ILLINOIS UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN PRODUCTION NOTE University of Illinois at Ufbana-Champaign Library Brittle Books Project, 2017.COPYRIGHT NOTIFICATION In Public Domain. Published 1923-1977 in the U.S. without printed copyright notice. This digital copy was made from the printed version held by the University of Illinois at Urbana-Champaign. It was made in compliance with copyright law. Prepared for the Brittle Books Project, Main Library, University of Illinois at Urbana-Champaign by Northern Micrographics Brookhaven Bindery La Crosse, Wisconsin 2017515.35 R21d cop.2iliiDirichlet's Problem A DISSERTATION presented to the Faculty of Princeton University in Candidacy for the Degree of Doctor of Philosophy by GEORGE E. RAYNOR PRESS OF THE NEW ERA PRINTING COMPANY LANCASTER, PA 19233** Accepted by the Department of Mathematics, June-1923.[Reprinted from Annals of Mathematics, Vol. XXIII., No. 3, March, 1922.j DIRICHLET'S PROBLEM. By George E. Raynor.* 1. The main object of the following paper is to give a solution of Dirichlet's problem valid for less restricted types of boundaries than those hitherto considered. On the whole, the argument follows the classical lines closely and involves a compromise between the Schwarz alternating process and the Poincare "Methode du balayage." A large part of the paper may, therefore, be regarded as a simplified expository development of certain well-known theorems on potential theory. Although the problem is treated in three dimensions only, the method is equally appli- cable to n. The writer here wishes to acknowledge his indebtedness to Professor J. W. Alexander, who has assisted him with numerous suggestions through- out the preparation of the paper. 2. For the purposes of this paper, a region R will be a set of points in three-space such that (1) to each point of the set there corresponds a sphere which encloses no point not of the set, (2) there exists a sphere enclosing all the points of the set, (3) given any two points Pi and P2 of the set, there is always a continuous arc P(t), h ^ t ^ t2, made up of points of the set and joining Pi to P2: Pi = P(ti), P2 = P(t2). The boundary B of the region R will be the set of all limit points of the region which are not themselves points of the region. The set R + B consisting of the points and boundary points of a region will thus be a closed set. A func- tion F is said to be continuous on a set of points C if it has a finite value at -every point of C and if to every point P of C and every number e > 0 there exists a number 5eP > 0 such that if P' be any point of C within a distance deP of P, \F(P) — F(Pr) | < e. A function V(x, yrz) is said to be harmonic in a region if at every point of the region it possesses first and second derivatives and if its second derivatives satisfy Laplace's equation I d2F d2V d2V i A27 = ^4 + ^ = 0. ! dx2 dy2 dz2 * Presented to the American Mathematical Society, December 28, 1922. 183184 george e. raynor. Dirichlet's Problem. Let U(x, y, z) be a function defined on the boundary B of a region R and continuous on B. The problem will be, if possible, to find a function V(x, y, z) which is continuous over the domain R + B, harmonic in R, and identical with U(x, y, z) on B. During the course of the discussion, we shall also have occasion to deal with the following slight extension of this problem, though only in the case where the boundary B of the region R consists of a finite number of analytic surface elements. Let U(x9 y, z) be a function bounded on B and continuous at all points of B except along a finite number of analytic arcs A where U{x, y, z) need not be defined. The problem will then be to find a function V(x, y, z) which is bounded and continuous over the domain R + B — A, harmonic in R and identical with U(x, y, z) in B — A. 3. In this section we shall prove a number of fundamental theorems concerning harmonic functions. Dirichlet's problem may be solved for the region interior to a sphere by means of Poisson's integral* which defines the value of the required function V at any point within the sphere. In this formula the integral is extended over the surface of the sphere, R is the radius of the sphere, p the distance from the center to the point (a, by c),r the distance from (a, 6, c) to a variable point on the surface of the sphere and U is a continuous function of position on the surface of the sphere. The above integral is harmonic in a, 6, c and is such that as (a, by c) approaches a point of the surface in any manner whatever, V(a, by c) will approach the value of U at that point. The same integral also solves the extended Dirichlet problem when U is a bounded function continuous except perhaps along a finite number of analytic arcs. In fact, provided merely that the function U be bounded and integrable in the. sense of Lebesgue, the integral (1) will define a function V harmonic within the sphere and such that as an interior point P approaches a point P0 of the sphere at which U is continuous, the value V(P) will approach, the value U(Po). If in formula (1) we put p = 0, r will become constant and equal to R , and we shall obtain Gauss' mean value theorem ! * For a derivation of this formula see, for example, Goursat, Cours d'Analyse Mathematique,, vol. 3, Chap. 28. (i) (2)i dirichlet's problem. 