III. Boundary Value Problems of Function Theory. Their Basic Concepts
We have already said that the theory of functions of a complex variable and especially its geometric part - the theory of conformal mapping - arose and was developed on the basis physical ideas. Leonhard Euler and Jean d'Alembert came to the analyticity of such functions from hydro-mechanical work. Bernhard Riemann in his studies constantly used interpretations of analytic functions, linked to plane flow of fluids and heat flow.
On the other hand, conversely, the development of the theory of functiosn of a complex variable allowed to extablish new methods of solution of important practical problems from diiferent sections of mathematical sciences ( hydro- and aero-dynamics, elasticity theory, electro-statics, magnetic and hear flows, etc.). We should note that a lot of the relevant work was done by Russian scientists.
Niklai Egorovich Joukowski and Serbei Aleksandrovich Chaplygin (1869 - 1942) obtained many of the basic results when applying function theory to hydro- and aero-dynamics. The methods of the theory of functions of a complex variable had a very important role in their important papers as well as in Joukowsky's book "The Theoretical Foundations of Aeronautics" (1911). Also in the second part of the 20-th Century, important applications of function theory to hydro- and aero-dynamics were published by M.V.Keldysh, S.A.Christiianovich, V.V. Golubev, L.I. Sedov, et.al.
G.V.Kolosov (1867 - 1936) initiated in 1909 the use of function theory in the plane theory of elasticity. The brilliant solution of these problems by methods, based on function theory, was found in the Twenties by N.I.Muskhelishvili. He explained these methods in his book , which appeared first in 1932 and was translated by Rainer Radok into English in 1951. The methods of the function theory also occupy a large part in thudies of different fields of physics (V.A.Fok, G.A.Hildert, et. al.).
We will consider in this chapter the basic physical ideas, linked to the theory of functions of a complex variable and its simplest applications. We start with the theory of harmonic functions of two variables, closely linked to the potentials of plane vector fields, the basic boundary value problems of the theory of harmonic and analytic functions and then, on the basis of the developed theory, concern ouraselves with the basic questions of applications.
3.1 Harmonic functions
Functions, harmonic in the domain D are called the real functions u(x, y) of the two real variables, which have in this domain continuous real continuous second second order partial derivatives and satisfy the differential equation*
( is the symbol of the differential operator). As a rule, this equation is called Laplace's equation. However, Laplace considered it only in 1782, while Euler used earlier it in his work on hydrodynamics and other branches of mathematical physics. We note immediately that, since the equation is linear, any linar combination
of harmonic functions uk(x, y) with real constant coefficients ak is again a harmonic function.
* We will always be here concerned with harmonic functions of two variables, because they are closely linked to analytic functions. In applications, harmonic functions of three variables u(x, y, z) are just as important; they satisfy the equation
which, however, we will not deal with.
As we will see in thsi chapter, the potentials of the very important vector fields considered in Physics are harmonic functions and any harmonic function may be represented physically as the potential of a vector field. Hence, also in the general case, harmonic functions are often called potentials and the theory of harmonic functions potential theory.
41. Properties of harmonic functions First of all, we will explain the links between the concepts of analytic and harmonic functions. They are expressed in the following two theorems:
Theorem 1 The real and imaginary parts of an arbitrary function f(z) = u(x, y) + iv(x, y) which is single-vakued and analytic in a domain D are in this domain harmonic functions.
The proof follows directly from the D'Alembert-Euler conditions
In fact, since analytic functions have derivatives of all orders, Equation (1) my be differentiated with respect to x and y. Differentiating the first of these with resepct to x, the second with respect to y and employing the theorem of the equality of mixed derivatives, we find
The proof for the function v(x, y) is analogous.
Two functions u(x, y) + iv(x, y), linked by the D'Alembert-Euler conditions, harmonic in the domain D, are said to be conjygate.
Theorem 2 For any function u(x, y), harmonic in a simply-connected domain D one may find a harmonic function v(x, y) conjugate with it.
In fact, we consider the integral
where z0 = x0 + iy0 is a fixed and z = x + iy is a variable point in the domain D. By Laplace's equation,
this integral does not depend on the path of integration and is a function of z only; we denote this function by v0(x, y). Using the properties of curvi-linear integrals
(we may take the integral from z to z + h along the horizontal segment, on which dy = 0); analogously, we have Hence, v0(x, y) is also a known function, the conjugate to the function u(x, y). Since the function is determined exactly by its partial derivatives apart froma constant term, the set of all harmonic functions, conjugate to u(x, y), is given by
where C is an arbitary (real) constant.
We note that in a multiply-connected domain D Integral (2)
is, generally speaking, a multi-valued function. It may assume different values along two paths L and joining the points z0 and z, if it is impossible to deform these paths from one into the other without leaving the domain D(i.e., if inside the domain, bouneed by L and there are points which do not belong to D). Obviously, we can follow in this case the corresponding reasoning in 13. and can assert that in multiply-connected domains the general formula for the values of the function v(x, y), determined by (2), has the form
where Nk are arbitrary integers and Gk are integrals along closed contours gk, each of which contains inside it one connected part of the boundary of D:
(cf. Equations (2) and (3) in 13.), The constants Gk are called the periods of the integral (2) or its cyclic constants.
