Solutions for the prescribing mean curvature equation

By variational methods, for a kind of Yamabe problem whose scalar curvature vanishes in the unit ball BN and on the boundary S^N-1 the mean curvature is prescribed, we construct multi-peak solutions whose maxima are located on the boundary as the parameter tends to 0^＋ under certain assumptions. We also obtain the asymptotic behaviors of the solutions.     

Solutions of the heat equation in domains with singularities

We study the asymptotics of solutions to the Dirichlet problem for the heat equation in time-dependent domains with singular points.Translated fromMatematicheskie Zametki, Vol. 64, No. 2, pp. 163–179, August, 1998.This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-00504 and by INTAS under grant No. 93-351.     

Critical cases of stability. Converse implicit function theorem for dynamical systems with cosymmetry

Stability criteria of boundary equilibria for dynamical systems in the three critical cases, (n, k)=(3, 0), (2, 1), and (1, 1), are obtained.Translated fromMatematicheskie Zametki, Vol. 63, No. 4, pp. 572–578, April, 1998.The author wishes to thank V. I. Yudovich for useful discussions.This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-01791 and by the International Science Foundation under grant NRQ000.     

The algebraic structure ofH-dissipative operators in a finite-dimensional space

We study properties of Jordan representations ofH-dissipative operators in a finite-dimensional indefiniteH-space. An algebraic proof is given of the fact that such operators always have maximal semidefinite invariant subspaces. Translated fromMatematicheskie Zametki, Vol. 63, No. 2, pp. 163–169, February, 1998. The authors axe grateful to Professor A. A. Shkalikov for useful discussions. The research of the first author was supported by INTAS under grant No. 93-0249. The research of the second author was supported by the International Science Foundation and the Russian Government under grant No. NZP300.     

Conditions for the spatial flatness and spatial injectivity of an indecomposable CSL algebra of finite width

This paper is concerned with the connection between the geometric properties of the latticeL of subspaces of a Hilbert spaceH and homological properties (flatness and injectivity) ofH regarded as a natural module over the reflexive algebra AlgL that consists of all operators leaving invariant each element of the latticeL. It follows from these results that the cohomology groups with coefficients inB(H) are trivial for a broad class of reflexive algebras. Translated fromMatemalicheskie Zametki, Vol. 63, No. 1, pp. 9–20, January, 1998. The author gladly expresses his sincere gratitude to A. Ya. Khelemskii and Yu. V. Selivanov for their assistance in this work. In particular, A. Ya. Khelemskii indicated how to simplify the proof of Theorem 1. This research was supported by the Russian Foundation for Basic Research under grant No. 93-01-00156, by the International Science Foundation under grant No. M95000, and by the INTAS fund under grant No. 93-1376.     

Some conic bundless

A representation of the anticanonical K3 surface of a singular pencil of conics is described. This generalizes the well-known Shokurov theorem.Translated fromMatematicheskie Zametki, Vol. 63, No. 6, pp. 903–910, June, 1998.The author is greatly indebted to V. A. Iskovskikh, Yu. G. Prokhorov, and I. A. Chel'tsov for fruitful discussions.This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-00820 and by INTAS under grant No. 93-2805.00-00-00.     

Reconstruction of Sturm-Liouville differential operators with singularities inside the interval

The inverse spectral problem for Sturm-Liouville differential operators on a finite interval is studied for an arbitrary and finite number of regular singular points inside the interval. A uniqueness theorem is proved; necessary and sufficient conditions and a procedure for the solution of the inverse problem are obtained.Translated fromMatematicheskie Zametki, Vol. 64, No. 1, pp. 143–156, July, 1998.This research was supported by the Ministry of Education (KTsFE) under grant No. 96-1.7-4 and by the Russian Foundation for Basic Research under grant No. 97-01-00566.     

The class of groups all of whose subgroups with lesser number of generators are free is generic

It is shown that, in a certain statistical sense, in almost every group withm generators andn relations (withm andn chosen), any subgroup generated by less thanm elements (which need not belong to the system of generators of the whole group) is free. In particular, this solves Problem 11.75 from the Kourov Notebook. In the proof we introduce a new assumption on the defining relations stated in terms of finite marked groups. Translated fromMatematicheskie Zametki, Vol. 59, No. 4, pp. 489–496, April, 1996. The research of the second author was partially supported by the Russian Foundation for Basic Research under grant No. 94-0101541 and by the International Science Foundation under grant No. MID000.     

Holomorphic functions and embedded real surfaces

The paper is devoted to the study of necessary and sufficient topological conditions for an embedded real surface to lie in a strictly pseudoconvex domain on a complex surface. These results are used to construct Stein domains on algebraic manifolds and to describe envelopes of holomorphy of real surfaces in P ² and in some other complex surfaces.Translated fromMatematicheskie Zametki, Vol. 63, No. 4, pp. 599–606, April, 1998.The author is grateful to A. Vitushkin for the statement of the problem and for constant encouragement.This research was partially supported by the Russian Foundation for Basic Research under grant No. 96-01-01002 and by the International Science Foundation under grant No. a97-1850.     

