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Fundamental Existence Theorems

  • Oliver Dimon Kellogg
Part of the Die Grundlehren der Mathematischen Wissenschaften book series (volume 31)

Abstract

As we saw in § 3 of Chapter IX (p. 237), Green, in 1828, inferred the existence of the function which bears his name from the assumption that a static charge could always be induced on a closed grounded con- ducting surface by a point charge within the conductor, and that the combined potential of the two charges would vanish on the surface. From this, he inferred the possibility of solving the Dirichlet problem. Such considerations could not, however, be accepted as an existence proof. In 1840, GAUSS gave the following argument. Let S denote the boundary of the region for which the Dirichlet problem is to’ be solved.

Keywords

Harmonic Function Boundary Point Dirichlet Problem Normal Derivative Closed Region 
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References

  1. Picard, Trai U d’Analyse, 3rd ed. Paris 1922, Vol. I, pp. 226—233Google Scholar
  2. Lebesgue: Sur le Probleme de Dirichlet, Comptes Rendus de l’Academie de Paris, Vol. 154 (1912), p. 335Google Scholar
  3. Lebesgue, Comptes Rendus, Vol. 178 (1924), p. 352Google Scholar
  4. Bouligand, Bulletin des sciences mathematiques, Ser. 2, Vol. 48 (1924), p. 205.Google Scholar
  5. Wiener, N., Journal of Mathemat’s and Physics of the Massachusetts Institute of Technology, Vol. III (1924), p. 49Google Scholar
  6. Kellogg, Comptes Rendus de T Academie de Paris, Vol. 187 (1928), p. 526Google Scholar

Copyright information

© Verlag Von Julius Springer 1929

Authors and Affiliations

  • Oliver Dimon Kellogg
    • 1
  1. 1.Mathematics in Harvard UniversityCambridgeUSA

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