If one may single out in a certain domain D', lying in D, a single-valued and continuous branch of the function v(x, y), determined by (3), then this branch is obviously a harmonic function, conjugate to u(x, y), whence the function v(x, y) is a multi-valued harmonic function. Note that the partial derivatives of thsi function are single-valued:
this follows from (3).
Obviously, Theorem 2 may be reformualted as follows:
Theorem 3' One may consider any function, harmonic in D, as teh real or imaginary part of some analytic function f(z); it is determined apart from a constant term which is imaginary or real, respectively.
We will not exclude the case of multiply-connected domains, whence the analytic function f(z) may turn out to be multi-valued.
Example A calculation of its partial derivatives shows that the function
is harmonic in the ring 0 < |z| < ¥. Integral (2) has the form
and represents in the ring 0 < |z| < ¥ an infinitely-valued function. The corresponding analytic function
is also infinitely-valued.
Theorem 3 Any harmonic function u(x, y) is an analytic function of its arguments x and y, i.e., in the neighbourhood of every point z0 = x0 + iy0 of the domain D it can be represented in the form of the sum of an aboslutely converging series
In fact, by Theorem 2', u(x, y) can be viewed as the real part of the function f(z), single-valued and analytic in some neighbourhood |z - z0| < R of the point z0. Let in this neighbourhood
where cn = an + ibn. The absolute value of the
real part of the general term of Series (6)
does not exceed
and since by Abel's Theorem of 19. it converges absolutely in any circle |z - z0| £ r < R, i.e., the series converges for r < R, also the series with the general term (7) will converge absolutely for |x - x0| + |y - y0| < R. This series also is one for u(x, y). After regrouping its terms (which is allowed due to the proved absolute convergence), we obtain the required series (5), and the theorem has been proved.
In particular, it follows from the proved theorem that harmonic functions have partical drivatives of all orders. It is readily proved that also harmonic functions have the last property.follows (cf. Theorem 1 of 17.)
Based on Theorem 3, one may obtain practically convenient methods for the establishment of an analytic function f(z) from its known real part. Transforming in an elementary manner the general term of Series(7) for u(x, y), we obtain the representation of this function in the neighbourhood of the point z0:
By Abel's Theorem, this series converges also for complex values x and y, sufficiently close to x0 and y0, whence we may set in it
where z is a point sufficiently close to z0, and we obtain:
Replacing here z by z, we find after simple transformations
Fomula (8) has been obtained for points z, close to z0, but, by the Uniqueness Theorem, it obviously remains true in the entire domain of definition of f(z), because in thsi doamin both parts of (8) are analytic functions of z.
In particular, if f(z) is analytic at the origin of co-ordinates, then one may set z0 = 0, and (8) assumes the specially simple form:
We now present several examples of the application of (8) and (9):
by (8), z0 = 1;
by (8), z0 = p/2.
In these three formulae, C is a purely imaginary constant.
We now will consider the properties of harmonic functions. On the basis of Theroems 1 and 2, these properties are easily obtained from the corresponding properties of analytic functions. For the sake of convenience, we will write again u(z) instead of u(x, y), u(P) instead of u(x1, x2, ··· , xn) for functions of several variables, undersatnding by P the point with the co-ordinates (x1, x2, ··· , xn).
Theorem 4 (mean value) If the function u(z) is continuous in the closed circle with radius r with centre at the point z and harmonic inside this circle, then
The proof follows directly from (5) in 14. by separation the of real part.
Theorem 5 A harmonic function other than a constant cnnot attain an extreme value at an internal point of its domain of definition.
It is sufficient to prove the theorem for the case of a maximum, because the point of a minimum of a harmonic function u(z) is a the point of the maximum of the function -u(z) which is also harmonic. Assuming the contrary, we we propose that the harmonic function u(z) attains a maximum at the internal point z0 of the domain.
We construct in the neighbourhood of the point z0 a single-valued function f(z) such that u = Re f(z). The function ef(z) is analytic and its modulus eu(z), by our assumption, attainsa maximum at an internal point z0 of the domain. This contradicts the maximum principle in 15., whence the theorem has been proved.
One might prove Theorem 5 directly on the basis of the mean value theorem just as the maximum principle has been proved in 15.
Theorem 6 If the function u(z), harmonic in the entire open plane is bounded, be it above or below, it is constant.
In fact, let u(z) be bounded above: u(z) < M. We construct a function f(z), analytic in the entire open plane, such that u(z)=Re f(z). By the condition of the theorem, all values of the function w = f(z) lie in the half-plane u < M, whence, by the Note at the end of 28., the function f(z) is constant, i.e., also u(z) is constant.
