Vekua / Sneddon / Ulam Generalized Analytic Functions

1. Auflage 2014
ISBN: 978-1-4831-8467-8
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band Volume 25, 698 Seiten, Web PDF

Reihe: International Series in Pure and Applied Mathematics

Generalized Analytic Functions is concerned with foundations of the general theory of generalized analytic functions and some applications to problems of differential geometry and theory of shells. Some classes of functions and operators are discussed, along with the reduction of a positive differential quadratic form to the canonical form. Boundary value problems and infinitesimal bendings of surfaces are also considered. Comprised of six chapters, this volume begins with a detailed treatment of various problems of the general theory of generalized analytic functions as as well as boundary value problems. The reader is introduced to some classes of functions and functional spaces, with emphasis on functions of two independent variables. Subsequent chapters focus on the problem of reducing a positive differential quadratic form to the canonical form; basic properties of solutions of elliptic systems of partial differential equations of the first order, in a two-dimensional domain; and some boundary value problems for an elliptic system of equations of the first order and for an elliptic equation of the second order, in a two-dimensional domain. The final part of the book deals with problems of the theory of surfaces and the membrane theory of shells. This book is intended for students of advanced courses of the mechanico-mathematical faculties, postgraduates, and research workers.

Autoren/Hrsg.

Vekua, I. N.

Sneddon, I. N.

Ulam, S.

Stark, M.

