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Lebesgue’s criticism of Carl Neumann’s method in potential theory

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Abstract

In the 1870s, Carl Neumann proposed the so-called method of the arithmetic mean for solving the Dirichlet problem on convex domains. Neumann’s approach was considered at the time to be a reliable existence proof, following Weierstrass’s criticism of the Dirichlet principle. However, in 1937 H. Lebesgue pointed out a serious gap in Neumann’s proof. Curiously, the erroneous argument once again involved confusion between the notions of infimum and minimum. The objective of this paper is to show that Lebesgue’s sharp criticism of Neumann’s proof was only partially justified.

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Notes

  1. For more detailed information on Neumann’s career we refer to Disalle (1993), Hölder (1922, 1925, 1927), Liebmann (1927), Salié (1981), Schlote (2001, 2004a, b, 2005). A list of Neumann’s 176 publications is provided in Hölder (1925).

  2. For an analysis of Neumann’s work in this direction see, in particular, Schlote (2004a, b, 2005) as well as Disalle (1993), Hölder (1922, 1925, 1927), and Salié (1981).

  3. A detailed exposition on the Neumann series in Banach spaces may be found, for example, in Heuser (1986,  106, 117, 523, 604), where it is proved (see p. 523) that the Neumann series \(\sum _{n=0}^{\infty } (-1)^nT^n\) converges in the norm topology, if and only if all eigenvalues of T are contained in the interior of the unit circle. However, the integral operator of the Dirichlet problem has 1 as an eigenvalue. Heuser (1986, 523) says: “Das Dirichletsche Problem gehört also nicht zum Machtbereich der Neumannschen Reihe—obwohl diese doch gerade zu seiner Bewältigung geschaffen wurde. Wir begegnen hier einer merkwürdigen mathematischen Ironie, die auch sonst in unserer Wissenschaft nicht eben selten ist.” (So the Dirichlet problem does not fall within the domain of Neumann series—even though it was created precisely to solve it. We have here a remarkable mathematical irony, which is indeed not so rare in our science.)

    We note that, for a given \(y \in X\), the condition \(||T|| < 1\) is not indispensable for the convergence of the Neumann series \(\sum _{n=0}^{\infty } (-1)^nT^ny\); see Suzuki (1976).

    The literature dealing with the Neumann series representation of solutions to the boundary value problems for the Laplace equation is extensive. We refer, for example, to Král (1976, 1980), Medková (1998, 1999, 2008), where other relevant papers are quoted.

  4. The literature on the history of classical potential theory and the Dirichlet principle to the beginning of the twentieth century is quite rich. We offer here a selection of historical studies on the subject: Bottazzini (1986, Appendix), Gray (2015, Chapters 13, 14, 16, 18), Jahnke (2003, Chapters 7, 11), and Kline (1972, Chapters 22, 28, 45). See also the articles by Archibald (1996, 2003), Archibald and Tazzioli (2005, 2014), and Birkhoff and Kreyszig (1984) .

  5. It is reproduced in Maz’ya and Shaposhnikova (1998, 376); see the commentary on pages 373–377.

  6. The example is discussed, for example, in Archibald (2016), Bottazzini (1986, 300), and Monna (1975).

  7. So with Beer’s method one problem is merely reduced to another, while by means of the method of the arithmetic mean that I have given an actual solution of both problems is made possible.

  8. A sketch of Neumann’s method of the arithmetic mean appeared in Neumann (1870). For a discussion of A. Beer’s investigations around 1860, see Dieudonné (1981, 42). Concerning the history of the Neumann series, A. Pietsch remarks in Pietsch (2007,  28): “The beginning, which dates back at least to Beer’s and Neumann’s investigations in potential theory” (Neumann 1877b, Chapters 5, 6) “was much more cumbersome. In modern terminology, Neumann looked at a concrete integral operator \(T: X \rightarrow X\) with \(||T|| = 1\) whose special properties imply that the series \(x + Tx + T^2x + \cdots \) converges for all \(x \in X\).” A detailed exposition was given in the book (Neumann 1877b) with an erroneous proof of contractivity; see also Neumann (1877a). A corrected proof was given in Neumann (1887). Neumann returned to his favourite subject in Neumann (1901). Here we refer to Bacharach (1883, §12), Burkhardt and Meyer (1899–1916, 471), Dieudonné (1981, 41), Lichtenstein (1909–1921, 204), and Sologub (1975, 98). In Neumann’s book (Neumann 1877b), Chapters 4 and 7 are devoted to a detailed study of a double-layer potential. The reader may also consult Maz’ya (1991, 133), Simon (2015a, 275) and (2015b, 115).

