Variational Methods for Boundary Value Problems for Systems of Elliptic Equations

by M. A. Lavrent'ev, J.R.M. Radok (Translator)

Overview

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

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ISBN-13:	9780486160283
Publisher:	Dover Publications
Publication date:	01/14/2016
Series:	Dover Books on Mathematics
Sold by:	Barnes & Noble
Format:	eBook
Pages:	160
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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.

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Variational Methods for Boundary Value Problems for Systems of Elliptic Equations

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

Dover Publications, Inc.

Copyright © 1963 P. Noordhoff, Ltd.
All rights reserved.
ISBN: 978-0-486-16028-3

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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 in D(Γ), the following results hold:

1°. for any r< 1, the region D([??]r) is contained in D(γr), where yr and [??]r can only coincide when [??] and Γ coincide;

2°. at the point z0

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.2)

where equality occurs only for [??] [equivalent to] Γ;

3°. if Γ and [??] have a point z1in common and at this point f' and [??] exist, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; (1.3)

4°. if Γ and [??] are annular with respect to z0, then at the point where the upper bound of the largest deformation is attained (λ > 1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.4)

In other words, if we consider regions containing the fixed point z0 and their conformal mappings on to the unit circle for which z0 goes over into the centre of the circle, then in the case of indentation of the boundaries:

1°. all equipotential lines move;

2°. the extension at the point z0 increases;

3°. at points of the boundaries which are not deformed the extension decreases;

4°. at points of maximum deformation the extension increases.

Proof. In the region D(Γ), consider the harmonic function p(x, y) = log [absolute value of f(z)/z]; by assumption, this function is regular in D and assumes on Γ the value log 1/[absolute value of z. If D([??]) belongs to D(Γ), one has on [??]

log [absolute value of f(z)] ≤ 0, log [absolute value of [??](z)] = 0;

consequently, by the maximum principle for harmonic functions, one has everywhere inside D([??]) (if [??] [not equivalent to Γ)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.5)

On the other hand,

log [absolute value of f(z)] = [absolute value of [??](z)] = log r

render the equations of the equipotential lines γ and [??]r; taking into consideration (1.5), we conclude therefore that D([??]r) is contained in D(γr) (for [??] [not equivalent to Γ).

The results 2°. and 3°. of Theorem 1.1 are obtained as limiting cases for r -> 0 and r -> 1, respectively; in order to derive 4°., it is sufficient to increase, in addition, the region D([??])λ times and to use 3°.

It is not difficult to derive from Lindelöf's principle another principle, which is also useful for applications:

Theorem 1.2 (Montel's principle). Let the regions D(Γ) and D([??]) contain the point z0and let Γconsist of arcs γ1and γ2such that γ1lies in D([??]) and γ2outside D([??]); in addition, let [??] be obtained from Γ by replacement of γ1by the curve [??]1lying outside D(Γ), and of γ2by [??]2lying in D(Γ) (Fig. 2). Further, let the transformations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

establish correspondence between the arcs γ1and [??]1and the arcs φ1and [??]1of the unit circle. Then

φ1 ≥ [??]1, (1.6)

where the equality sign is only possible for [??] [equivalent to] Γ.

In the case when the arcs γ1, [??]1 are sufficiently smooth so that f' and [??] exist on the boundary, Theorem 1.2 follows from 3°. of Theorem 1.1; in the general case, the theorem may be derived either by considering an arbitrary region as a limiting case of smooth regions or by reducing the problem directly to the case of distortion of the unit circle.

1.1.2. Mapping on to a strip. Let Γ0 and Γ be two smooth arcs the only common points of which are their ends a1 and a2; we will not exclude cases when one or both ends coincide with the point at infinity z = ∞. We will denote by D(Γ0, Γ) the region bounded by Γ0 and Γ, and by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.7)

the function mapping the region D(Γ0, Γ) conformally on to the strip 0 < υ < 1. Under these conditions the function f will be determined apart from a real constant which is unimportant for the following considerations. Now let γυ be the curve which becomes under the transformation (1.7) the straight line υ = const.

For the mapping function f(z, Γ0, Γ) one has a principle which is analogous to Lindelöf's principle:

Theorem 1.3. If the region D([??]0, Γ) is contained in D(Γ0, Γ), then

1°. for any υ, 0 < υ < 1, the region D([??]υ, Γ) is contained inside D(γυ, Γ);

2°. at any point z0of the curve Γ

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

3°. if the curves [??]0and Γ0have a common point z1, then one has at this point

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(in all tree cases the equality sign occurs only for [??]0 [equivalent to] Γ0);

4°. in addition, let it be known that all the curves Γ0, [??]0, Γ are represented by single-valued functions y = f0(x), y = [??]0(x), y = f(x); then we have at the point z2 = x2 + iy0(x2), where f0(x) = [??]0(x) attains a maximum, and at the corresponding point [??]2 = x2 + i]??]0(x2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. Consider the harmonic function

V = Im f(z, Γ0, Γ).

