Boundary Value Problems of Finite Elasticity: Local Theorems on Existence, Uniqueness, and Analytic Dependence on Data

In this book I present, in a systematic form, some local theorems on existence, uniqueness, and analytic dependence on the load, which I have recently obtained for some types of boundary value problems of finite elasticity. Actually, these results concern an n-dimensional (n ~ 1) formal generalization of three-dimensional elasticity. Such a generalization, be­ sides being quite spontaneous, allows us to consider a great many inter­ esting mathematical situations, and sometimes allows us to clarify certain aspects of the three-dimensional case. Part of the matter presented is unpublished; other arguments have been only partially published and in lesser generality. Note that I concentrate on simultaneous local existence and uniqueness; thus, I do not deal with the more general theory of exis­ tence. Moreover, I restrict my discussion to compressible elastic bodies and I do not treat unilateral problems. The clever use of the inverse function theorem in finite elasticity made by STOPPELLI [1954, 1957a, 1957b], in order to obtain local existence and uniqueness for the traction problem in hyperelasticity under dead loads, inspired many of the ideas which led to this monograph. Chapter I aims to give a very brief introduction to some general concepts in the mathematical theory of elasticity, in order to show how the boundary value problems studied in the sequel arise. Chapter II is very technical; it supplies the framework for all sub­ sequent developments.

Table of Contents:
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.