×

Lebesgue’s criticism of Carl Neumann’s method in potential theory. (English) Zbl 1444.01014

Quoting extensively from mathematical publications and supplying footnoted translations, the author examines the formulation and reception of Carl Neumann’s (1832–1925) method of the arithmetic mean from the 1870s until the author’s own contribution in 1977 [with J. Kral, J. Math. Anal. Appl. 61, 607–619 (1977; Zbl 0372.31004)]. Neumann’s method addressed a criticism of solutions to the Dirichlet problem in potential theory that confused the concepts of infimum and minimum. The author focuses especially on Neumann’s own presentations of the method and their critical appraisal by Henri Lebesgue a half century later, arguing that aspects of Lebesgue’s criticism missed their mark. All told, this story of proofs and corrections shows the notable persistence of subtle points of confusion through successive efforts to remedy a significant argument and method, somewhat belying the typical image of meticulous rigorisation associated with the examined period of mathematical analysis. Rather than develop such potentially suggestive historiographical implications, the author maintains a mathematical focus on the historical succession of formulations and criticisms.

MSC:

01A55 History of mathematics in the 19th century
01A60 History of mathematics in the 20th century
31-03 History of potential theory
35-03 History of partial differential equations

Biographic References:

Lebesgue, Henri; Neumann, Carl

Citations:

Zbl 0372.31004
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Archibald, T., From attraction theory to existence proofs: The evolution of potential-theoretic methods in the study of boundary-value problems, 1860-1890, Revue d’histoire des mathématiques, 2, 1, 67-93 (1996) · Zbl 0855.01010
[2] Archibald, T. 2003. Analysis and physics in the nineteenth century: The case of boundary-value problems. In A history of analysis. Hans Niels Jahnke (ed.). History of Mathematics, Vol. 24, American Mathematical Society, Providence, RI, London Mathematical Society, London, 197-211. · Zbl 1088.01001
[3] Archibald, T., Counterexamples in Weierstraß’s work, Karl Weierstraß (1815-1897), 269-285 (2016), Wiesbaden: Springer Spektrum, Wiesbaden · Zbl 1330.01044
[4] Archibald, T., and Tazzioli, R. 2005. The reception of Fredholm’s results on integral equations: preliminary report. Real Analysis Exchange, 29th Summer Symposium Conference, 113-136. · Zbl 1103.45001
[5] Archibald, T.; Tazzioli, R., Integral equations between theory and practice: the cases of Italy and France to 1920, Archive for History of Exact Sciences, 68, 5, 547-597 (2014) · Zbl 1346.01014
[6] Armitage, Dh; Gardiner, Sj, Classical potential theory (2001), Springer: Springer Monographs in Mathematics, Springer · Zbl 0972.31001
[7] Bacharach, M., Abriss der Geschichte der Potentialtheorie (1883), Göttingen: Vanderhoeck & Ruprecht’s Verlag, Göttingen · JFM 15.0861.01
[8] Beer, A., Allgemeine Methode zur Bestimmung der elektrischen und magnetischen Induction, (Poggendorffs) Annalen der Physik und Chemie, 98, 137-142 (1856)
[9] Bernkopf, M., The development of function spaces with particular reference to their origins in integral equation theory, Archive for History of Exact Sciences, 3, 1, 1-96 (1996) · Zbl 0149.33703
[10] Bernstein, Sn; Petrowsky, Ig, On the first boundary value problem (Dirichlet’s problem) for equations of the elliptic type and on the properties of functions satisfying such equations, Uspekhi Matem. Nauk, 8, 8, 8-31 (1941) · Zbl 0063.00338
[11] Birkhoff, G.; Kreyszig, E., The establishment of functional analysis, Historia Mathematica, 11, 3, 258-321 (1984) · Zbl 0549.01001
[12] Bliedtner, J.; Hansen, W., Potential theory. An analytic and probabilistic approach to balayage: Universitext (1986), Berlin: Springer, Berlin · Zbl 0706.31001
[13] Bôcher, M. 1913. Boundary problems in one dimension. In Proceedings of the Fifth International Congress of Mathematicians (Cambridge, 22-28 August 1912), I, E.W. Hobson and A. E. Love (eds.), Cambridge University Press, Cambridge, 163-195. · JFM 43.0374.05
[14] Bottazzini, U., The higher calculus: A history of real and complex analysis from Euler to Weierstrass (1986), Berlin: Springer-Verlag, Berlin · Zbl 0597.01011
[15] Brelot, M. 1970. Historical introduction. In Potential Theory (C.I.M.E., I Ciclo, Stresa, 1969). Edizioni Cremonese, Rome 1970: 1-21. · Zbl 0201.13901
[16] Brelot, M., Les étapes et les aspects multiples de la théorie du potentiel, Enseignement Mathématique, 18, 1, 1-36 (1972) · Zbl 0235.31002
[17] Brelot, M. 1985. Le balayage de Poincaré et l’épine de Lebesgue. In Proceedings of the 110th national congress of learned societies (Montpellier), Com. Trav. Hist. Sci., Paris, 141-151.
