Journal of the Korean Society for Industrial and Applied Mathematics

Korean Society for Industrial and Applied Mathematics (한국산업응용수학회)
권호 표지
DOI QR코드

DOI QR Code

COMPACTNESS AND DIRICHLET'S PRINCIPLE

  • Seo, Jin Keun (Department of Computational Science and Engineering, Yonsei University) ;
  • Zorgati, Hamdi (Department of Mathematics, Faculty of Sciences of Tunis, University of Tunis El Manar)
  • Received : 2014.04.19
  • Accepted : 2014.05.27
  • Published : 2014.06.25
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper we explore the emergence of the notion of compactness within its historical beginning through rigor versus intuition modes in the treatment of Dirichlet's principle. We emphasize on the intuition in Riemann's statement on the principle criticized by Weierstrass' requirement of rigor followed by Hilbert's restatement again criticized by Hadamard, which pushed the ascension of the notion of compactness in the analysis of PDEs. A brief overview of some techniques and problems involving compactness is presented illustrating the importance of this notion. Compactness is discussed here to raise educational issues regarding rigor vs intuition in mathematical studies. The concept of compactness advanced rapidly after Weierstrass's famous criticism of Riemann's use of the Dirichlet principle. The rigor of Weierstrass contributed to establishment of the concept of compactness, but such a focus on rigor blinded mathematicians to big pictures. Fortunately, Poincar$\acute{e}$ and Hilbert defended Riemann's use of the Dirichlet principle and found a balance between rigor and intuition. There is no theorem without rigor, but we should not be a slave of rigor. Rigor (highly detailed examination with toy models) and intuition (broader view with real models) are essentially complementary to each other.

