Three-dimensional mathematical problems of the interaction between thermoelastic and scalar oscillation fields in a general physically anisotropic case are studied by the boundary integral equation methods. Uniqueness and existence theorems are proved by the reduction of the original interface problems to equivalent systems of boundary pseudodifferential equations. In the non-resonance case the invertibility of the corresponding matrix pseudodifferential operators in appropriate functional spaces is shown on the basis of the generalized Sommerfeld-Kupradze type thermoradiation conditions for anisotropic bodies. In the resonance case the co-kernels of the pseudodifferential operators are analysed and the efficient conditions of solvability of the original interface problems are established.

Three‐dimensional mathematical problems of interaction between elastic and scalar oscillation fields are investigated. An elastic field is to be defined in a bounded inhomogeneous anisotropic body occupying the domain Ω¯1⊂ℝ3 while a physical (acoustic) scalar field is to be defined in the exterior domain Ω¯2=ℝ3\Ω1 which is filled up also by an anisotropic (fluid) medium. These two fields satisfy the governing equations of steady‐state oscillations in the corresponding domains together with special kinematic and dynamic transmission conditions on the interface ∂Ω1. The problems are studied by the so‐called non‐local approach, which is the coupling of the boundary integral equation method (in the unbounded domain) and the functional‐variational method (in the bounded domain). The uniqueness and existence theorems are proved and the regularity of solutions are established with the help of the corresponding Steklov–Poincaré type operators and on the basis of the Gårding inequality and the Lax–Milgram theorem. In particular, it is shown that the physical fluid–solid acoustic interaction problem is solvable for arbitrary values of the frequency parameter. Copyright © 1999 John Wiley & Sons, Ltd.

Abstract. This is the survey of the applications of the potential methods to the problems of continuum mechanics. Historical review, new results, prospects of the development are given.This survey paper is dedicated to the 90th birthday of Victor Kupradze. Therefore we shall cover here mainly questions connected with his scientific interests and dealt with by his pupils and followers. We wish to note specially that V. Kupradze's old works on the application of potential methods to the study of wave propagation, radiation and diffraction problems that had greatly contributed to the progress in these directions will hardly be mentioned.Eight years have passed since our previous survey of the field in question. 1That was the period of great events in our life, change of the outlook, revaluation of many results, the arising of new difficulties in the development of science. The potential method keeps on developing and we do have results obtained in these years which are worthwhile being told about.

where the mixed interface problems were formulated. With the aid of the explicit solution for the rigid contact problem, here the mixed interface problems are reduced to pseudodifferential equations on manifolds with boundary. We obtain uniqueness and existence in Bessel-potential and Besov spaces. Embedding theorems yield Holder continuitÿ Ž . with any exponent ␣ g 0, 1r2 . ᮊ

Forniulierung der ProblemeDie Grundgleichungen der linearen Thermoelastizitatstheorie fur einen homogenen isotropen Korper sind 8. u(1.1)Ilabei ist u = 2~ ( P , 1) der Verschiek)ungsvektor an der Gtelle P = (2, , x?, x:!)zur Zeit t , 0 = 0 ( P , t ) = 17 ( P , t ) -TI, die Temperaturdifferenz zur konstantenTenil)eraturleitf&higkeit, c, die spezifische Warme bei konstanten Deformationen unti q ( P , t ) = c, Q ( P , t ) die pro Volumen und Zeit hervorquellende Warmemenge. Eine Herleitung der Warmeleitungsgleichung (1.2) ~u l s thermodynamischen y TI, t t c, 01 Prinzipien undder Annahme -'< 1 findet man z. R. in [9], S. 9-22. To I Wir suchen zeitlich periodische Losungen u(P, t ) = u ' ( P ) cos w t + u"(P) sin u) t , O ( P , t ) = O'(P) cos u) t + O"(P) sin u) t unter mit der Frequenz u) zeitlich periodischen Randbedingungen und f ( P , t ) = f ' ( P ) cos w t + f"(P) sin OJ t, Q ( P , t ) = Q' ( P ) cos (r) t + Q" (P) sin cc) t . I m weiteren sei u = u ( P ) = u ' ( P ) + iu"(P), 0 = O ( P ) = O'(P) + i.O"(P), f = f ( P ) = f ( P ) + if"(P), Q =&(PI = Q'(P) + iQ"(P).