Table of contents

A smooth non-linear map is studied in a neighbourhood of an abnormal (degenerate) point. Inverse function and implicit function theorems are proved. The proof is based on the examination of a family of constrained extremal problems; second-order necessary conditions, which make sense also in the abnormal case, are used in the process. If the point under consideration is normal, then these conditions turn into the classical ones.

For subsets of a Banach space the notions of a generating set and an -strongly convex set are introduced. The latter can be represented as the intersection of sets of the form , which are translates of the generating set . A generating set must satisfy a condition that ensures a special support principle, as shown in the paper. Using this support principle a new area of convex analysis is constructed enabling one to strengthen classical results of the type of the Caratheodory and Krein-Milman theorems. Various classes of generating sets are described and the properties of -strongly convex sets are studied.

Two kinds of new mathematical model of variational type are put forward: non-linear analytic and coanalytic problems. The formulation of these non-linear boundary-value problems is based on a decomposition of the complete scale of Sobolev spaces into the "orthogonal" sum of analytic and coanalytic subspaces. A similar decomposition is considered in the framework of Clifford analysis. Explicit examples are presented.

The following theorem is established. Theorem. Let a function and a sequence satisfy the following condition: the function is non-decreasing on , and as . Then there is a function such that

and for all here is the -th partial sum of the trigonometric Fourier series of .

Many-dimensional non-strictly hyperbolic systems of conservation laws with a radially degenerate flux function are considered. For such systems the set of entropies is defined and described, the concept of generalized entropy solution of the Cauchy problem is introduced, and the properties of generalized entropy solutions are studied. The class of strong generalized entropy solutions is distinguished, in which the Cauchy problem in question is uniquely soluble. A condition on the initial data is described that ensures that the generalized entropy solution is strong and therefore unique. Under this condition the convergence of the "vanishing viscosity" method is established. An example presented in the paper shows that a generalized entropy solution is not necessarily unique in the general case.