Keywords

References

  1. Alexandrov P and Urysohn P (1923). Sur les espaces topologiques compacts, Bull. Intern. Acad. Pol. Sci. Ser. A. , pp. 5-8.
  2. Arzela C (1895). Sulle funzioni di linee, Mem. Accad. Sci. Ist. Bologna Cl. Sci. Fis. Mat. 5 (5), pp.55-74.
  3. Ascoli G (1883). Le curve limiti di una varieta data di curve, Atti della R. Accad. Dei Lincei Memorie della Cl. Sci. Fis. Mat. Nat. 18 (3), pp. 521-586.
  4. Bolzano B (1930). Functionlehre, pp.1833-1841 (from Bolzano's manuscripts in Spisy B. Bolzana 1,Prague .
  5. Bolzano B (1842). Zahlenlehre (from Bolzano's manuscripts in Spisy B. Bolzana 2, Prague 1931.
  6. Bottazzini U.(2002) "Algebraic truths" vs "geometric fantasies": Weierstrass' Response to Riemann, Proceedings of the ICM, Beijing 2002, (3), arXiv:math/0305022, pp. 923-934.
  7. Buhler, WK (1987). Gauss: a biographical study. Springer-Verlag, ISBN 0-387-10662-6, pp. 144-145.
  8. Cantor G (1883) Uber unendliche lineare Punktmannigfaltigkiet, Math. Annalen 21, 51-58, pp. 545-591. https://doi.org/10.1007/BF01446819
  9. Carbery A(1997). Harmonic analysis of the Calderon-Zygmund school, 1970-1993, Bulletin of the London Mathematical Society, 1997, 30(01):11 - 23.
  10. Cech E (1937). On bicompact spaces, Annals of Math. 38, pp. 823-844. https://doi.org/10.2307/1968839
  11. Coifman RR, McIntosh A and Meyer Y (1982). L'integrale de Cauchy definit un operateur bournee sur $L^2$ pour courbes lipschitziennes. Ann. of Math. 116, pp. 361-87. https://doi.org/10.2307/2007065
  12. David G and Journe JL (1984). A boundedness criterion for generalized Calderon-Zygmund operators. Ann. of Math. 120, pp. 371-97. https://doi.org/10.2307/2006946
  13. De Giorgi E (1975). Sulla convergenza di alcune successioni d'integrali del tipo dellaerea, Rend. Mathematica 8, pp. 277-294.
  14. Fabes E, Jodeit M and Riviere N (1978). Potential techniques for boundary value problems on $C^1$ domains. Acta Math., 141, pp. 165-186. https://doi.org/10.1007/BF02545747
  15. Feferman S (2000). Mathematical intuition vs. mathematical monsters, Synthese 125 (3), pp. 317-332. https://doi.org/10.1023/A:1005223128130
  16. Frechet M (1910). Les dimensions d'un ensemble abstrait, Math. Ann. 68, pp. 145-168. https://doi.org/10.1007/BF01474158
  17. Hildebrandt TH (1925). The Borel Theorem and its Generalizations. In J. C. Abbott (Ed.), The Chauvenet Papers: A collection of Prize-Winning Expository Papers in Mathematics. Mathematical Association of America.
  18. Hilbert D (1900). Uber das Diche Prinzip. Jahresbericht Deut. Math.-Ver. 8, pp. 184-188.
  19. Gauss CF (1839). Allgemeine Lehrsatze in Beziehung auf die im verkehrten Verhaltnisse des Quadrats der Entfernung wirkenden Anziehungs- und Abstossungs-Krafte. Werke 5(1867), pp. 197-244.
  20. Gauss CF(1838). Allgemeine Theorie des Erdmagnetismus, Werke 5(1867), pp. 127-193.
  21. Gauss CF (1840). Atlas des Erdmagnetismus, Werke 12 (1929) pp. 326-408.
  22. Hadamard J (1902). Sur les problemes aux derivees partielles et leur signification physique.
  23. Hadamard J (1906). Sur le principe de Dirichlet, Bull. Soc. Math. France, 34, pp. 135-138.
  24. Hilbert D (1904). Uber das Dirichletsche Prinzip. Math. Ann. 59, pp. 161-184. https://doi.org/10.1007/BF01444753
  25. Hilbert D (1905). Uber das Dirichletsche Prinzip. J. Reine Angew. Math. 129, pp. 63-67.
  26. Jordan C (1881). Sur la serie de Fourier, C. R. Acad. Sci. Paris, 92, pp. 228-230.
  27. Kleiner I (1991). Rigor and Proof in Mathematics: A Historical Perspective, Mathematics Magazine, 64(5), pp. 291-314. https://doi.org/10.2307/2690647
  28. Kondrachov W (1945). Certain properties of functions in the space Lp, Dokl. Akad. Nauk SSSR, 48, pp. 535-538.
  29. Murat F (1978). Compacite par compensation, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 5(4), pp. 489-507.
  30. Rellich F (1930). Ein Satz uber mittlere Konvergenz, Gottingen Nachr., Acta Math. 141, pp. 165-186.
  31. Russ S. (2004). The Mathematical Works of Bernard Bolzano. Oxford: Oxford University Press.
  32. Seo JK and Woo EJ (2012). Nonlinear inverse problems in imaging, Wiley Press.
  33. Shannon CE (1949). Communication in the presence of noise, Proc. Institute of Radio Engineers, 37(1), pp. 10-21.
  34. Sobolev SL (1938). On a theorem of functional analysis Math. Sb. 46, pp. 471-496.
  35. Stone MH (1937). Applications of the Theory of Boolean Rings to General Topology, Trans. Amer. Math. Soc. 41 (3), pp. 375-781. https://doi.org/10.1090/S0002-9947-1937-1501905-7
  36. Tartar L (1979). Compensated compactness and applications to partial differential equations, in Nonlinear Analysis and Mechanics, Heriott Watt Symposium, ed. R. J. Knops. IV, Research Notes in Mathematics, 39 Pitman, pp. 136- 212.
  37. Taylor A (1982). A Study of Maurice Frechet: I. His Early Work on Point Set Theory and the Theory of Functionals. Archive for History of Exact Sciences, 27 (3), pp. 233-295. https://doi.org/10.1007/BF00327860
  38. Taylor A (1985). A Study of Maurice Frechet: II. Mainly about his Work on General Topology 1909-1928. Archive for History of Exact Sciences 34 (3), pp. 279-380. https://doi.org/10.1007/BF00411640
  39. Tietze H (1923). Beitrage zur allgemeinen topologie I, Math. Ann. 88, pp. 280-312.
  40. Tonelli L (1926) Sulla quadratura delle superficie I,II,III, Rend. Acc. Naz. Lincei. 3 pp. 375-362.
  41. Verchota G (1984). Layer potentials and boundary value problems for Laplace's equation in Lipschitz domains. J. of Func. Anal. 59, pp. 572-611. https://doi.org/10.1016/0022-1236(84)90066-1
  42. Weierstrass K (1885). Uber die analytische Darstellbarkeit sogenannter willkurlicher Functionen einer reellen Veranderlichen. Sitzungsberichte der Koniglich Preuischen Akademie der Wissenschaften zu Berlin (II).