The following two theorems establish the character of level lines of harmonic functions, i.e., the sets of the points at which u(z) = const.
Theorem 7 If not a constant the harmonic function u(z) has a closed level line u(z) = u0, then insode this line lies at least one special point* of this function.
* At such a point, the function is not harmonic.
In fact, otherwise u(z), continuous in a closed domain, bounded by a level line, must attain a largest value u(z1) and a smallest value u(z2). By Theorem 5, the points z1 and z2 must lie on the boundary of the domain, i.e., on a level line; consequently, u(z1) = u(z2) and the function u(z) is constant.
Theorem 8 Any sufficiently small neighbourhood of a point z0 of a level line u(z) = u0 subdivides this line into an even number 2n (n ³ 1) of sections in which u(z) assumes in turn values larger and less than u0 .
The function u(z) - u0 vanishes at the point u0 ; the function v(z) becoming there equal to it so that v(z0) = 0, we obtain the analytic function
which also vanishes at z0. We denote by n the order of this zero when we have in the neighbourhood of the point z0
where z - z0 = reij, A ¹ 0, B some constant and o(rn) denotes a small quantity of order higher than rn as r ® 0. Hence, it is seen that for sufficiently small r, as j ranges from 0 = 2p, u - u0 vanishes 2n times, changing irs sign, and the theorem has been proved.
In exactly the same way one proves that the level lines of the harmonic function v(z), conjugate to u(z), passing through the point z0 , are subdivided in the neighbourhood of thsi point into n branchestouching in z0 the bisectors, mentioned in Theorem *.
It follows from Theorem 8 that the level lines of harmonic functions heve only simple points (n = 1) or multiple points * with different tangents (n > 1) - the case of isolated points, end points or points of return are excluded.
* In any closed domain, the harmonic property of functions u(z) yield a finite number of multiple points of level lines (at each such point f'(z)=0); other wise, by the uniqueness theorem (20.), one must have f '(z) º 0.
In the sequel, it will be useful to note the following proposition - the inverse of the mean value theorem.
Theorem 9 If the function u(z) is continuous in the domain D and at any point z for sufficiently small r
then the function u(z) is harmonic in D.
Our proof is based on the existence theorem of harmonic functions, taking on the boundary of a simply-connected domain given values; this theorem will be proved in 43. Let z0 - an arbitary point in D and 0 - a closed simply-connected domain which belongs to D and contains the point z0. By the existence theorem, we construct the harmonic function u0(z) which assumes on the boundary C0 of 0 the same valueas also the function u(z) and denote U(z) = u0(z) - u(z).
By the dsign and conditions of the proved theorem, U(z) is continuous in 0 and equal to zero on the boundary of this domain. Besides, the value of U(z) at the centre of any circle, belonging to 0 is equal to the arithemtic mean of its values on circumefrence of this circle, because both the functions u(z) and u0(z) have this property: the first, by assumption, the second by the mean value theorem.
Hence it follows that the function U(z) cannot attain an extreme value at internal points of D0; the proof of thsi proposition applies only to the continuity of the functions and the mean value theorem (cf., the comments after Theorem 5). However, since a function U(z), continuous in a closed domain, must attain its extreme values, it attains them on the boundary of 0. But, since on the boundary everywhere U(z) = 0, then the largest and smallest values of U(z) equal zero, whence U(z) º 0 everywhere on 0. This means that everywhere in 0 the function u(z) coincides with the jarmonic function u0(z) and, in particular, is harmonic at the point z0. Since z0 is an arbitary point of D, the theorem has been proved.
We proceed next to a theorem, analogous to Weierstrass' Theorem in 19.
Theorem 10 Let there be given a sequence of functions u0, u1, ··· , un, ···, harmonic in the domain D and continuous in . If the series converges uniformly on the boundary of D, then it also converges uniformly inside D, where its sum is a function, harmonic in D.
The uniform convergence of the series inside D follows from the extremum principle. In fact, by a known Cauchy cobvergence test *, there follows from the uniform convergence of the series on the boundary of the domain D that for any e > 0 one can find an integer N such that for any n > N any positive integer p and all points z of the boundary
Since the absolute sum of the terms here is harmonic, by the extremum principle, also for all points of the domain
Hence follows the uniform convergence of the series . There remains to prove that the sum of this series u(z) is a ahrmonic fucntion. We use for this purpose Theorems 9 and 4. For any sufficiently sma;; r, we have
(thsi integration of the series is admissible due to its uniform convergence). By Theorem 3, the integrals on the right hand side are equal to 2puk(z), whence
and, by Theorem 9, the function u(z) is harmonic at the point z. The theorem has been proved, since z is an arbitary point of the domain D.
In conclusion, we note yet two theroems, useful in the sequel. The first of them expresses that the property of fucntions to be harmonic is not violated by analytic transformation of the independent variable.