic effect, explosive boiling of the superheated liquid surface layer, and cavitation processes developing after laser exposure.     

Some pathological examples of solutions to a Beltrami equation

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 disk. These examples show that, in the general case of the Hardy spaces of solutions to a uniformly elliptic Beltrami equation (and to more general noncanonical first-order elliptic systems), the usual statement of boundary value problems used for holomorphic and generalized analytic functions is ill-posed.     

On a priori estimates of the derivatives of the radius-vector of a surface of positive curvature

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 convex surface of positive curvature with boundary.Translated from Itogi Nauki i Tekhniki, Seriya Problemy Geometrii, Vol. 19, pp. 186–212, 1987.     

Representations of the “second kind” for the hardy classes of solutions to the Beltrami equation

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.