185 which gives the value of a harmonic function at the center (a0, b0j c0) of a sphere as the average of its values on the surface. This formula shows at orice that, if we consider the value of a function at an interior point of a re,gion in which the function is harmonic, the values of the function in every small neighborhood of this point cannot be all greater or all less t^an the value of the function at that point. Hence we have at once the following theorem. ; Theorem 1. If a function is harmonic in a region R, it can have rjeither a maximum nor a minimum in R. ; Here, we are, of course, using the terms maximum and minimum in the restricted sense. Theorem 2. If a function V is harmonic in a region R with boundary B and continuous in the domain R + B, the greatest and least values of V in + B are attained on the boundary B. j For a function which is continuous on a closed set of points is bounded and actually attains its least upper and greatest lower bounds. Theorem 3. If the function V, harmonic in R and continuous in U + B, is constant (positive, negative) on B, it is constant (positive, negative) in R + B. Theorem 4. If V1 and V2 be functions harmonic in R and continuous in R + B cfnd if V\ = V2 (Fi > F2, Fx < V2) at every point of B, then V\ = F2 (Fi > Vt, Vi < V2) at every point of R + B. This is seen on putting V = Vi — V2 in the previous theorem. In other words, we have Theorem 5. If a solution of Dirichlet's problem exists, the solution is unique. We can also prove without difficulty that the extended Dirichlet problem referred to at the end of § 2 never admits of more than one solution. In other words, Theorem 6. If Vi and V2 be two functions which are bounded and continuous in the domain R + B — A, harmonic in R and equal in B — A, the functions are identical in R + B — A. To prove the theorem we have only to show that the function V = Vi — V2 which is bounded and continuous in R + B — A, harmonic in R and zero on B — A must vanish at all points of R. This we do with the aid of a comparison function. Let P<> be any point of R and V(P0) the value of V at this point. We may assume without loss of generality that V(Po) S 0, for if F(P0) were negative we could work equally well with the function — F instead of F. Moreover, since F is bounded, there exists a constanti 186 george e. raynor. M such that l; V(P) < M « at all points of R. Now let ju be a positive constant and r the distance from a point of the system of arcs A to an arbitrary point (x, y, z) of space. Then the integral (5) I{P) = evaluated over all the arcs A defines the potential field due to a line distri- bution of density /z over A. Thus, I(x, y, z) is positive and harmonic ait every point P not of A and approaches infinity as the point P approaches a point of A. Suppose now that the value of jjl be chosen so small that at the point Pp, I(Pq) < M. i Then the points P of R such that » I{P) < M \ \ will form one or more sub-regions of R, one of which R' will contain the point P0. Moreover, each point of the boundary of Rf will either be a point of B — A or a point of the equipotential surface * \ I{P) = M \ of the function I(P). Now at a boundary point of the first sort I(P) > 0 and V{P) = 0, while at a point of the second sort I(P) = M, V(P) ^ M. Thus on the entire boundary of R' we shall have I(P) ^ V(P). Therefore, by Theorem 4, since the functions I (P) and V (P) are both con- tinuous on the boundary of Rthis last relation is valid at every point within R' and in particular at the point P0. It follows at once that the value of V(Po) cannot be positive; otherwise, by choosing the constant ^ sufficiently small we could make J(Po) < F(Po) and thus be led to a contradiction. As an immediate corollary to Theorem 6, we have the following theorem which will be needed later on in the discussion. Theorem 7. If a function V be bounded and continuous in R + B — Ay harmonic in R and non-negative on B — A, it is non-negative in R + B ~ A.dirichlet's problem. 