Showing papers in "Russian Mathematical Surveys in 1987"

Some problems of the qualitative theory of non-linear degenerate second-order parabolic equations

Abstract: CONTENTSIntroduction Chapter I. Existence and uniqueness of generalized solutions of the Cauchy problem and of initial boundary-value problems ??1.1. Theorems of existence and non-existence ??1.2. Uniqueness theorems Chapter II. Estimates and differential properties of generalized solutions ??2.1. The character of the dependence on the data ??2.2. Regularity and non-regularity Chapter III. Behaviour of generalized solutions for small values of time ??3.1. Finite rate of propagation of perturbations ??3.2. Inertia ??3.3. Shrinking of the size of the support Chapter IV. Behaviour of generalized solutions for large values of time ??4.1. Stabilization in finite time ??4.2. Localization of perturbations ??4.3. Asymptotic formulae References

The symmetry approach to the classification of non-linear equations. Complete lists of integrable systems

Abstract: CONTENTSIntroduction Chapter I. Integrability conditions § 1. Formal symmetries and conservation laws § 2. The technique of formal series § 3. Canonical conservation laws and divergency conditions Chapter II. Evolution equations of the second order § 4. Invertible transformations § 5. The classification theorem Chapter III. Enlargement of the module of invertible substitutions § 6. Generalized contact transformations § 7. Partial differentiations and potentiations § 8. Transformations of symmetric systems Chapter IV. Integrable systems of Schrodinger type § 9. The list of integrable equations § 10. A description of symmetric systems. Tests for integrability § 11. A brief bibliographical comment References

The $ \bar\partial$-equation in the multidimensional inverse scattering problem

Abstract: CONTENTSChapter I. A survey of the results § 1.1. The method of Faddeev in the inverse scattering problem for the Schrodinger equation § 1.2. The results of Newton, Ablowitz, and Nachman § 1.3. Necessary and sufficient properties of the scattering data. Generalized dispersion relations § 1.4. Methods of solution of the inverse problem with non-overdetermined data. The results of Moses and Prosser and their generalizations § 1.5. The inverse problem on a fixed energy level for the two-dimensional Schrodinger operator and non-linear equations. The methods of S. P. Novikov and Manakov and their further development § 1.6. The inverse problem on a fixed energy level for the three-dimensional Schrodinger operator. The results of Beals and Coifman. New resultsChapter II. Necessary properties of the scattering data § 2.1. The Green-Faddeev function and its properties. Analysis of the integral equation § 2.2. Generalized scattering data . The non-linear -equation . Corollaries § 2.3. Properties of zeros of the Fredholm determinant for the equation § 2.4. Solution of the inverse problem on the basis of generalized dispersion relationsChapter III. Characterization of scattering data. Preliminary results § 3.1. Liouville's theorem for solutions of the non-linear -equation § 3.2. Formulae for solutions of the -equation in a concave domain of § 3.3. Estimates for the form § 3.4. Estimates for solutions of the -equation § 3.5. Proof of Liouville's theorem for the non-linear -equation Chapter IV. Characterization of scattering data. Final results § 4.1. Theorems concerning the characterization of scattering data in physical and non-physical domains § 4.2. A separate analyticity theorem for the non-linear -equation § 4.3. Continuability of solutions of the non-linear -equation from the physical to the non-physical domain § 4.4. Reconstruction of the Schrodinger operator from minimal scattering data § 4.5. Solution of the inverse scattering problem on a fixed energy level for the multidimensional Schrodinger operatorChapter V. Further results and problems § 5.1. The inverse scattering problem for the Schrodinger equation in a magnetic field § 5.2. Inverse scattering problems for the acoustic equationReferences

The theory of extensions and explicitly-soluble models

Abstract: CONTENTSIntroduction § 1. Boundary values for abstract operators and the Krein formula for generalized resolvents § 2. Zero-radius potentials in diffraction problems § 3. Quantum-mechanical problems with energy-dependent potentials Conclusion References

L2-Theory of the Maxwell operator in arbitrary domains

Abstract: CONTENTSIntroduction § 1. Classes of functions. Boundary conditions § 2. Definition of the Maxwell operator. Inclusion in the elliptic theory § 3. Bounded domains with Lipschitz boundary § 4. Domains with piecewise-smooth boundary § 5. Asymptotic properties of the discrete spectrum of the operator § 6. Comments and bibliographic notes References

On the determination of minimal global attractors for the Navier-Stokes and other partial differential equations

