×

On the calculus of variations in Hilbert’s works to analysis. (Über die Variationsrechnung in Hilberts Werken zur Analysis.) (German) Zbl 0924.01012

The author considers the changes that occurred in the calculus of variations at the end of the 19th century and places emphasis on the influence of Dirichlet’s principle. The proof of the principle led Hilbert to the problems of the existence in a generalized sense and the regularity of solutions of elliptic partial differential equations (his famous Paris address of 1900). The history of these issues is closely tied up with the rise of modern analysis. Also, Hilbert’s theorem of independence of the calculus of variations is briefly discussed. When Hilbert published his first essential results, he was turning his attention to integral equations, thus, coming closer to his goal of a unified methodological approach to analysis.

MSC:

01A60 History of mathematics in the 20th century
49-03 History of calculus of variations and optimal control
01A55 History of mathematics in the 19th century

Biographic References:

Hilbert, D.
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Alexandrov, Pavel (Hrsg.):Die Hilbertschen Probleme (deutsche Übersetzung der russischen Ausgabe von 1969). Ostwalds Klassiker Nr. 525. Akademische Verlagsgesellschaft Geest & Portig. Leipzig21979.
[2] Courant, Richard (Hrg.): “David Hilbert zur Feier seines sechzigsten Geburtstages”. SonderheftDie Naturwissenschaften 10 (1922) 4, 67-103.
[3] Giaquinta, Marian; Hildebrandt, Stefan:Calculus of variations. Vol. 1-2. J. Springer: Berlin 1996. · Zbl 0853.49002
[4] Goldstine, Hermann:A history of the calculus of variations from the 17th century through the 19th century. J. Springer. New York 1980. · Zbl 0452.49002 · doi:10.1007/978-1-4613-8106-8
[5] Hilbert, David:Gesammelte Abhandlungen. Bd. 3. J. Springer: Berlin 1935.
[6] Nirenberg, Louis; Pier, v. J.-P. (ed.), Partial differential equations in the first half of the century (1994), Basel · Zbl 0807.01017
[7] Schröder, Kurt: “Hilberts Beiträge zur Analysis und Physik”.Mitteilungen der Mathematischen Gesellschaft der DDR (1968) 2, 47-66.
[8] Thiele, Rüdiger; Behara (ed.); Fritsch (ed.); Lintz (ed.), Gauß’ Arbeiten über kürzeste Linien aus der Sicht der Variationsrechnung, 167-178 (1995), Berlin · Zbl 0882.01005
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.