  9. The progress of the method is summarized in the introduction of Neumann (1905), where contributions of Poincaré, Korn, Liapounoff, Stekloff, and Zaremba are mentioned. Among many accounts see also Zaremba’s papers (1901, 1904). The historical place of Neumann’s method is discussed in Dieudonné (1981, Chapter II), Kellogg (1929, 281), Hellinger’s survey in Hilbert (1970, 94), Heuser (1986, 601), Král and Netuka (1977), and Král (1977, 207), as well as in many of the papers already cited.

  10. In this section, page numbers refer to Neumann’s book (Neumann 1877b).

  11. If \(\sigma \) is a given curve or surface with positive normal \(\nu \), and if on \(\sigma \) there is a double layer with moment \(\mu \), then the potential of this double layer at an arbitrary point x has the value [...]. Here E denotes the distance of the point from the element \(\mathrm{d}\sigma \), further \(\vartheta \) denotes the angle the line E (\(\mathrm{d}\sigma \mapsto x\)) makes with the normal \(\nu \), and accordingly the expression represents the visible quantity of the element \(\mathrm{d}\sigma \) for an observer at x multiplied by \(\varepsilon \), where \(\varepsilon = + 1\) or \(= -1\), depending on whether the observer has the positive or negative side of the element before their eyes.

  12. The quantity \(\varepsilon _s\) connected with \({\widetilde{\omega }}_s\) through the relation

    $$\begin{aligned} {\widetilde{\omega }}_s + \varepsilon _s = {\widetilde{\omega }} \end{aligned}$$

    may be called the supplement of the angle measure, or more briefly the supplementary angle measure.

  13. If \(\sigma \) denotes a closed curve or surface with positive inner side, and one thinks of a double layer on \(\sigma \) whose moment \(\mu \) is everywhere continuous, then the potential exerted at a variable point x by the double layer will have the following properties: \(W_s + \varepsilon _s\mu _s \) as a function of s is everywhere continuous on \(\sigma \). The values \(W_a\) form a continuous system whose limiting value \(W_{as}\) is connected with the direct value \(W_s\) by the relation \(W_{as} = \big (W_s + \varepsilon _s\mu _s\big ) - {\widetilde{\omega }}\mu _s.\) The values \(W_i\) form a continuous system whose limiting value \(W_{is}\) is connected with the direct value \(W_s\) by the relation \(W_{is} = \big (W_s + \varepsilon _s\mu _s\big ) + {\widetilde{\omega }}\mu _s.\)

  14. If we think of drawing a sphere \(\kappa \) of unit radius around i then \((\mathrm{d}\sigma )_i\) is an element on this sphere marked out by the cone. If, moreover, we now think of i as the centre of perspective and project the surface \(\sigma \) with all the values of f that are on it, then the arithmetic mean of all the values of f now on \(\kappa \) is given by \([\ldots ]\) for \((\mathrm{d}\sigma )_i\) is, as we have already said, an element of the sphere \(\kappa \), and the sum of all these elements is equal to the spherical surface itself, so \(= 2 {\widetilde{\omega }}\).

    But, after \((*)\), the fraction \((**)\) represents the value of \(\frac{1}{2} W_i\). And accordingly one can call \(\frac{1}{2}W_i\) the arithmetic mean of all the values of f on \(\sigma \) with respect to the point i.