[18] Browder, Fe, The relation of functional analysis to concrete analysis in 20th century mathematics, Historia Mathematica, 2, 4, 577-590 (1975) · Zbl 0344.01007
[19] Burkhardt H., Meyer, F. 1899-1916. Potentialtheorie. In Encyklopädie der Mathematischen Wissenschaften II A 7 b, B. G. Teubner, Leipzig, 464-503.
[20] Cheng, Alexander H.-D.; Cheng, Daisy T., Heritage and early history of the boundary element method, Engineering Analysis with Boundary Elements, 29, 3, 268-302 (2005) · Zbl 1182.65005
[21] Dieudonné, J. 1981. History of functional analysis. North-Holland Mathematics Studies, 49, Notas de Matemática, 77. North-Holland Publishing Co., Amsterdam. · Zbl 0478.46001
[22] Disalle, R., Carl Gottfried Neumann, Einstein in context, 345-353 (1993), Cambridge: Cambridge University Press, Cambridge · Zbl 1181.01042
[23] Fredholm, I. 1900. Sur une nouvelle méthode pour la résolution du problème de Dirichlet. Förhandlingar Öfversigt af Kongliga Svenska Vetenskaps-Akademiens, (Stockholm) 57: 39-46. · JFM 32.0435.02
[24] Fredholm, I., Oeuvres complètes (1955), Litos Reprotryck, Malmö: Institut Mittag-Leffler, Litos Reprotryck, Malmö · Zbl 0068.08601
[25] Gaier, D., Konstruktive Methoden der konformen Abbildung (1964), Berlin: Springer: Springer Tracts in Natural Philosophy, Berlin · Zbl 0132.36702
[26] Goursat, É., Cours d’analyse mathématique (1942), Gauthier-Villars, Paris: Tome III, Gauthier-Villars, Paris · JFM 68.0098.04
[27] Gray, Jj, The real and the complex: A history of analysis in the 19th century (2015), Berlin: Springer: Springer Undergraduate Mathematics Series, Berlin · Zbl 1330.01001
[28] Harnack, A., Die Grundlagen der Theorie des logarithmischen Potentiales und der eindeutigen Potentialfunktion in der Ebene (1887), Leipzig: B. G. Teubner, Leipzig · JFM 19.1026.05
[29] Hellinger, E., and Toeplitz. O. 1923-1927. Integralgleichungen und Gleichungen mit unendlichvielen Unbekannten. In Encyklopädie der Mathematischen Wissenschaften, II C 13, B.G. Teubner, Leipzig, 1335-1601. · JFM 53.0350.01
[30] Heuser, H., Funktionalanalysis. Theorie und Anwendung: Mathematische Leitfäden (1986), Leipzig: B.G. Teubner, Leipzig · Zbl 0653.46002
[31] Hilbert, D., Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen (1912), Leipzig, B.G: Teubner, Leipzig, B.G · JFM 43.0423.01
[32] Hilbert, D., Gesammelte Abhandlungen. Band III (1970), Berlin: Springer, Berlin
[33] Hölder, O., Carl Neumann zum 90. Geburtstag, Mathematische Annalen, 86, 3-4, 161-162 (1922) · JFM 48.0025.07
[34] Hölder, O., Carl Neumann (Nachruf), Berichte der Sächs, Akademie Mathematisch-Naturwissenschaftliche Klasse, 78, 154-180 (1925)
[35] Hölder, O., Carl Neumann, Mathematische Annalen, 96, 1, 1-25 (1927) · JFM 52.0027.03
[36] Hsiao, G.C., and Wendland, W.L. 2004. Boundary element methods: Foundation and error analysis. In Encyclopaedia of Computational Mechanics, Vol. I, John Wiley & Sons, 339-373.
