Volume 2, Issue 6

The basic three-dimensonal boundary-contact dynamic problems are considered for a piecewise-homogeneous isotropic elastic medium bounded by several closed surfaces. Using the Fourier method, the considered problems are proved to be solvable under much weaker restrictions on the initial data of the problems as compared with other methods.

In the present work there has been posed and studied a general nonlinear optimal problem and a quasi-linear optimal problem with fixed time and free right end. It contains absolutely continuous monotone delays in phase coordinates and absolutely continuous monotone and distributed delays in controls. For these problems the necessary and, respectively, sufficient conditions of optimality in the form of the maximum principle have been proved.

In this paper we prove that a higher-order differential equation with one middle term has every bounded solution oscillatory. Moreover, the behavior of unbounded solutions is given. Two other results dealing with positive solutions are also given.

An interpolation inequality of Nirenberg, involving Lebesgue-space norms of functions and their derivatives, is modified, replacing one of the norms by a Hölder norm.

Theorems are proved as regards strong differentiability of integrals in different directions.

Three-dimensional mathematical problems of the elasticity theory of anisotropic piecewise homogeneous bodies are discussed. A mixed type boundary contact problem is considered where on one part of the interface, rigid contact conditions are given (jumps of the displacement and the stress vectors are known), while on the remaining part screen or crack type boundary conditions are imposed. The investigation is carried out by means of the potential method and the theory of pseudodifferential equations on manifolds with boundary.

Some relations between cardinal invariants of X and C ( X ) are established in the weak topology, where C ( X ) is the space of continuous real-valued functions on X in the compact-open topology.