  15. At the same time we will say that a given curve or surface is of rank R if it never has more than R points in common with an infinitely long straight line in an arbitrary position ...a closed curve or surface of the second rank can be called an everywhere convex curve or surface...

  16. If on a given curve two marked points M, N are in such a position that any tangent to the curve goes through one of these two points then the curve may be called two-starred and M, N its stars.

  17. We divide the given curve or surface \(\sigma \) into two parts \(\alpha \) and \(\beta \), \(\sigma = \alpha + \beta \), in which each of these may be divided into arbitrary many individual pieces, and in this way construct the expressions [...], where the integral \(\int \big (\mathrm{d}\alpha \big )_s\) is to be taken over all the elements of the part \(\alpha \) and the integral \(\int \big (\mathrm{d}\beta \big )_s\) over all the elements of the part \(\beta \).

  18. If, for example, \(\sigma \) is a spherical surface and we think of this spherical surface as carrying a map of the surface of the Earth, then we can, if we wish, understand by \(\alpha \) everywhere that is covered with water and by \(\beta \) the continents and islands.

  19. Of particular importance for our later researches is the question of whether \(\xi \), \(\eta \) can truly attain their lower bound of zero.

  20. In formula  (4), the affixed (sic!) character [on the inequality sign] should draw attention to the character above it, which does not say \(\ge \) but rather > .

  21. If one divides a closed curve or surface \(\sigma \) with positive inner side in two parts \(\alpha \) and \(\beta \) (each of which can have arbitrarily many individual pieces) and understands by s, \(s_1\) two freely movable points on \(\sigma \), then the quantity [...] varies in a way that depends in the art and manner of the division as well as also with the positions of the points s, \(s_1\).

    But if we now assume that the curve or surface \(\sigma \) is of the second rank and not two-starred then in the space considered \(\zeta \) will be subject to \(0 \le \zeta < 1.\) What is true of the variable \(\zeta \) is necessarily also true of every special value of this variable. So if one denotes the maximum value of \(\zeta \) by \(\lambda \) then it follows at once that \(0 \le \zeta \le \lambda < 1.\) This \(\lambda \), which is always positive and less than 1, represents a constant appropriate to the given curve or surface and may be called the configuration constant.

  22. Carl Neumann was dealing successfully with these questions; notably, he proposed a method for solving the Dirichlet problem that rightly remains famous; he confined his study to convex domains. Poincaré justified the method for an extended class of non-convex domains; Fredholm’s research increased our understanding of the importance of this method and the reasons for its success. The criticism I want to address here concerns only its classical legitimacy for the case of convex domains.

  23. Neumann’s reasoning, which was intended to replace that of Riemann, is thus based on exactly the same confusion between the least upper bound and maximum, justly criticized by Weierstrass.

    This mistake is due to Neumann; however, it is also a mistake of the writers who contrast Riemann with Neumann, of those professors who presented Neumann’s reasoning, and of all those who have read these arguments without objection. In brief, we have all made this mistake; it thus merits that we pause for a moment to analyse it.

    It is quite astonishing that everyone, beginning with Weierstrass, accepted the validity of Neumann’s reasoning, and that contemporary textbooks continue to contrast Riemann’s reasoning, declared as erroneous, with that of Neumann, proclaimed entirely correct.

  24. Clearly, it is unconditionally essential for the approach that the designated configuration constant is less than 1. The proof of this given in the study of 1877 is not without difficulties and without doubt did not completely satisfy Neumann himself. In a work published later (Neumann 1887) he came back to the issue and after a reproduction of the old one gave a new and completely satisfying proof of the inequality \(\lambda < 1\) based on elementary intuitions.

  25. These results in my previous work are to be considered as absolutely rigorous. And I shall presently prove with equal rigour (in §6) that every configuration constant \(\lambda \) is less than one by a given amount for any given closed curve or surface, in the cases when they are everywhere convex and are not two-starred. Indeed, I have already tried to prove this. More precisely considered, I based my previous work on Weierstrass’s theorem that any continuous function on a given domain must take its maximum value somewhere in the domain—and it therefore seems (this is the point of contention) questionable whether Weierstrass’s theorem is truly applicable in the very peculiar circumstances of the present work.