[37] Hsiao, Gc; Wendland, Wl, Boundary integral equations. Applied mathematical sciences (2008), Berlin: Springer, Berlin · Zbl 1157.65066
[38] Jahnke, H.N (ed.) 2003. A history of analysis. History of Mathematics, Vol. 24, American Mathematical Society, Providence, RI, London Mathematical Society, London. · Zbl 1088.01001
[39] Kantorovich, L.V., and V.I. Krylov. 1958. Approximate methods of higher analysis. Interscience. · Zbl 0083.35301
[40] Kellogg, O.D. 1929, 1967. Foundations of potential theory. Reprint from the 1st edition of 1929. Die Grundlehren der Mathematischen Wissenschaften, Band 31, Springer. · JFM 55.0282.01
[41] Khavinson, D.; Putinar, M.; Shapiro, Hs, Poincaré’s variational problem in potential theory, Archive for Rational Mechanics and Analysis, 185, 1, 143-184 (2007) · Zbl 1119.31001
[42] Kleinman, R.E. 1976. Iterative solutions of boundary value problems. In Function theoretic methods for partial differential equations. (Proceedings of the International Symposium Held at Darmstadt, 1976), Lecture Notes in Math., Vol. 561, Springer, 298-313. · Zbl 0341.35030
[43] Kleinman, Re; Wendland, Wl, On Neumann’s method for the exterior Neumann problem for the Helmholtz equation, Journal of Mathematical Analysis and Applications, 57, 170-202 (1977) · Zbl 0351.35022
[44] Kline, M., Mathematical thought from ancient to modern times (1972), Oxford: Oxford University Press, Oxford · Zbl 0277.01001
[45] Korn, A., Lehrbuch der Potentialtheorie. Allgemeine Theorie des Potentials und der Potentialfunktionen im Raume (1899), Berlin: Ferd. Dümmlers Verlagsbuchhandlung, Berlin · JFM 30.0690.05
[46] Korn, A., Lehrbuch der Potentialtheorie II. Allgemeine Theorie des logarithmischen Potentials und der Potentialfunktionen in der Ebene (1901), Berlin: Ferd Dümmlers Verlagsbuchhandlung, Berlin · JFM 31.0728.04
[47] Král, J. 1976. Potential theory and Neumann’s method. Mitt. Math. Ges. DDR 3-4: 71-79. · Zbl 0349.31008
[48] Král, J., Integral operators in potential theory. Lecture notes in mathematics (1980), Berlin: Springer, Berlin · Zbl 0431.31001
[49] Král, J.; Netuka, I., Contractivity of C. Neumann’s operator in potential theory, Journal of Mathematical Analysis and Applications, 61, 607-619 (1977) · Zbl 0372.31004
[50] Král, J., Netuka. I., Veselý, J. 1977. Potential theory IV. Státní pedagogické nakladatelství, Praha, (Czech).
[51] Kreyszig, E., Zur Entwicklung der zentralen Ideen in der Funktionalanalysis, Elemente der Mathematik, 41, 2, 25-35 (1986) · Zbl 0596.46001
[52] Lebesgue, H. 1937. Sur la méthode de Carl Neumann. Journal de Mathématiques Pures et Appliqués, \(9^{\text{e}}\) série, XVI, 205-217 and 421-423. · Zbl 0017.01603
[53] Lebesgue, H., 1963a. En marge du calcul des variations. Une introduction au calcul des variations et aux inégalités géométriques. Monographies de “L’Enseignement Mathématique”, No. 12., Institut de Mathématiques, Université Geneva, Imprimerie Kundig, Geneva. · Zbl 0136.09701
[54] Lebesgue, H., En marge du calcul des variations, Enseignement Mathématique, 2, 9, 209-326 (1963) · Zbl 0136.09701
[55] Lebesgue, H., Oeuvres Scientifiques (1972), Geneva: Institut de Mathématiques de l’Université de Genève, Geneva · Zbl 0253.01018
[56] Leis, Rolf, Zur Entwicklung der angewandten Analysis und mathematischen Physik in den letzten hundert Jahren, Ein Jahrhundert Mathematik 1890-1990, 491-535 (1990), Wiesbaden: Vieweg+Teubner Verlag, Wiesbaden · Zbl 0813.35002
[57] Lichtenstein, L. 1909-1921. Neuere Entwicklung der Potentialtheorie. Konforme Abbildung. In Encyklopädie der Mathematischen Wissenschaften II C3, B. G. Teubner, 177-377. · JFM 47.0450.01
[58] Liebmann, H., Zur Erinnerung an Carl Neumann, Jahresbericht der Deutschen Mathematiker-Vereinigung, 36, 175-178 (1927) · JFM 53.0028.01
[59] Mawhin, J. 2010. Henri Poincaré and the partial differential equations of mathematical physics. In The scientific legacy of Poincaré, Hist. Math., Vol. 36, American Mathematical Society, Providence, RI, 257-277. · Zbl 1198.01011
