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Rüdiger Thiele1

  • 1Karl-Sudhoff-Institut der Universität Leipzig, Augustusplatz 10-11, D-04109, Leipzig.

NTM
|October 16, 2016
PubMed
Summary
This summary is machine-generated.

This paper explores David Hilbert's foundational work in the Calculus of Variations and integral equations, highlighting his influence on modern analysis and partial differential equations.

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Area of Science:

  • Mathematical Analysis
  • Calculus of Variations
  • Integral Equations

Background:

  • Focuses on the transformation in the Calculus of Variations at the end of the 19th century.
  • Emphasizes the significant influence of Dirichlet's principle and its proof on Hilbert's work.
  • Contextualizes Hilbert's contributions within the rise of modern mathematical analysis.

Purpose of the Study:

  • To examine key developments in analysis stemming from Hilbert's contributions.
  • To trace the impact of Dirichlet's principle on Hilbert's research and subsequent mathematical problems.
  • To discuss Hilbert's theorem on independence in the Calculus of Variations and his work on integral equations.

Main Methods:

  • Historical analysis of mathematical developments.

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  • Examination of Hilbert's foundational papers and problems.
  • Discussion of the influence of Dirichlet's principle and Hilbert's theorem.

    • Main Results:

    • Hilbert's proof of Dirichlet's principle led to crucial questions on the existence and regularity of solutions to elliptic partial differential equations.
    • Hilbert's work established significant contributions to the formal apparatus and field theory of the Calculus of Variations.
    • Hilbert recognized integral equations as a flexible area for achieving a unifying methodological approach in analysis.

    Conclusions:

    • Hilbert's research, particularly concerning Dirichlet's principle, profoundly shaped modern analysis and the study of partial differential equations.
    • Hilbert's theorem of independence advanced the Calculus of Variations.
    • Hilbert's exploration of integral equations demonstrated his pursuit of unifying principles in mathematical analysis.