  26. The cardinal property of the configuration constant \(\lambda \) resides in the formula...Here in §6 I derive this property in a truly more rigorous and at the same time as general a way as possible, in that I take as my starting point for these considerations a straight-sided pentagon in the plane, and in the spatial case on the other hand a polyhedron bounded by seven plane faces, and in this way follow a path that is completely independent of the previously expected theorem of Weierstrass.

  27. It could rather be that here [the transition from (26.) to (27.)] the scope of Weierstrass’s principles is exceeded. I therefore consider the given proof as only provisional, and in the following § replace it with another and indeed absolutely rigorous proof.

  28. To avoid an unduly long footnote, Neumann’s original words will be found in an Appendix.

  29. Extension to higher-dimensional space of Carl Neumann’s method for solving problems concerning functions of real variables satisfying the differential equation \(\triangle F=0\).

  30. The eminent geometer [Neumann] was dealing with problems about finding functions that attain prescribed values on a closed curve or a closed surface (depending on whether it is a function of two or three variables), and he was able to resolve them in considerable generality by a method that is both very rigorous and very elegant. As he was dealing mainly with applications of his theories to physics, C. Neumann did not seek to extend them to the case of an arbitrary number of variables; the main aim of our work is precisely to show that the results remain valid independently of this number, and that they can be applied with the same restrictions to closed surfaces of p-dimensional space.

  31. In our opinion, some of Neumann’s proofs, as they are presented, leave a few doubts in the mind of a reader; we strove everywhere to achieve clarity and the greatest rigour which the method could offer. See, in particular, Nos. 28 and 29.

  32. Both points s and \(s_1\) have a determined particular position on the surface.

  33. If one gives the points s and \(s_1\) a determined particular position on the surface, one can assign a positive quantity below which the value of the expression in question never falls, when \(\alpha \) and \(\beta \) vary in a completely arbitrary manner.

  34. If now all possible positions of the points s and \(s_1\) on the surface \(\sigma \) are considered, it is clear that one can assign a positive number \(\Theta \) such that:

    $$\begin{aligned} \Theta \le \int \big (\mathrm{d}\alpha \big )_s + \int \big (\mathrm{d}\beta \big )_{s_1}, \end{aligned}$$

    for arbitrary s and \(s_1\) on the one hand, and \(\alpha \) and \(\beta \) on the other.

  35. As s and \(s'\) vary on the surface, this lower bound is actually a minimum; denoting it by \(\lambda \), it obviously lies between zero and one.

  36. This conclusion cannot pass as absolutely rigorous; however, clearly, it is precise for convex curves, as are usually considered.

  37. Knowledgeable expositions of Poincaré’s contribution to potential theory during the last decade of the nineteenth century are presented in Archibald and Tazzioli (2005, 2014), Mawhin (2010), Dieudonné (1981, Chapter I), Heuser (1986, Chapter XIX), Hellinger and Toeplitz (1923–1927, Chapter I), and Lichtenstein (1909–1921, Chapter III). For a modern presentation of Poincaré’s variational problem in potential theory, see Khavinson et al. (2007). See also, for example, Dieudonné (1981, 40), Burkhardt and Meyer (1899–1916, 502), Kellogg (1929, 283, 318, 322), Simon (2015a, 275). On the history as well as the importance of Poincaré’s méthode du balayage see Brelot (1970, 1972, 1985). The central role of balayage in contemporary potential theory is described in Armitage and Gardiner (2001, 129), and Bliedtner and Hansen (1986).