[60] Maz’ya, V.G. 1991. Boundary integral equations. In Analysis, IV, Encyclopaedia Math. Sci., Band 27, Springer, 127-222. · Zbl 0780.45002
[61] Maz’ya, V., and Shaposhnikova, T. 1998. Jacques Hadamard, a universal mathematician. History of Mathematics, Vol. 14, American Mathematical Society, Providence, RI, London Mathematical Society. · Zbl 0906.01031
[62] Medková, D., The boundary-value problems for Laplace equation and domains with nonsmooth boundary, Archivum mathematicum (Brno), 34.1, 173-181 (1998) · Zbl 0910.35038
[63] Medková, D., Solution of the Dirichlet problem for the Laplace equation, Applications of Mathematics, 44, 2, 143-168 (1999) · Zbl 1060.35041
[64] Medková, D., The successive approximation method for the Dirichlet problem in a planar domain, Applications of Mathematics (Warsaw), 35.2, 177-192 (2008) · Zbl 1151.31001
[65] Medková, D., The Laplace equation. Boundary value problems on bounded and unbounded Lipschitz domains (2018), Berlin: Springer, Berlin · Zbl 1457.35002
[66] Monna, A.F. 1975. Dirichlet’s principle. A mathematical comedy of errors and its influence on the development of analysis. Oesthoek, Scheltema & Holkema, Utrecht. · Zbl 0312.31001
[67] Neumann, C. 1870. Zur Theorie des logarithmischen und des Newtonschen Potentials. Berichte über die Verhandlungen der Königlich Sächsischen Gesellschaft der Wissenschaften zu Leipzig 22: 49-56 and 264-321. · JFM 03.0493.01
[68] Neumann, C., Zur Theorie des logarithmischen und des Newton’schen Potentials, Mathematische Annalen, 11, 558-566 (1877) · JFM 09.0689.02
[69] Neumann, C., Untersuchungen über das logarithmische und Newtonsche Potential (1877), Leipzig: Teubner, Leipzig · JFM 10.0658.05
[70] Neumann, C. 1887. Über die Methode des arithmetischen Mittels. S. Hirzel, Leipzig, Erste Abhandlung, 1887, Zweite Abhandlung, 1888.
[71] Neumann, C., Über die Methode des arithmetischen Mittels, inbesondere über die Vervollkommnungen, welche die betreffenden Poincaréschen Untersuchungen in letzter Zeit durch die Arbeiten von A. Korn und E.R. Neumann erhalten haben, Mathematische Annalen, 54, 1-48 (1901) · JFM 31.0416.03
[72] Neumann, Er, Studien über die Methoden von C. Neumann und G. Robin zur Lösung der beiden Randwertaufgaben der Potentialtheorie (1905), Leipzig: Teubner, Leipzig · JFM 36.0819.01
[73] Picard, É., Traité d’Analyse, I (1922), Paris: Gauthier-Villars, Paris · JFM 48.0046.04
[74] Pietsch, A., History of Banach spaces and linear operators (2007), Boston: Birkhäuser Boston Inc, Boston · Zbl 1121.46002
[75] Plemelj, J., Potentialtheoretische Untersuchungen (1911), Leipzig: B.G Teubner, Leipzig · JFM 42.0828.10
[76] Poincaré, H., Sur les équations aux derivées partielles de la physique mathématique, American Journal of Mathematics, 12, 3, 211-294 (1890) · JFM 22.0977.03
[77] Poincaré, H., La méthode de Neumann et le problème de Dirichlet, Acta Mathematica, 20, 1, 59-142 (1897) · JFM 27.0316.01
[78] Prym, Fe, Zur Integration der Differentialgleichung \(\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2} =0\), Journal für die reine und angewandte Mathematik, 73, 340-373 (1871) · JFM 03.0182.02
[79] Radon, J., Über Randwertaufgaben beim logarithmischen Potential. Sitzber. Akad. Wiss. Wien, 128, 1123-1167. Also in Gesammelte Abhandlungen, Band 1, Verlag der Österreichischen Akademie der Wissenschaften, Vienna, Birkhäuser Verlag, Basel, 1987, 228-272 (1919)
[80] Riesz, F.; Sz.-Nagy, B., Leçons d’analyse fonctionnelle, 2ème éd (1953), Budapest: Akadémiai Kiadó, Budapest · Zbl 0051.08403
[81] Riquier, C. 1886. Extension à l’hyperespace de la méthode de M. Carl Neumann pour la résolution de problèmes relatives aux fonctions de variables réelles qui vérifient l’équation différentielle \(\Delta F = 0\). Thèse à la Faculté des Sciences de Paris, A. Hermann.
