On some elliptic transmission problems

Abstract. Boundary value problems for second order linear elliptic equations with coefficients having discontinuities of the first kind on an infinite number of smooth surfaces are studied. Existence, uniqueness and regularity results are furnished for the diffraction problem in such a bounded domain, and for the corresponding transmission problem in all of R N . The transmission problem corresponding to the scattering of acoustic plane waves by an infinitely stratified scatterer, consisting of layers with physically different materials, is also studied. 0. Introduction. In this work we study boundary value problems for linear equations of elliptic type whose coefficients have discontinuities of the first kind on an infinite number of smooth surfaces that divide a bounded domain in R N into nested layers. On those surfaces, the so-called "transmission (conjugacy, matching, linking) conditions" are imposed, that express the continuity of the medium and the equilibrium of the forces acting on it. The discontinuity of the coefficients of the equations corresponds to the fact that the medium consists of several physically different materials. From the point of view of the theory of generalized solutions-which we employ in our approach-such problems can be considered as special cases of usual boundary value problems. On the contrary, the investigation of these problems by classical methods requires the theory of integral equations, and in this context they differ essentially from the usual boundary value problems where the medium has smoothly varying characteristics. Boundary value problems with discontinuous coefficients (also known as diffraction problems) have been treated by many authors, employing a variety of approaches. I

Uniform stability and semi-stability of motions in dynamical systems on metric spaces

Abstract. Some stability properties of motions in pseudo-dynamical systems and semi-systems are studied. Introduction. The purpose of the present paper is to study some properties of motions in dynamical and pseudo-dynamical systems (see definitions below) on metric spaces. We consider uniform stability and semistability (see Section 2) of motions in nonempty subsets of the phase space and discuss in particular properties of mappings of the types