Inhaltsverzeichnis

1;Front Cover;1
2;Generalized Analytic Functions;6
3;Copyright Page;7
4;Table of Contents;10
5;ANNOTATION;8
6;FOREWORD;28
7;PART ONE: FOUNDATIONS OF THE GENERAL THEORY OF GENERALIZED ANALYTIC FUNCTIONS AND BOUNDARY
VALUE PROBLEMS;32
7.1;CHAPTER I. SOME CLASSES OF FUNCTIONS AND
OPERATORS;36
7.1.1;§1. Classes of functions and functional spaces;36
7.1.2;§2. Classes of curves and domains. Some properties of conformal
mapping;48
7.1.3;§3. Some properties of Cauchy type integrals;52
7.1.4;§4· Non-homogeneous Cauchy-Riemann system;54
7.1.5;§5. Generalized derivatives in the Sobolev sense and their
properties;58
7.1.6;§6. Properties of the operator
Tg;69
7.1.7;§7. Green's formula for the class of functions D1p. Areal
derivative;83
7.1.8;§8. On differential properties of functions of the form TGf. Operator
II;87
7.1.9;§9. Extension of the operator
II;95
7.1.10;§10. Some other properties of functions of the classes DZ(G)
and D-Z(G;103
7.2;CHAPTEE II. REDUCTION OF A POSITIVE DIFFERENTIAL QUADRATIC FORM TO THE CANONICAL FORM. BELTRAMFS EQUATION. GEOMETRIC
APPLICATIONS;107
7.2.1;§1. Introductory remarks. Homeomorphisms of a quadratic
form;107
7.2.2;§2. Beltrami's system of equations;109
7.2.3;§3. Construction of the basic homeomorphism of Beltrami's
equation;110
7.2.4;§4. Proof of existence of a local homeomorphism;113
7.2.5;§5. Proof of the existence of a complete homeomorphism;123
7.2.6;§6. Reduction of a positive quadratic differential form to the canonical form· Isometric and isometric-conjugate coordinate
systems on a surface;132
7.2.7;§7. Reduction of equations of elliptic type to the canonical
form;154
7.3;CHAPTER III. FOUNDATIONS OF THE GENERAL THEORY OF
GENERALIZED ANALYTIC FUNCTIONS;161
7.3.1;§1. Basic concepts, terminology and notations;161
7.3.2;§2. Integral equation for functions of the class;169
7.3.3;§3. Continuity and differentiability properties of functions of the
class;171
7.3.4;§4. Basic lemma. Generalizations of some classical theorems;175
7.3.5;§5. Integral representation of the second kind for generalized
analytic functions;187
7.3.6;§6. Generating pair of functions of the class
Derivative in the Bers sense;192
7.3.7;§7. Inversion of the non-linear integral
equation;195
7.3.8;§8. Systems of fundamental generalized analytic functions and
fundamental kernels of tbe class;198
7.3.9;§9. Adjoint equation. Green's identity. Equations of the
second order;200
7.3.10;§10. Generalized Cauchy formula;203
7.3.11;§11. Continuous continuations of generalized analytic functions.
Generalized principle of symmetry;208
7.3.12;§12. Compactness;210
7.3.13;§13. Representation of resolvents by means of kernels;214
7.3.14;§14. Representation of generalized analytic functions by means
of generalized integrals of the Cauchy type;219
7.3.15;§15. Complete systems of generalized analytic functions.
Generalized power series;223
7.3.16;§16. Integral equations for the real part of a generalized
analytic function;233
7.3.17;§17. Properties of solutions of elliptic systems of equations
of the general form;235
7.4;CHAPTER IV.
BOUNDARY VALUE PROBLEMS;252
7.4.1;§1. Formulation of the generalized Riemann-Hilbert problem.
Continuity properties of the solution of the problem;252
7.4.2;§2. The adjoint boundary value Problem Â'. Necessary and sufficient conditions
of solubility of Problem A;259
7.4.3;§3. Index of Problem A. Reduction of the boundary condition
of Problem A to the canonical form;269
7.4.4;§4· Properties of the zeros of the solutions of the homogeneous Problem Â. Criteria of solubility of the Problems
 and A;273
7.4.5;§5. Investigation of special classes of boundary value problems of
the type A in the case 0 < n < m — 1;288
7.4.6;§6. On the conditions of correctness of Problem A;313
7.4.7;§7. Solution of Problem A by means of two-dimensional
integral equations. Application of the generalized principle of symmetry. Generalized Schwarz integral;323
7.4.8;§8. The boundary value problem of inclined derivative for an
elliptic equation of the second order;347
7.4.9;§9. Application of two-dimensional singular integral equations
to the boundary value problems;364
7.4.10;§10. Remarks concerning certain papers on Problem A. Some
formulations of more general problems;392
7.4.11;APPENDIX TO CHAPTER IV;397
8;PART TWO: SOME APPLICATIONS TO PROBLEMS OF THE THEORY OF SURFACES AND THE
MEMBRANE THEORY OF SHELLS;420
8.1;CHAPTEE V. FOUNDATIONS OF THE GENERAL THEORY OF INFINITESIMAL
BENDINGS OF SURFACES;422
8.1.1;§1. Equations of infinitesimal bending in vectorial form;424
8.1.2;§2. Equation of infinitesimal bending with respect to a Cartesian
coordinate system. The first proof of the rigidity of ovaloids;426
8.1.3;§3. The system of equations for the components of the displacement
field in an arbitrary coordinate system on the surface. Some criteria of rigidity;436
8.1.4;§4. A property of surfaces of the second order;450
8.1.5;§5. The rotation field. The characteristic equation of infinitesimal
bending;455
8.1.6;§6. Bending fields. Static field;463
8.1.7;§7. Variations of various geometrical quantities under infinitesimal
bending of a surface. Some criteria of rigidity;474
8.1.8;§8. Conjunction conditions on the contact lines. Some criteria
of rigidity of surfaces with edges. Bush constraints. Perfect clamping;493
8.1.9;§9. Some classes of rigid closed sectionally regular surfaces;519
8.1.10;§10. Some classes of rigid convex surfaces with edges;530
8.1.11;§11. Infinitesimal bendings of surfaces of revolution;555
8.2;CHAPTEK VI. PROBLEMS OF THE MEMBRANE THEORY OF
SHELLS;594
8.2.1;§1. Forces and moments due to the stress field;597
8.2.2;§2. Basic system of equilibrium equations of a shell;603
8.2.3;§3. System of equations of the membrane state of stress of
shells. Geometric interpretation;615
8.2.4;§4. New derivation of the characteristic equation;625
8.2.5;§5. Conditions of existence of the state (Ã). Boundary value
problems;626
9;REFERENCES;678
10;SUBJECT INDEX;690