Synopses & Reviews
In this famous work, a distinguished Russian mathematical scholar presents an innovative approach to classical boundary value problems and#8212; one that may be used by mathematicians as well as by theoreticians in mechanics. The approach is based on a number of geometric properties of conformal and quasi-conformal mappings. It employs the general basic scheme for the solution of variational problems first suggested by Hilbert and developed by Tonnelli. The method lies on the boundary between the classical methods of analysis, with their concrete estimates and approximate formulae, and the methods of the theory of functions of a real variable with their special character and general theoretical quantitative aspects.
The first two chapters cover variational principles of the theory of conformal mapping and behavior of a conformal transformation on the boundary. Succeeding chapters address hydrodynamic applications and quasi-conformal mappings, as well as linear systems and the simplest classes of non-linear systems.
Mathematicians will find the method of the proof of the existence and uniqueness theorem of special interest. Theoreticians in mechanics will consider the approximate formulae for conformal and quasi-conformal mappings highly useful in solving many concrete problems of the mechanics of continuous media. This classic work is also a valuable resource for researchers in the fields of mathematics and physics.
Synopsis
A famous monograph with an innovative approach to classical boundary value problems, using the general basic scheme for the solution of variational problems first suggested by Hilbert and developed by Tonnelli. Directed to both mathematicians and theoreticians in mechanics.
Synopsis
A famous monograph with an innovative approach to classical boundary value problems, directed to both mathematicians and theoreticians in mechanics. 1963 edition.
Synopsis
In this work, a distinguished Russian mathematical scholar presents an innovative approach to classical boundary value problems. The first two chapters cover variational principles of the theory of conformal mapping and behavior of a conformal transformation on the boundary. Succeeding chapters address hydrodynamic applications, quasi-conformal mappings, linear systems and the simplest classes of non-linear systems.
Synopsis
Famous monograph by aand#160;distinguished mathematicianand#160;presents an innovative approach to classical boundary value problems. The treatment employs the basic scheme first suggested by Hilbert and developed by Tonnelli.and#160;1963 edition.
Synopsis
In this famous monograph, a distinguished mathematician presents an innovative approach to classical boundary value problems that employs the basic scheme first suggested by Hilbert and developed by Tonnelli.and#160;The treatment covers variational principles of the theory of conformal mapping, hydrodynamic applications and quasiconformal mappings, linear systems, and other subjects.
About the Author
M. A. Lavrent'ev (1900and#8211;80) was a prominent Soviet mathematician, associated with Moscow State University and the Steklov Institute of Mathematics.
Table of Contents
and#160; Introduction
1. Variational principles
2. Sufficient conditions
3. Generalizations
Chapter 1. Variational principles of the theory of conformal mapping
and#160; 1.1 The principles of Lindeland#246;f and Montel
and#160;and#160;and#160; 1.1.1 The case of the circle
and#160;and#160;and#160; 1.1.2 Mapping on to a strip
and#160; 1.2 Mechanical interpretation
and#160; 1.3 Quantitative estimates
Chapter 2. Behaviour of a conformal transformation on the boundary
and#160; 2.1 Derivatives at the boundary
and#160; 2.2 Narrow strips
and#160; 2.3 Behaviour of the extension at points of maximum inclination and extreme curvature
Chapter 3. Hydrodynamic applications
and#160; 3.1 Stream line flow
and#160; 3.2 Generalizations
and#160; 3.3 Stream line flow with detachment
and#160; 3.4 Wave motions of a fluid
and#160; 3.5 The linear theory of waves
and#160; 3.6 Rayleigh waves
and#160; 3.7 The exact theory
and#160; 3.8 Generalizations
and#160;and#160;and#160; 3.8.1 Motion of a fluid over a submarine trench
and#160;and#160;and#160; 3.8.2 Motion over a bottom with a ridge
and#160;and#160;and#160; 3.8.3 Spillway with singularities
Chapter 4. Quasi-conformal mappings
and#160; 4.1 The concept of the quasi-conformal map
and#160; 4.2 Derivative systems
and#160; 4.3 Strong ellipticity
Chapter 5. Linear systems
and#160; 5.1 Transformations with bounded distortion
and#160;and#160;and#160; 5.1.1 Equi-graded continuity
and#160;and#160;and#160; 5.1.2 Almost conformal mappings
and#160; 5.2 The simplest class of linear systems
and#160;and#160;and#160; 5.2.1 Invariance with respect to conformal mappings
and#160;and#160;and#160; 5.2.2. Stability of conformal mappings
and#160;and#160;and#160; 5.2.3 Condition of smoothness of a transformation
and#160;and#160;and#160; 5.2.4 Application to arbitrary linear systems
and#160;and#160;and#160; 5.2.5 Existence theorem
Chapter 6. The simplest classes of non-linear systems
and#160; 6.1 Maximum principle
and#160; 6.2 The principle of Schwarz-Lindeland#246;f
and#160; 6.3 Quantitative estimates
and#160; 6.4 Inductive proof of Lindeland#246;f's principle
and#160; 6.5 The existence theorem
and#160; 6.6 Generalizations
and#160; 6.7 Hydrodynamic applications
and#160; References; Index