  38. In his deep investigations on the convergence of the well-known method of Neumann in potential theory, H. Poincaré studied the Dirichlet problem as a special case of another problem, which he calls the Neumann problem. This problem can be expressed as follows: let W be a double-layer potential carried by a closed surface S, V be the limit value of W for a variable point M tending to a point \(M_0\) from inside S, and \(V'\) be the limit value of W for a variable point M tending to a point \(M_0\) from outside S. In such a situation, we want to find a double layer carried by S such that its potential satisfies the equation

    $$\begin{aligned} V - V' = \lambda (V + V') + 2 \Phi , \end{aligned}$$

    where \(\Phi \) is a given function of the parameters determining the position of a point on S. Neumann resolved the problem by developing the unknown function by the powers of the parameter \(\lambda \). But it follows from Poincaré’s investigations that W is a meromorphic function. Hence, it is clear that Neumann’s development cannot converge for all values of \(\lambda \). However, since we know that a meromorphic function can always be expressed as a ratio of two entire functions, it seemed to me natural to seek these entire functions directly.

  39. The journey from the “crypto-integral” equations (the term coined in Dieudonné 1981, 22) to Fredholm’s discovery and then to F. Riesz’s theory of compact operators is described in Siegmund-Schultze (2003), and in Dieudonné (1981, Chapters II, V, VII), where particular attention is devoted to the Beer–Neumann method (pp. 39–46). Valuable information on the contribution of the integral equation method to the creation of functional analysis can be found in Bernkopf (1996) and Heuser (1986, Chapters XII, XIX). On the history of functional analysis see Hellinger in Hilbert (1970) and the extensive contribution by Hellinger and Toeplitz (1923–1927), Browder (1975), and several articles already cited.

  40. For a nice description of the duality between the Dirichlet problem and the Neumann problem (“über Kreuz” konjugiert), see Heuser (1986, 429), and Maz’ya (1991, 139). For the case of non-smooth boundaries, see Král (1980) and Medková (2018).

  41. In the planar case, Radon pushed the generalization of the method towards what we consider its natural limit. The domains considered by Radon are those bounded by the curves of bounded rotation.

  42. No matter how elegant this method may be, it has the inconvenience of being applicable only under quite restrictive assumptions as far as the boundary of domains considered is concerned. These restrictions arise from the nature of the method; indeed, double layers consisting of oriented dipoles can only be distributed on sufficiently regular curves (or surfaces).

  43. For a history of the boundary element method, see Cheng and Cheng (2005). From an extensive literature we refer to Hsiao and Wendland (2004, 2008), Steinbach and Wendland (2001), and Kleinman (1976).

  44. We have had to replace his numbering of the equations 4, 5, 6p, 6q, 7, 8, 9, 10a by the numbering 9, 10, 11, 12, 13, 14, 15, 16.

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Appendix

Appendix

The text is from Neumann (1887).Footnote 44

Ist nun \(\mathrm{d}\sigma \) irgend ein Element der Linie \(a_0\), und p irgend ein Punkt auf \(A_0\) so wird [...]:

$$\begin{aligned} \big (\mathrm{d}\sigma \big )_p = \frac{\mathrm{d}\sigma \cdot \cos \delta }{E}, \end{aligned}$$
(9)

wo E den Abstand des Punktes p von \(\mathrm{d}\sigma \), und \(\delta \) den Winkel von E gegen die auf \(\mathrm{d}\sigma \) errichtete Normale vorstellt. Construirt man eine die Linie \(a_0\) in \(\mathrm{d}\sigma \) berührende und durch p gehende Kreisperipherie, so ergiebt sich für den radius R dieser Peripherie die Gleichung: \(2 R \cos \delta = E\); worduch die Formel (9) übergeht in:

$$\begin{aligned} \big (\mathrm{d}\sigma \big )_p = \frac{\mathrm{d}\sigma }{2R}. \end{aligned}$$
(10)

Welche Lage man nun dem Elemente \(\mathrm{d}\sigma \) auf \(a_0\) und dem Punkte p auf \(A_0\) zuertheilen mag, niemals kann dabei der Kreisradius R unendlich gross werden; denn der Punkt p kann bei einer Durchlaufung der gebrochenen Linie \(A_0\) niemals auf die Linie \(a_0\) oder auf die Verlängerung derselben fallen.

