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. LocalTheorems 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.