Boundary Value Problems of Finite Elasticity: Local Theorems on Existence, Uniqueness, and Analytic Dependence on Data. (= Springer tracts in natural philosophy, Volume 31). Tullio Valent Fundamental Structure.- 1.2. Special Subsets.- 1.3. Operators.- 1.4. Functionals.- §2. The Spectrum of an Operator.- 2.0. Preliminary Remarks.- 2.1. Basic Definitions.- 2.2. Functions of an Operator.- 2.3. Connection Between the Spectra of an Operator and its Inverse.- §3. Special Classes of Operators.- 3.0. Preliminary Remarks.- 3.1. CC Operator s. Definition and Basic Properties.- 3.2. CC Operators. Fredholm-Riesz Theory.- 3.3. Selfadjoint CC Operators.- 3.4. Selfadjoint, Normal, and Unitary Operators.- 3.5. Some Additional Conventions.- II. Function Spa

Tullio Valent

ISBN 10: 0387965505 / ISBN 13: 9780387965505

Editore: Springer, New York, 1988

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Let me begin by explaining the meaning of the title of this book. In essence, the book studies boundary value problems for linear partial differ ential equations in a finite domain in n-dimensional Euclidean space. The problem that is investigated is the question of the dependence of the nature of the solvability of a given equation on the way in which the boundary conditions are chosen, i.e. on the supplementary requirements which the solution is to satisfy on specified parts of the boundary. The branch of mathematical analysis dealing with the study of boundary value problems for partial differential equations is often called mathematical physics. Classical courses in this subject usually consider quite restricted classes of equations, for which the problems have an immediate physical context, or generalizations of such problems. With the expanding domain of application of mathematical methods at the present time, there often arise problems connected with the study of partial differential equations that do not belong to any of the classical types. The elucidation of the correct formulation of these problems and the study of the specific properties of the solutions of similar equations are closely related to the study of questions of a general nature. Detailangaben zum Buch - Partial Differential Equations: An Introduction to a General Theory of Linear Boundary Value Problems (Springer Series in Soviet Mathematics) EAN (ISBN-13): 9783540166993 ISBN (ISBN-10): 3540166998 Herausgeber: Springer ISBN/EAN: 3540166998 ISBN: 3-540-16699-8, 978-3-540-16699-3 Autor des Buches: boas Titel des Buches: partial differential equations boundary value problems, introduction the theory differential equations, linear theory, springer mathematics, general problems Autor/in: Aleksei A. Dezin Titel: Springer Series in Soviet Mathematics; Partial Differential Equations - An Introduction to a General Theory of Linear Boundary Value Problems Verlag: Springer; Springer Berlin 165 Seiten Erscheinungsjahr: 1987 Berlin; Heidelberg; DE Übersetzer/in: Ralph P. Boas (Russisch) Mathematik Analysis; Mathematische Analysis, allgemein; Verstehen; differential equation; partial differential equation; Boundary value problem; mathematical analysis; C; Analysis; Mathematics and Statistics I. Elements of Spectral Theory.- §0. Introductory Remarks.- §1. Basic Definitions.- 1.0. Introductory Remarks.- 1.1. Fundamental Structure.- 1.2. Special Subsets.- 1.3. Operators.- 1.4. Functionals.- §2. The Spectrum of an Operator.- 2.0. Preliminary Remarks.- 2.1. Basic Definitions.- 2.2. Functions of an Operator.- 2.3. Connection Between the Spectra of an Operator and its Inverse.- §3. Special Classes of Operators.- 3.0. Preliminary Remarks.- 3.1. CC Operator s. Definition and Basic Properties.- 3.2. CC Operators. Fredholm-Riesz Theory.- 3.3. Selfadjoint CC Operators.- 3.4. Selfadjoint, Normal, and Unitary Operators.- 3.5. Some Additional Conventions.- II. Function Spaces and Operators Generated by Differentiation.- §0. Introductory Remarks.- §1. The Space IH(F).- §2. Differential Operations and the Maximal Operator.- §3. The Minimal Operator and Proper Operators.- 3.1. The Minimal Operator.- 3.2. Transposed Operations and Adjoint Operators.- 3.3. Existence of Proper Operators.- §4. Weak and Strong Extensions of Differential Operations.- 4.0. Preliminary Remarks.- 4.1. Basic Definitions.- §5. Averaging Operators.- 5.0. Preliminary Remarks.- 5.1. Averaging on the Line.- 5.2. Averaging in a Multidimensional Domain.- 5.3. Averaging and the Differentiation Operation.- 5.4. The Friedrichs Lemma.- §6. The Identity of Weak and Strong Extensions of Differential Operations.- 6.1. The Case of Mainly Constant Coefficients.- 6.2. The Case of Variable Coefficients.- 6.3. Some Examples.- 6.4. Equivalence of Weak and Strong Extensions as a Corollary of the Unique Solvability Problems.- 6.5. The Case of an Ordinary Differential Operator.- §7. W Spaces.- 7.0. Introductory Remarks.- 7.1. Weak an.

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• Titolo: Boundary Value Problems of Finite Elasticity: Local Theorems on Existence, Uniqueness, and Analytic Dependence on Data. (= Springer tracts in natural philosophy, Volume 31). Tullio Valent Fundamental Structure.- 1.2. Special Subsets.- 1.3. Operators.- 1.4. Functionals.- §2. The Spectrum of an Operator.- 2.0. Preliminary Remarks.- 2.1. Basic Definitions.- 2.2. Functions of an Operator.- 2.3. Connection Between the Spectra of an Operator and its Inverse.- §3. Special Classes of Operators.- 3.0. ...
• Editore: Springer, New York
• Data di pubblicazione: 1988
• Lingua: Inglese
• ISBN 10: 0387965505
• ISBN 13: 9780387965505
• Rilegatura: Hardcover
• Edizione: 1988.
• Condizione: gut
• Peso: 375 grammi

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Riassunto

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.

Contenuti

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.