[82] Salié, H. 1981. Carl Neumann. In 100 Jahre Mathematisches Seminar der Karl-Marx-Universität Leipzig, VEB Deutscher Verlag der Wissenschaften, Berlin, 92-101.
[83] Schlote, K-H, Zur Entwicklung der mathematischen Physik in Leipzig. I. Der Beginn der Neumannschen Ära, NTM . (N.S.), 9, 229-245 (2001) · Zbl 0991.01012
[84] Schlote, K-H, Carl Neumanns Forschungen zur Potentialtheorie, Centaurus, 46, 2, 99-132 (2004) · Zbl 1067.01012
[85] Schlote, K-H, Carl Neumann’s contributions to electrodynamics, Physics in Perspective, 6, 252-270 (2004) · Zbl 1099.01014
[86] Schlote, K-H, Carl Neumann’s contributions to potential theory and electrodynamics, European mathematics in the last centuries, 123-140 (2005), Wrocław: Univ. Wrocław, Wrocław · Zbl 1156.01009
[87] Schober, G., Neumann’s lemma, Proceedings of the American Mathematical Society, 19, 306-311 (1968) · Zbl 0159.40601
[88] Schwarz, H.A. 1870. Über die Integration der partiellen Differentialgleichung \(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0\) unter vorgeschriebenen Grenz- und Unstetigkeitbedingungen. Monatsberichte der Königlichen Akademie der Wissenschaft zu Berlin, 767-795. · JFM 02.0214.03
[89] Siegmund-Schultze, R. 2003. The origins of functional analysis. In A history of analysis, Hans Niels Jahnke (ed.). History of Mathematics, Vol. 24, American Mathematical Society, Providence, RI, London Mathematical Society, London, 385-407. · Zbl 1088.01001
[90] Simon, B. 2015a. Harmonic analysis. A Comprehensive Course in Analysis, Part 3. American Mathematical Society, Providence, RI. · Zbl 1334.00002
[91] Simon, B. 2015b. Operator theory. A Comprehensive Course in Analysis, Part 4. American Mathematical Society, Providence, RI. · Zbl 1334.00003
[92] Sologub, V.S. 1975. The development of the theory of elliptic equations in the eighteenth and nineteenth centuries. Izdat. “Naukova Dumka”, Kiev, (Russian).
[93] Sretenskii, L.N. 1946. Theory of the Newtonian potential. OGIZ, Gos: Izdatelstvo tech.-teoret. liter., Moskva. (Russian).
[94] Steinbach, O.; Wendland, Wl, On C. Neumann’s method for second-order elliptic systems in domains with non-smooth boundaries, Journal of Mathematical Analysis and Applications, 262.2, 733-748 (2001) · Zbl 0998.35014
[95] Steklov, Va, Fundamental problems in mathematical physics (1983), Moscow: Nauka, Moscow
[96] Suzuki, N., On the convergence of Neumann series in Banach space, Mathematische Annalen, 220, 2, 143-146 (1976) · Zbl 0304.47016
[97] Weierstraß, K. T. W. 1870. Über das sogenannte Dirichletsche Princip. Mathematische Werke von Karl Weierstraß, 2, 49-54. Reprinted in Weierstraß, K. T. W. 1988. Ausgewählte Kapitel aus der Funktionenlehre. Vorlesung, gehalten in Berlin 1886, R. Siegmund-Schultze (ed.). Leipzig, Teubner Archiv zur Mathematik.
[98] Zaremba, S., On the theory of the Laplace equation and Neumann’s and Robin’s methods, Bulletin international de l’Académie des sciences de Cracovie, 41, 350-405 (1901)
[99] Zaremba, S., Les fonctions fondamentales de H. Poincaré et méthode de Neumann pour une frontière composée de polygones curvilignes, Journal de Mathématiques Pures et Appliquées V. Sér., 10.4, 395-444 (1904) · JFM 35.0355.03
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.