187 This may be seen at once with the aid of the comparison function (5), where fx is now taken as a negative quantity which is allowed to approach zero. 4. From Poisson's integral we can derive a well-known inequality* which will be useful in proving the next theorem. In the formula F(o'b-c) ~ii.f£u ~s/- d° let U ^ 0 everywhere on S. Then, by Theorem 3, V will be positive everywhere within S. Let F0 be the value of V at the center of S and Vp its value at the point P. The maximum value of r is evidently R + p and its minimum value R — p. We have then, replacing r by R + p, > W,um~+7)>d° - But by Gauss' formula (2), we have Uda = 47tW0, ft and hence finally o\ y ^ R{R — p) y 1 " > (R + p)t , 0- In a similar manner, replacing r by R — p, we obtain (4) K, < V, Harnack's Theorem. If a sequence of monotonic increasing functions, Ui(x, y, z), -• • •, un{x, y, z), • • •, all of which are harmonicin a regioji R, con- verges at one "point P of the region, it will converge at all points of R and the limit function will be harmonic in R. Let S be a sphere with center P and radius R lying entirely within the given region. Let A be any point in S at a distance p from P. If we con- sider the difference (un+v — un)P, we have from the inequality (4) for all values of the indices n and p (Un+p Un) A < ^ _ ^y2 (un+p Un) P) which proves the convergence of the sequence at the point A. The above expression also shows that within and on any sphere S'. with the same center P but with radius Rf < R our sequence Ui(x, y, z), • • •, un(x, y, z), • • • converges uniformly to a limit function u which may be written as a * Cf., for example, Goursat, loc. cit.188 george e. raynor. uniformly convergent series, u = ux + O2 - «i) + • • • + (un — un-i) + ■■■ = Z(«n - Un-1), U0 = 0. Replacing each term in this series by its value given by Poisson's integral for the sphere S' we have « = — u«-1) = f(Un' ~ Rr3P d*' =^I£ E(un' ~ Un-')R r/ d<7'' the signs £ and f f being interchangeable since — un-\) is uni- formly convergent on S'. By our previous discussion we know that the last integral above is harmonic and we have that the limit function u is harmonic within S Having established the theorem for the points in S' we shall now prove that it is true at any other point Q of the region R. Suppose that the theorem fails for the point Q. Join P and Q by a continuous arc A(t), h ^ t ^ t2, A(t 1) = P, A(t2) = Q, lying in the region R. Proceeding from P to Q along this arc we can then find a point R which is either the first point at which the theorem fails or is the last point which is such that the theorem is true for all points preceding it. But either of these situa- tions is impossible, for if we take a sphere S" lying entirely in the region R, having its center on the arc PR, and enclosing the point R, we have immediately by the first part of the proof that the theorem is true in this sphere and hence true for points immediately following R. Hence our supposition that the theorem fails at Q is false and the theorem is proved. Theorem 8. If the sequence of functions Ui(x, y,z), • • •, un(x, y, z), defined in R + B and harmonic in R converges uniformly everywhere on the boundary B of R, it will converge uniformly everywhere in R + B and the limit function will be harmonic in R. Let U1, U2, - j Un, - • • be the values which U\, u2, • • •, un, • • • take on the boundary B. Then by hypothesis if an e > 0 be given, we can find an m such that for n ^ m and for all positive values of p we will have at all points of B | TJn ZJn-\~p ] c. In particular this inequality holds for the maximum value of the left-hand member and hence we have by Theorem 2, at all interior points of Ry \un Ufi+p { < 6, which proves the uniform convergence.dirichlet's problem. 189 That the limit function is harmonic in R can now be proved precisely as in the latter part of the previous theorem. Schwarz's Alternating Process. 5. This is a method whereby it is shown that if Dirichlet's problem has a solution for each of two overlapping regions R and Rf, then, under suitable conditions, it has a solution for the entire region R + R' covered by the original pair of regions. It will be sufficient for our purposes to confine our attention to the case when the region R is the sum of the interiors of a finite number of spheres Si, S2, • • •, Sn, no two of which are tangent to one another, and when the region Rf consists of the interior of a single sphere Sf such that Sf is not tangent to any of the spheres Si, • • Sn. We shall also assume that the regions R and Rf overlap but that neither contains the other. It is then to be proved that if the ex- tended Dirichlet problem (§ 2) is always solvable for R it is always solvable for R + Rf- We know, of course (§3), that the extended problem is solvable for Rf. Let B be the boundary of the region R and C the set of curves in which the boundary B intersects the boundary S' of R'. Moreover, let E and I be the portions of B exterior and interior to Sf respectively and Er and /' the parts of Sr exterior and interior to B respectively. Under certain con- ditions it may happen that either E or