Abstract: CONTENTS ??1. Introduction ??2. Problems with dissipation of parabolic type (semigroups of class 1) ??3. Semigroups of class 2. Equations of hyperbolic type References

Combinatorial geometries and torus strata on homogeneous compact manifolds

Abstract: CONTENTSIntroduction § 1. Torus orbits and strata on the Grassmannian § 2. Matroids and strata on the Grassmannian § 3. The moment mapping and toric varieties § 4. The geometry of compact homogeneous spaces § 5. Definition of a stratum via the moment mapping § 6. Strata and Schubert cells § 7. Polytopes corresponding to strata § 8. The general (W,Q)-matroid § 9. Examples of strata References

Ergodic properties of certain systems of two-dimensional discs and three-dimensional balls

Abstract: CONTENTS § 1 Introduction § 2 Necessary information on scattering and semiscattering billiards § 3 Stable and unstable foliations § 4 Local ergodicity § 5 Ergodicity of certain systems of three discs References

Topological homogeneity. Topological groups and their continuous images

Abstract: CONTENTSIntroduction Terminology and notation § 1. Some concepts and estimates from the theory of cardinal invariants § 2. Topological homogeneity and the exponent of a space in the Vietoris topology § 3. Homogeneity, products and retracts of homogeneous spaces § 4. Pracharacter, predensity, integrability by character, and the existence of Gδ-points in compact spaces § 5. Continuous images of topological groups § 6. Some additional results on homogeneity and topological groups References

Semiclassical approximation for equations with periodic coefficients

Abstract: CONTENTS § 1. Introduction § 2. The effective Hamiltonian § 3. Turning points § 4. Comments and bibliographical indications References

Complexes of curves and the Teichmüller modular group

Abstract: CONTENTSIntroduction § 1. Complexes of curves and families of functions § 2. Singularities and non-degeneracy of functions § 3. Some variants of the complex of curves § 4. Corners and modular groups § 5. Generators and relations for modular groups Appendix to § 5. The Hatcher-Thurston and Harer complexes § 6. The virtual cohomological dimension of modular groups § 7. Stabilization of homology of modular groups § 8. The virtual Euler characteristic of modular groups References

A local index theorem for families of $ \bar\partial$-operators on Riemann surfaces

Abstract: CONTENTS Introduction § 1. Laplacians on a Riemann surface § 2. Variational formulae § 3. First variation of the function § 4. Second variation of the function § 5. Concluding remarks References

Regularity of solutions of variational inequalities

Abstract: CONTENTS Introduction § 1. Boundedness and Holder continuity of solutions of variational inequalities of general form § 2. Problems with obstacles in a domain § 3. Thin obstacles § 4. Diagonal systems § 5. Contact problems in elasticity theory § 6. Problems with convex constraints on the gradient of the solution References

Surgery on manifolds with finite fundamental groups

Abstract: CONTENTSIntroduction Chapter I. Wall groups and their computation § 1. Wall groups § 2. Wall groups of finite Abelian 2-groups § 3. The spinor norm Chapter II. Non-realizability theorems § 4. Free involutions on homotopy spheres § 5. Obstructions to splitting. Even dimension § 6. The two-row diagram § 7. Elements of the second type Resume References

Asymptotic formulae for the number of lattice points in Euclidean and Lobachevskii spaces

Abstract: CONTENTSIntroduction § 1. The asymptotic behaviour of the number of lattice points in the case of a Euclidean space § 2. The asymptotic behaviour of the number of lattice points in the case of a Lobachevskii space § 3. Another derivation of formula (1.10) § 4. Another derivation of formula (2.12) Appendix. The construction of the function in the case of a non-Euclidean spaceReferences

Periodic groups and Lie algebras

Abstract: CONTENTS ??1. Introduction ??2. Relations in an Engel Lie algebra and its associative enveloping algebra ??3. Nilpotency of homogeneous Engel Lie algebras with a chain of ideals of special type ??4. Quasi-identities in Engel Lie algebras ??5. Nilpotency of an Engel Lie algebra generated by finitely many elements of nil-index?2 ??6. Nilpotency of an Engel Lie algebra in general References

Asymptotic behaviour of the solutions of the Cauchy problem for non-linear first order equations

Abstract: CONTENTSIntroduction § 1. Survey of results on the asymptotic behaviour of solutions of first order non-linear equations § 2. Representation and asymptotic behaviour of the solution of the Cauchy problem for a quasilinear conservation law in the case of a monotonic initial function § 3. The stabilization to a constant of the solution of the Cauchy problem for a quasilinear conservation law References