Use of the Lagrange method in nonlinear analysis

The problem of the spectrum of a nonlinear system and its relation to ellipsoid transform widths are studied.Translated fromMatematicheskie Zametki, Vol. 59, No. 3, pp. 334–342, March, 1996.This research was partially supported by the Russian Foundation for Basic Research under grant No. 93-01-00237 and by the International Science Foundation under grant No. MP1000.     

A weakened asymptotic monogeny condition

Functions of one complex variable that satisfy a weakened asymptotic monogeny condition are proved to be holomorphic. Translated fromMatematicheskie Zametki, Vol. 60, No. 6, pp. 902–911, December, 1996. The author thanks Proffessor Dolzhenko for discussing the results of the paper. The research was supported by the Russian Foundation for Basic Research under grant No. 93-01-00236.     

The prison yard problem

Given a polygon II withn vertices whose sides arewalls. Guards, located at vertices can see all directions, but cannot see beyond walls. We prove that at most [n/2] guards suffice to see everywhere the whole plane. If II is not convex, then [n/2] suffice.The research was done while this author visited the Department of Mathematics at Rutgers University. Research supported in part by the Hungarian National Science Foundation under grant No. 1812Supported in part by NSF grant DMS 86-06225 and AF grant OSR-86-0078     

Asymptotics ast → ∞ of the solutions of nonlinear equations with nonsmall initial perturbations

A new method is proposed for finding asymptotics ast → ∞ of the solutions of the Cauchy problem for nonlinear evolution equations with nonsmall initial data. Translated fromMatematicheskie Zametki, Vol. 59, No. 6, pp. 855–864, June, 1996. This research was supported by the Russian Foundation for Basic Research under grant No. 93-011-134 and by the International Science Foundation under grant No. BX000.     

Real hypersurfaces in complex two-plane Grassmannians with commuting normal Jacobi operator

We give a complete classification of -invariant real hypersurfaces in complex two-plane Grassmannians G ₂(C ^m+2) with commuting normal Jacobi operator . The first author was supported by MCYT-FEDER grant BFM 2001-2871-C04-01, the second author by grant Proj. No. KRF-2006-351-C00004 from Korea Research Foundation and the third author by grant Proj. No. R14-2002-003-01001-0 from Korea Research Foundation, Korea 2006 and Proj. No. R17-2007-006-01000-0 from KOSEF.     

Differentiability of convex measures

In this paper we consider convex measures on finite-dimensional spaces. We prove the differentiability of convex measures in the Skorokhod sense (and under some natural conditions, in the Fomin sense also). Simultaneously we give some additional results on differentiability of convex measures.Translated fromMatematicheskie Zametki, Vol. 58, No. 6, pp. 862–871, December, 1995.This research was partially supported by the Russian Foundation for Basic Research under grant No. 94-01-01556, by the Ministry of Science under grant No. M38300 and by the International Science Foundation under grant No. M38000.     

Weight of faces in plane maps

Precise upper bounds are obtained for the minimum weight of minor faces in normal plane maps and 3-polytopes with specified maximum vertex degree. Translated fromMatematicheskie Zametki, Vol. 64, No. 5, pp. 648–657, November, 1998. The research of the first named author was supported in part by the Visiting Fellowship Research Grant GR/K00561 from the Engineering and Physical Sciences Research Council and by the Russian Foundation for Basic Research under grant No. 96-01-01614 and No. 97-01-01075.     

Integrable boundary conditions for many-component burgers equations

Infinite series of boundary conditions that are consistent with even-order higher symmetries and ensure the integrability of a Burgers type equation are constructed. Translated fromMatematicheskie Zametki, Vol. 60, No. 6, pp. 888–901, December, 1996. This research was supported by the Russian Foundation for Basic Research under grant No. 93-011-16008 and by the International Science Foundation under grant Nos. MLY00 and RK2000.     

A singularly perturbed boundary value problem for a second-order equation in the case of variation of stability

A boundary value problem for a second-order nonlinear singularly perturbed differential equation is considered for the case in which there is variation of stability caused by the intersection of roots of the degenerate equation. By the method of differential inequalities, we prove the existence of a solution such that the limit solution is nonsmooth. Translated fromMatematicheskie Zametki, Vol. 63, No. 3, pp. 354–362, March, 1998. This research was partially supported by the Russian Foundation for Basic Research under grant No. 96-01-00694.     

Strong means and oscillation of multiple Fourier series in multiplicative systems

In this paper we establish a general assertion relating the oscillation of the sequence of rectangular partial sums of a multiple Fourier series in a multiplicative system to the strong summability of this series. The systems of group generators are assumed to be uniformly bounded. Earlier these assertions were obtained by the author for the Walsh and Chrestenson series.Translated fromMatematicheskie Zametki, Vol. 63, No. 4, pp. 607–616, April, 1998.This research was supported by the Russian Foundation for Basic Research under grant No. 95-01-00135.