Variational Methods for Boundary Value Problems for Systems of Elliptic Equations (Dover Books on Advanced Mathematics) - Softcover

Lavrent'ev, M. A.

Softcover

ISBN 10: 0486661709 ISBN 13: 9780486661704

Publisher: Dover Publications Inc., 2016

Synopsis

In this famous monograph, a distinguished mathematician presents an innovative approach to classical boundary value problems ? 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 and employs the general basic scheme for solution of variational problems first suggested by Hilbert and developed by Tonnelli.
The first two chapters cover variational principles of the theory of conformal mapping and behavior of a conformal transformation on the boundary. Chapters 3 and 4 explore hydrodynamic applications and quasiconformal mappings, and the final two chapters address linear systems and the simplest classes of non-linear systems. Mathematicians will take particular interest in the method of the proof of the existence and uniqueness theorems as well as the general theory of quasi-conformal mappings. Theoreticians in mechanics will find the approximate formulas for conformal and quasi-conformal

About this title

About the Author

M. A. Lavrent'ev (1900–80) was a prominent Soviet mathematician, associated with Moscow State University and the Steklov Institute of Mathematics.

Excerpt. © Reprinted by permission. All rights reserved.

Variational Methods for Boundary Value Problems for Systems of Elliptic Equations

By M. A. Lavrent'ev, J. R. M. Radok

Dover Publications, Inc.

Copyright © 1963 P. Noordhoff, Ltd.
All rights reserved.

ISBN: 978-0-486-66170-4

Contents

Introduction, 9, 
Chapter I. Variational principles of the theory of conformal mapping, 16, 
Chapter II. Behaviour of a conformal transformation on the boundary, 26, 
Chapter III. Hydrodynamic applications, 42, 
Chapter IV. Quasi-conformal mappings, 72, 
Chapter V. Linear systems, 90, 
Chapter VI. The simplest classes of non-linear systems, 118, 
References, 146, 
Index, 151, 

CHAPTER 1

VARIATIONAL PRINCIPLES OF THE THEORY OF CONFORMAL MAPPING

1.1. The principles of Lindelöf and Montel

Let there be given in the plane of the complex variable z two simply connected regions D and [??], bounded by curves Γ and [??], and let w = f(z) and w = [??](z) be functions which map the regions D and D on to one of the standard regions: circle, half-plane or strip. As supplementary conditions which determine the mapping uniquely let us require that for [??] [equivalent to] Γ we have [??] [equivalent to] f.

At the basis of the application of variational methods to the theory of functions of a complex variable or to problems of mechanics which can be solved by the methods of this theory we have the following problem.

Assuming w = f(z) to be known and [??] to be infinitely close to Γ find the variation of f(z), i.e., the principal linear part of the variation of f(z) as Γ -> [??].

We will consider the cases of mappings on to the circle, half-plane and strip separately for the corresponding, most frequently encountered normalisations.

1.1.1. The case of the circle. Let D = D(Γ) denote the simply connected region bounded by the line Γ. Select in D some fixed point z0 and map D conformally on to the unit circle [absolute value of w< 1 so that the point z0 corresponds to the point w = 0:

[w = f(z, Γ), f(z0, Γ) = 0. (1.1)

There will be infinitely many functions having this property, but all of them will differ by factors eiθ, where θ is an arbitrary, real number. The multiplier eiθ will not play any role in what follows and we will understand by f any function of this class.

We will denote the closed curve corresponding to the transformation (1.1) to the circle [absolute value of w] = r< 1 by γr; on replacing Γ by [??], the corresponding function f and curve γr will be denoted by [??] and [??]r, respectively.

Consider the polar coordinate system r, φ with the pole at the point z0 and assume that in this system of coordinates the radii r and [??] of the points of Γ and [??] are single-valued functions of φ (i.e., the regions D(Γ) and D([??]) form rings around z0). We will call the points z1 = r1eiθ1 of the contour Γ, at which λ = = r(φ)/[??](φ) attains a maximum and z2 = r2eiθ2 where λ attains a minimum points of largest deformation of Γ (with respect to z0); the corresponding numbers λ1 and λ2 will be called upper and lower bounds of deformation.

We will now formulate

Theorem 1.1 (Lindelöf's principle). If the region D([??]) is contained

Edition details

• Publisher: Dover Publications Inc.
• Publication date: 2016
• Language: English
• ISBN 10: 0486661709
• ISBN 13: 9780486661704
• Binding: Paperback
• Number of pages: 160