Table of contents

For elliptic operators with infinitely many variables, having a large parameter for the zero-order term, it is proven that the Dirichlet problem has a unique solution on CL-manifolds with boundary. The Green kernel of the associated invertible operator is a measure which depends on the point of observation as well as on the parameter. The existence of a unique solution of the first boundary value problem for a second-order parabolic operator with infinitely many variables on the direct product of a CL-manifold with boundary and the semi-axis t≥0 is proved. Bibliography: 7 items.

In this paper, topological invariants of dynamical systems given on a two-dimensional manifold M² of genus p>1 are selected which allow one to distinguish topologically inequivalent systems which have nonclosed, Poisson stable trajectories and non-null-homotopic closed trajectories. A necessary and sufficient condition for the topological equivalence of transitive dynamical systems on M² is established. Figures: 6. Bibliography: 20 items.

In the present paper a coercive inequality is established for an infinite-dimensional elliptic differential operator of order 2m, and a theorem on smoothness of a generalized solution of the Dirichlet problem for an equation containing such an operator is established. Bibliography: 7 items

Billiards are considered within domains in the plane or on the two-dimensional torus with the euclidian metric, where the boundaries of these domains are everywhere convex inward. It is shown that the flow {S_t} generated by such a billiard is a K-system. A fundamental place is here assigned to the proof of the theorem showing that transversal fibers for the flow {S_t} consist "on the whole" of sufficiently long regular segments. From this theorem follow assertions on the absolute continuity of transversal fibers for the billiards in question. Bibliography: 6 items.

In this article is discussed a stable lower estimate for gradients of functions near a critical point. It is stable in the sense that if the function belongs to a family of functions depending smoothly on a finite-dimensional parameter, then the estimate will be correct for functions with close parameter values. Related to this estimate is a stratification of the space of jets of functions; the strata are semialgebraic sets of increasing codimension. Bibliography: 6 items.