Boundary Value Problems of Finite Elasticity

I. A Brief Introduction to Some General Concepts in Elasticity.- 1. Some Notations.- 2. Deformations and Motions.- 3. Mass. Force.- 4. Euler's Axiom. Cauchy's Theorem.- 5. Constitutive Assumptions. Elastic Body.- 6. Frame-Indifference of the Material Response.- II. Composition Operators in Sobolev and Schauder Spaces. Theorems on Continuity, Differentiability, and Analyticity.- 1. Some Facts About Sobolev and Schauder Spaces.- 2. A Property of Multiplication in Sobolev Spaces.- 3. On Continuity of Composition Operators in Sobolev and Schauder Spaces.- 4. On Differentiability of Composition Operators in Sobolev and Schauder Spaces.- 5. On Analyticity of Composition Operators in Sobolev and Schauder Spaces.- 6. A Theorem on Failure of Differentiability for Composition Operators.- III. Dirichlet and Neumann Boundary Problems in Linearized Elastostatics. Existence, Uniqueness, and Regularity.- 1. Korn's Inequalities.- 2. A Generalization of a Theorem of Lax and Milgram.- 3. Linearized Elastostatics.- 4. The Dirichlet Problem in Linearized Elastostatics. Existence and Uniqueness in W1,p(?, ?n).- 5. The Neumann Problem in Linearized Elastostatics. Existence and Uniqueness in W1,p(?, ? n).- 6. Some Basic Inequalities for Elliptic Operators.- 7. Regularity Theorems for Dirichlet and Neumann Problems in Linearized Elastostatics.- IV. Boundary Problems of Place in Finite Elastostatics.- 1. Formulation of the Problem.- 2. Remarks on Admissibility of a Linearization.- 3. A Topological Property of Sets of Admissible Deformations.- 4. Local Theorems on Existence, Uniqueness, and Analytic Dependence on f for Problem ((1.1), (1.3)).- 5. Stronger Results on Existence and Uniqueness for Problem ((1.1), (1.3)).- 6. Local Theorems on Existence and Uniqueness for Problem ((1.1), (1.2)).- V. Boundary Problems of Traction in Finite Elastostatics. An Abstract Method. The Special Case of Dead Loads.- 1. Generality on the Traction Problem in Finite Elastostatics.- 2. Preliminary Discussion.- 3. A Basic Lemma.- 4. Critical Infinitesimal Rigid Displacements for a Load.- 5. A Local Theorem on Existence, Uniqueness, and Analytic Dependence on a Parameter.- 6. The Case of Dead Loads.- 7. Some Historical Notes.- VI. Boundary Problems of Pressure Type in Finite Elastostatics.- 1. Preliminaries.- 2. The Case When the Load Is Invariant Under Translations.- 3. The Case When the Load Is Invariant Under Rotations.- 4. The Case of a Heavy Elastic Body Submerged in a Quiet Heavy Liquid.- Appendix I. On Analytic Mappings Between Banach Spaces. Analytic Implicit Function Theorem.- Appendix II. On the Representation of Orthogonal Matrices.- Index of Notations.