Problems in the Whole- and Half-Space

Abstract

Among comparatively recent contributions to elastostatic uniqueness theorems are those concerned with problems of the whole- and half-space. Their late arrival may appear surprising in view of the practical importance of such problems and the length of time they have been studied, yet because in the half-space the extension of Kirchhoff’s classical theorem is more difficult than in exterior domains, the appropriate uniqueness theorems have accordingly taken longer to establish. On the other hand, the situation in the whole space is almost trivial and probably for this reason it has never been recorded. Of course, it is easy to state the requisite order conditions that the displacement and stress must satisfy at infinity in order to make the pertinent integrals converge to zero in the application of Kirchhoff’s theorem, but as outlined in Section 4.2, these prescriptions are at best artifical. The difficulty arises in determining the rate of growth or decay implied by suitably mild restrictions on the asymptotic behaviour of the displacement and stress components. Actually, we shall show that once this knowledge is obtained, the classical energy arguments become unnecessary, and instead we may use, for instance, Duffin’s reflexion principles to derive uniqueness.