L. E. Payne

Abstract: 1 Introduction.- 2 Basic Equations.- 2.1 Formulation of Initial-Boundary Value Problems.- 2.2 The Classical and Weak Solutions.- 2.3 The Homogeneous Isotropic Body. Plane Elasticity.- 2.4 Definiteness Properties of the Elasticities.- 3 Early Work.- 4 Modern Uniqueness Theorems in Three-Dimensional Elastostatics.- 4.1 The Displacement Boundary Value Problem for Bounded Regions.- 4.1.1 General Anisotropy.- 4.1.2 A Homogeneous Anisotropic Material.- 4.1.3 A Homogeneous Isotropic Material.- 4.1.4 The Implication of Strong Ellipticity for Uniqueness.- 4.1.5 The Non-Homogeneous Isotropic Material with no Definiteness Assumptions on the Elasticities.- 4.1.6 The Displacement Boundary Value Problem for a Homogeneous Isotropic Sphere.- 4.1.7 Fichera's Maximum Principle.- 4.2 Exterior Domains.- 4.3 The Traction Boundary Value Problem.- 4.3.1 General Anisotropy.- 4.3.2 A Homogeneous Isotropic Material.- 4.3.3 The Traction Boundary Value Problem for a Homogeneous Isotropic Elastic Sphere.- 4.3.4 Necessary Conditions for Uniqueness in the Traction Boundary Value Problem for Three-Dimensional Homogeneous Isotropic Elastic Bodies.- 4.4 Mixed Boundary Value Problems.- 4.4.1 General Anisotropy.- 4.4.2 A Homogeneous Isotropic Material.- 5 Uniqueness Theorems in Homogeneous Isotropic Two-Dimensional Elastostatics.- 5.1 Kirchhoff's Theorem in Two-Dimensions. The Displacement and Traction Boundary Value Problems.- 5.2 Uniqueness in Plane Problems with Special Geometries.- Appendix: Uniqueness of Three-Dimensional Axisymmetric Solutions.- 6 Problems in the Whole- and Half-Space.- 6.1 Specification of the Various Boundary Value Problems. Continuity onto the Boundary and in the Neighbourhood of Infinity.- 6.2 Uniqueness of Problems (a)-(d). Corollaries for the Space EN.- 6.3 Uniqueness for the Mixed-Mixed Problem of Type (e).- 6.3.1 A Complete Representation of the Biharmonic Displacement in a Homogeneous Isotropic Body Occupying the Half-Space.- 6.3.2 Uniqueness in the Mixed-Mixed Problem (e).- 7 Miscellaneous Boundary Value Problems.- 7.1 Problems for a Sphere.- 7.2 The Cauchy Problem for Isotropic Elastostatics.- 7.3 The Signorini Problem. Other Problems with Ambiguous Conditions.- 8 Uniqueness Theorems in Elastodynamics. Relations with Existence, Stability, and Boundedness of Solutions.- 8.1 The Initial Displacement and Mixed-Boundary Value Problems. Energy Arguments.- 8.2 The Initial-Displacement Boundary Value Problem. Analyticity Arguments.- 8.3 The Initial-Mixed Boundary Value Problem for Bounded Regions. Further Arguments.- 8.4 Summary of Existing Results in the Uniqueness of Elastodynamic Solutions.- 8.5 Non-Standard Problems, including those with Ambiguous Conditions.- 8.6 Stability, Boundedness, Existence and Uniqueness.- References.

TL;DR: In this paper, it is shown that if the nonlinearity and initial data in A satisfy certain restrictions then no classical (or weak) solution of A can exist for all time.