Ja noch mehr: Man erkennt, dass der grösste Werth, den jener radius R für alle auf \(a_0\) befindlichen Elemente \(\mathrm{d}\sigma \) und für alle auf \(A_0\) befindlichen Punkte p annehmen kann, ein bestimmter endlicher ist, der, falls die Seiten und Winkel des Fünfecks in bestimmter Weise gegeben sind, sofort und ohne Mühe berechnet werden könnte. Bezeichnet man nun diesen bestimmten endlichen Werth mit \(R_A\), so ist R stets \(\le R_A\); so dass sich also aus (10) ergiebt:

$$\begin{aligned} \big (\mathrm{d}\sigma \big )_p \ge \frac{\mathrm{d}\sigma }{2R_A}, \ \text {(vorausgesetzt} \, \mathrm{d}\sigma \ \text {auf} \ a_0, \ \text {und} \ p \ \text {auf} \ A_0). \end{aligned}$$
(11)

Nimmt man statt p irgend einen anderen, ebenfalls auf \(A_0\) gelegenen Punkt q, so wiederholt sich dieselbe Formel:

$$\begin{aligned} \big (\mathrm{d}\sigma \big )_q \ge \frac{\mathrm{d}\sigma }{2R_A}, (\mathrm{d}\sigma \ \text {auf} \ a_0, \ \text {und} \ q \ \text {auf} \ A_0). \end{aligned}$$
(12)

Wir zerlegen jetzt die ganze Peripherie des Fünfecks [genau wie früher ...] in beliebiger Weise in zwei Theile \(\alpha \) und \(\beta \), deren jeder aus beliebig vielen einzelnen Stücken bestehen kann, und bezeichnen die auf \(a_0\) fallenden Theile von \(\alpha \) und \(\beta \) respective mit \(\alpha _0\) und \(\beta _0\); so das also \(a_0\) durch die Gesammtheit von \(\alpha _0\) und \(\beta _0\) dargestellt ist:

$$\begin{aligned} a_0 = \alpha _0 + \beta _0. \end{aligned}$$
(13)

Alsdann wird offenbar, weil \(\alpha _0\) einen Theil won \(\alpha \), möglicherweise auch das ganze \(\alpha \) repräsentirt [und weil überdies die \((\mathrm{d}\alpha )_p\) durchweg positiv sind; vergl. [ ...]], die Relation stattfinden:

$$\begin{aligned} \int \big (\mathrm{d}\alpha \big )_p \ge \int \big (\mathrm{d}\alpha _0\big )_p. \end{aligned}$$

Ebenso ergiebt sich

$$\begin{aligned} \int \big (\mathrm{d}\beta \big )_q \ge \int \big (\mathrm{d}\beta _0\big )_q. \end{aligned}$$

Demgemäss folgt aus (3) [...] sofort:

$$\begin{aligned} \xi \ge \int \big (\mathrm{d}\alpha _0\big )_p + \int \big (\mathrm{d}\beta _0\big )_q. \end{aligned}$$
(14)

Nach (11) ist aber jedwedes \( \big (\mathrm{d}\alpha _0\big )_p \ge \frac{\mathrm{d}\alpha _0}{2R_A}\), und nach (12) jedwedes \( \big (\mathrm{d}\beta _0\big )_q \ge \frac{\mathrm{d}\beta _0}{2R_A}\). Somit folgt aus (14):

$$\begin{aligned} \xi \ge \frac{\int \mathrm{d}\alpha _0 + \int \mathrm{d}\beta _0}{2R_A} = \frac{\alpha _0 + \beta _0}{2R_A}, \end{aligned}$$
(15)

d. i. mit Rücksicht auf (13):

$$\begin{aligned} \xi \ge \frac{a_0}{2R_A}, \ \text {({vorausgesetzt}} \ p \ \text {und} \ q \ \text { beide auf} \ A_0). \end{aligned}$$
(16)

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Netuka, I. Lebesgue’s criticism of Carl Neumann’s method in potential theory. Arch. Hist. Exact Sci. 74, 77–108 (2020). https://doi.org/10.1007/s00407-019-00233-z

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