It is valid in D(Γ0, Γ) and assumes on Γ0 and Γ values which are equal to 0 and 1, respectively. By virtue of the maximum principle, the function V is positive everywhere inside D(Γ0, Γ) and 0 < V< 1; hence everywhere inside D([??]0, Γ)

V < [??]. (1.8)

On the other hand, one has everywhere along the lines γυ and [??]υ

V = υ, [??] = υ. (1.9)

Comparing (1.8) and (1.9), we obtain 1°. In order to derive 2°., it is sufficient to note that on Γ

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [partial derivative]/[partial derivative]n is the derivative in the direction of the normal to Γ.

1.2. Mechanical interpretation

The above principle has a very obvious mechanical interpretation. The function w = f(z, Γ0, Γ) may be considered as the complex potential of a velocity field of the steady motion of an ideal incompressible fluid. The motion is assumed to be two-dimensional, the liquid being contained between curvilinear walls Γ0 and Γ. The width of the strip of the w-plane on to which the function f maps the region of flow is determined by the amount of flow of the fluid, i.e., by the quantity of fluid passing through the section in unit time; in our case this amount is equal to unity. The curves γυ are the stream lines of the flow where the amount of flow between Γ0 and γυ is equal to υ; in future, we will say that γυ is the stream line corresponding to the amount υ.

In correspondence with established hydrodynamic terminology the variational principle may be given the following formulation:

If the wall Γ0, bounding the flow of a liquid, is pressed against the flow, then:

1°. all stream lines are brought closer to the other wall;

2°. the velocity of the flow along the untouched wall is everywhere increased;

3°. the velocity of the flow near the undeformed part Γ0is decreased;

4°. on Γ0, at the location of the greatest contraction, the flow velocity is increased.

1.3. Quantitative estimates

The variational principle studied above permits us to estimate the sign of the variation as a function of the character of the variation of the boundary of the mapped region. In this way we can construct upper bounds for different functionals. We will begin with applications of the principle to the theory of approximate mappings, to the construction of variations and estimates of the remainder term.

Thus, assume that we know the function w = f(z, Γ0, Γ) and let it be required to construct its variation for the transition from D(Γ0, Γ) to D(Γ0, [??]). Without restricting generality we can assume that [??] is contained inside D(Γ0, Γ); in addition, we will consider only regions D(Γ0, Γ) for which f'(z, Γ0, Γ) exists and is continuous in D, including its boundary (the conditions which must be satisfied by Γ0 and Γ for this case will be formulated in the following chapter).

In the case of the transformation ω = f(z, Γ0, Γ), ω = [xi] + iη, the curve [??] goes over into the curve L which lies close to the straight line η = 1; let the function w = F(ω, L), F(± ∞, L) = = ± ∞ bring about the mapping of the region D(L), bounded by the straight line η = 0 and L, on to the strip 0 < υ < 1; then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.10)

The function F maps the region "close" to the strip 0 < η < 1 on to the strip 0 < υ < 1; hence

F = ω + φ(ω, L), (1.11)

where φ is a "small" function. Substituting in (1.10) the expression for F from (1.11), we find

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.12)

Our problem has been reduced to the determination of the linear part of the function φ. Differentiating (1.12), one can obtain the variation for f'

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

or

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.13)

We will now compute the imaginary part of F to be denoted by V. On the curve L: η = 1 = λ([xi]), the function V must assume the value unity, on the axis η = 0, the value zero. Let [??] denote the harmonic function which assumes for η = 0 the value zero, and on the straight line η = 1 the value 1 + λ([xi]); then [??] has on the line L the value

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

however,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where ε is small for sufficiently small and smooth functions λ([xi]). Assume that V = [??]. By a well-known formula of the theory of conformal mapping (M. A. Lavrent'ev and B. V. Shabat, 1958, p. 213), one has

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.14)

and hence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In particular, for η = 0,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.15)

Thus, we obtain for the required variations formally the formulae: inside the region:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.16)

for points where η = 0:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.17)

for points of L:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.18)

(Continues...)

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Excerpted from Variational Methods for Boundary Value Problems for Systems of Elliptic Equations by M. A. Lavrent'ev, J. R. M. Radok. Copyright © 1963 P. Noordhoff, Ltd.. Excerpted by permission of Dover Publications, Inc..
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Table of Contents

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,