E' contains no points at all, that is to say, that the boundary of one of the two regions lies wholly within the other region. This will not invalidate the argument, however. On the boundary of the region R + Rf, a set of values W{P) is given such that W(P) is bounded, | W(P) | < M, and continuous over all of the boundary B with the possible exception of certain analytic arcs A. The problem is then to determine a solution of the extended Dirichlet problem for the region R + Rr corresponding to the arbitrary boundary value W(P). Evidently we may assume without loss of generality that the function W(P) is positive everywhere on B — A. For, as the function W is bounded, there exists a constant C such that W + C is positive on B — A, Moreover, if we can solve the problem corresponding to the positive boundary values W + C, the required solution for the boundary values W will be obtained by merely subtracting the constant C from the previous solution. Now let uL be a function harmonic in R, taking the assigned value W on E and the value zero on /. Then by Theorem 3 if the boundary values are continuous, or by Theorem 7 if they are discontinuous, the function will take a system of positive values on Now let Ui be a solution for190 GEORGE E. RAYNOR. the region R' taking the values of ui on /' and the required values] TF on E The function u\ will have a certain set of positive values on /, and we can form a new harmonic function for R taking the required values W on E and the values of U\ on I. Proceeding in this manner by alternating back and forth from region R to region R' we obtain two sequences of functions, 1^1, 11%, * " ' j Un) ' j Ui, U2', Un, • • -, the first set being harmonic and positive in R and taking the required values on E, and the second harmonic in Rf and taking the proper values on E'. Now we see from the manner in which these functions are obtained that at any point P in R the functions u are continually increasing. Furthermore, they are all bounded and hence approach a limit at P. Thus, by Harnack's Theorem, the functions Ui, u2, • • • converge to a limit function u which is harmonic at all points of R. By the same argument we see that the sequence of u"s converges to a harmonic func- tion u' in Rf. In the region or regions bounded by I, I' and C the limits of the u and uf sequences must coincide with the limit of the monotonic increasing sequence UL', j • ' ' and hence in this region we have u = u\ Thus, we may regard the limit function in Rf as a continuation of the one in R and we have thus obtained a single function V harmonic in the region R + R'. It now remains to be shown that the limit function V(P) approaches the value W(P) as the point P of R + Rf approaches a point P0 of the bound- ary, provided P0 is not on one of the arcs A. We first consider the case where the boundary point P0 is a point of E. Since the boundary of the region R + Rf is made up of portions of spheres, no two of which are tangent to one another, a sufficiently small sphere aS0 about the point P0 will certainly pass through points that are not of the region R + R', as well as through points of the region itself. Moreover, if the sphere aS0 is made to shrink to the point P0 by allowing its radius to approach zero, the ratio p between the area of the part of aS0 interior to R + Rr and the total area of So will remain, from a certain point on, less than some constant q less than unity. If two of the spherical portions on the bound- ary of R + R' were allowed to be tangent at P0, the ratio in question would approach unity instead of remaining less than q, but the case of tangency we have explicitly ruled out. We now construct a comparison function U0 defined in the followingdirichlet's problem. 191 manner. Let S0 be a sphere with center at P0 and radius so small that the ratio po of the portion of S0 interior to R +* R' to the total area is less than q. Moreover, let N denote the least upper bound of the assigned boundary values W at points of B — A within the sphere S0 and N + N' (N' ^ 0) the least upper bound of the values of W at the points of B — A as a whole. The comparison function U0 is then to be such that at points of S0 within R + R' it takes on the value N + N', at the re- maining points of So it takes on the value N, at points within S0 it is harmonic and defined by means of a Poisson integral, using the boundary values just assigned on this surface S0 itself. Thus at points P of SQ interior to R + R\ we have Uo(P) = N + N' S Un(P), since no value of the function un can exceed the least upper bound of the assigned boundary values on B — A. Moreover, at points of B — A interior to S0, Uq(P) S N ^ un(P) n = 1, 2, Therefore, by Theorem 7, the inequality (6) U0(P)^un(P) n = l,2,... holds at all points of the region or regions composed of the points