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AuthorValent, Tullio. author
TitleBoundary Value Problems of Finite Elasticity [electronic resource] : Local Theorems on Existence, Uniqueness, and Analytic Dependence on Data / by Tullio Valent
ImprintNew York, NY : Springer New York, 1988
Connect tohttp://dx.doi.org/10.1007/978-1-4612-3736-5
Descript XII, 191 p. online resource

SUMMARY

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, beยญ sides being quite spontaneous, allows us to consider a great many interยญ 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 Part of the matter presented is unpublished; other arguments have been only partially published and in lesser generality.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 exisยญ 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 subยญ sequent developments


CONTENT

I. A Brief Introduction to Some General Concepts in Elasticity -- ยง1. Some Notations -- ยง2. Deformations and Motions -- ยง3. Mass. Force -- ยง4. Eulerโ{128}{153}s Axiom. Cauchyโ{128}{153}s Theorem -- ยง5. Constitutive Assumptions. Elastic Body -- ยง6. Frame-Indifference of the Material Response -- II. Composition Operators in Sobolev and Schauder Spaces. Theorems on Continuity, Differentiability, and Analyticity -- ยง1. Some Facts About Sobolev and Schauder Spaces -- ยง2. A Property of Multiplication in Sobolev Spaces -- ยง3. On Continuity of Composition Operators in Sobolev and Schauder Spaces -- ยง4. On Differentiability of Composition Operators in Sobolev and Schauder Spaces -- ยง5. On Analyticity of Composition Operators in Sobolev and Schauder Spaces -- ยง6. A Theorem on Failure of Differentiability for Composition Operators -- III. Dirichlet and Neumann Boundary Problems in Linearized Elastostatics. Existence, Uniqueness, and Regularity -- ยง1. Kornโ{128}{153}s Inequalities -- ยง2. A Generalization of a Theorem of Lax and Milgram -- ยง3. Linearized Elastostatics -- ยง4. The Dirichlet Problem in Linearized Elastostatics. Existence and Uniqueness in W1,p(?, ?n) -- ยง5. The Neumann Problem in Linearized Elastostatics. Existence and Uniqueness in W1,p(?, ? n) -- ยง6. Some Basic Inequalities for Elliptic Operators -- ยง7. Regularity Theorems for Dirichlet and Neumann Problems in Linearized Elastostatics -- IV. Boundary Problems of Place in Finite Elastostatics -- ยง1. Formulation of the Problem -- ยง2. Remarks on Admissibility of a Linearization -- ยง3. A Topological Property of Sets of Admissible Deformations -- ยง4. Local Theorems on Existence, Uniqueness, and Analytic Dependence on f for Problem ((1.1), (1.3)) -- ยง5. Stronger Results on Existence and Uniqueness for Problem ((1.1), (1.3)) -- ยง6. 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 -- ยง1. Generality on the Traction Problem in Finite Elastostatics -- ยง2. Preliminary Discussion -- ยง3. A Basic Lemma -- ยง4. Critical Infinitesimal Rigid Displacements for a Load -- ยง5. A Local Theorem on Existence, Uniqueness, and Analytic Dependence on a Parameter -- ยง6. The Case of Dead Loads -- ยง7. Some Historical Notes -- VI. Boundary Problems of Pressure Type in Finite Elastostatics -- ยง1. Preliminaries -- ยง2. The Case When the Load Is Invariant Under Translations -- ยง3. The Case When the Load Is Invariant Under Rotations -- ยง4. 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


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