of R + Rf interior to aS0. Consequently, a similar inequality holds for the limit function V(P) also, (7) Uo(P) S V(P) at all points of R + Rf interior to S0. Now by Gauss' mean value theorem, the value of Uo(P) at the center Po of So is given by (8) Uo(Po) = po(N + N') + (1 - Po)N = N + p,N' U„(P). Consequently, for points of R + R' interior or on Si we shall have the result (6') U1(P) S un(P) and therefore, also, (7') U1(P) S V(P), similar to (6) and (7) respectively. Moreover, the value of Ui(P) at the center P0 of Si is Ui(P) = pi (N + qN') + (1 - pi)N = N + piqN' V(P). Now, given any positive value e, the initial sphere S0 may be chosen so small that N < W (Po) + | • Moreover, the integer i may be chosen so large that Thus Ui-i(P) < TF(Po) + e within Si, and (9) V(P) < W(Po) + « at all points of R + R' within Si. In precisely the same way, using greatest lower bounds where before we used least upper bounds, we can prove the existence of a sphere S/dirichlet's problem. 193 about P0 within which (10) V(P) > W(P0) - 6. Relations (9) and (10) establish the continuity of V(P) at P0. In similar fashion we can establish the continuity of the function V(P) at a point P0 of the portion E' of the boundary such that P0 is not on an arc A. It only remains, therefore, to prove the continuity of V(P) at a point P0 of C which is not a point of A. This we do by constructing a series of spheres aS0, Si, • • ■ about the point P0 and the corresponding comparison functions U0, Ui, • • •, just as before. The only difference is that in treating this case, we have already established the continuity of the function V(P) at all boundary points of R + Rf except those of A and C. Therefore, at each step we obtain the relation Ui{P) > V(P) directly from Theorem (7) without having to consider the approximating functions un{P) or vn(P) at all. Thus, the extended Dirichlet problem for the region R + R' is solved. 6. We are now in a position to establish Dirichlet's problem for a very general type of region R. It will be sufficient to assume that the boundary B of the region R is such that if a sphere So of variable radius be drawn with center at any point P0 of B, then the ratio p of the Lebesgue surface measure of the portion of the sphere interior to and on B to the total area of the sphere remains less than some constant q(P0) < 1 as soon as the radius of the sphere is less than some value r(P0). This condition throws out of consideration a region R bounded by a surface B possessing an inward pointing spur of too sharp a type, though an inward pointing conical point is perfectly legitimate, or an outward pointing spur of any degree of sharpness. As a matter of fact, it is easy to prove that given a sufficiently sharp inward pointing spur on the boundary, the problem admits of no solution continuous at the tip of the spur. In the course of the discussion we shall see that the radius of S0 need not shrink to zero continuously, it being sufficient merely that we can find for each point of B at least one denumerably infinite set of spheres satisfying the above conditions. Before proceeding further we shall prove the following well-known lemma.* Lemma. A three-dimensional region R can be covered by the interiors of a denumerably infinite set of spheres. * See, for example, Poincare, "Sur les Equations aux Derivees Partielles cle la Physique Mathematique," in the Am. Jour, of Math., vol. 12, p. 211.194 GEORGE E. RAYNOR. For, given any e > 0, the set S2 do not intersect, by Poisson's integral we can obtain two functions v2 and v2n harmonic in Si and S2 respectively, v2 taking the same values as F(x, y, z) on Si and v2" the same values as F(x, y, z) on S2. Then the function v2 will be taken as equal to v2 and v2" in and on Si and S2 respectively and identical with F(x, y, z) in the remaining portion of R + B. Proceeding in this manner, step by step, we obtain a sequence of functions, Vl) V2j 'j Vny ' j vn being harmonic in the regions covered by the interiors of the first n spheres, taking on' the boundaries of those regions the same values as F(x, y, z) and identical with Fix, y, z) in the remaining portion of I' + B. We shall now prove that as n increases the function vn approaches a limit function v(x, y, z) which will be continuous in R + B, harmonic in R and identical with W(x, y, z) on B, or, in other words, that the solution of Dirichlet's problem exists for the domain R + B. Consider then a point P0 of B and let Wo be the value of W(x, y, z) at this point. Let So be a sphere with center P0 and radius so small that the ratio p of the Lebesgue surface measure of the portion of S0 within or on B to its total area is less than some constant q < 1. Let Bn' be the boundary of the region R' within which the approximating function v„ is harmonic and M be the least upper bound of Fix, y, z) in R + B. Within Bn' we know, by Theorem 2, that v„ is less than the greatest value of Fix, y, z) on Bn'. Hence, since vn is identical with Fix, y, z) in the portion of R + B on and exterior to Bn', we have (11) Vnix, y, z) ^ M at all points of R + B. Let M' be the least upper bound of Fix, y, z) within or on So. Let us now construct, by means of Poisson's integral, a comparison function U0 harmonic within So and taking on the portion of So interior to B the value M and on the portion exterior to B the value M'. We then have at once that within S0 (12) P.gl'S Fix, y, z). Now, if Bn' intersects S0, the portion of R' within So will be made up of regions bounded partly by So and partly by Bn' on which vn = Fix, y, z).196 GEORGE E. RAYNOR. Hence on the boundaries of these regions we have (13) U0 ^vn, and by Theorem 4 the same relation will subsist within these regions. In the portions of the region R — R' within S0 we have (14) vn = F(x, y, z), and hence by (12) we find (13) holding for this region. On the other hand, if Bnf does not intersect S0 we once more obtain (13) directly from (14) and (12). Hence in each case we see that the comparison function U0 is greater than all the approximating functions within aS0- We may now take a sequence of spheres Sn with center P0 and with radii decreasing to zero and set up, precisely as described in connection with the alternating process, a sequence of comparison functions Uo, Ui, ' ' ' , Un, ' such that in Sn, Un will be greater than all of the approximating functions. Since F(x, y, z) is continuous in R + B, given an e > 0 the initial sphere S0 may be taken so small that within this sphere Mf will differ from W0 by less than c/2. • Then, since our decreasing spheres are subject to exactly the same condition as in the preceding section, we can take n so large that ultimately in Sn, Un will differ from the value W0 by less than e. Hence all the approximating functions will be less than W0 + e in Sn. Now by using greatest lower bounds where before we used least upper bounds we obtain by an exactly analogous argument that all the approximating functions will be greater than W0 — e in some sphere Sn' with center at P0. Hence if we let Snp0" be a sphere with center at P0 and interior to both Sn and Sn', we have the result, given any e > 0 we can find for each point P of B a sphere SnPfr with P as center within which the oscillations of the approximating functions vn all remain less than 2e. From this set of spheres we can choose by the Heine-Borel Theorem a finite sub-set which will cover the boundary B. Consider now any point P; in the region R and draw about it a small sphere S' lying entirely in R. Let us now choose a value m of n so large that the boundary BJ' of the region in which the approximating function vn is harmonic lies entirely in the above sub- set of spheres and encloses the sphere S'. By the above argument we have that for all values of n ^ m the oscillations of the approximating functions will be less than e everywhere on Bnf and hence by Theorem 2 will be less than e on S'. Therefore, by Theorem 8 the approximating functions converge to a limit in which is harmonic in Sf. Hence in particular, we have that at any interior point P9 of the region R thedirichlet's problem. 197 approximating functions converge to a limit, and this limit function is harmonic at P'. It now remains to prove that our limit function v takes on the assigned boundary values W(x, y, z) on B. But this follows at once by precisely the same argument as was used in the case of the alternating process to show that the limit function there obtained approaches the proper bound- ary values. We have precisely the same inequalities subsisting between the comparison functions and the limit functions, and by the restriction on the boundary B made at the beginning of this section, we have the same condition on the decreasing set of spheres for each point of B. Hence we have finally the result, Theorem 9. Dirichlet's problem has a solution for every region R whose boundary B is such that, if a sphere be drawn about any point of B, the ratio of the measure of the points of the surface of the sphere interior to and on B to the whole area of the sphere will ultimately remain less than unity as the radius of the sphere approaches zero. The shrinking process need not be continuous but may be made by a denumerably infinite set of steps only.This book is a preservation facsimile produced for the University of Illinois, Urbana-Champaign. It is made in compliance with copyright law and produced on acid-free archival 60# book weight paper which meets the requirements of ANSI/NISO Z39.48-1992 (permanence of paper). Preservation facsimile printing and binding by Northern Micrographics Brookhaven Bindery La Crosse, Wisconsin 2017