Über die Variationsrechnung in Hilberts Werken zur Analysis

“Entsprechend meiner Auffassung, daß die Variationsrechnung noch ganz in den ersten Anfängen sich befindet, und daß ihr eigentliches Ziel die allgemeine Untersuchung der Beziehungen zwischen Funktionen ist, habe ich in meiner Vorlesung die Beziehungen zu allgemeinen Fragen der Analysis in den Vordergrund gestellt.” Hilbert in der Zusammenfassung der VorlesungVariationsrechnung, WS 1904/05. [SUB Cod. Ms. Hilbert 555, Beilage S. 2]

Abstract

The paper deals with some of the developments in analysis against the background of Hilbert's contributions to the Calculus of Variations.

As a starting point the transformation is chosen that took place at the end of the 19th century in the Calculus of Variations, and emphasis is placed on the influence of Dirichlet's principle. The proof of the principle (the “resuscitation”) led Hilbert to questions arising in the 19th and 20th problems of his famous Paris address in 1900: theexistence in a generalized sense, and theregularity of solutions of elliptic partial differential equations. By this new concept Hilbert pointed out two very important issues the history of which is closely tied up with the rise of modern analysis.

Further on, Hilbert'stheorem of independence of the Calculus of Variations, an important contribution to its formal apparatus as well as to its field theory, is briefly discussed.

But even as Hilbert published his first essential results, he was turning his attention to another area of mathematics that differs in an important respect from the Calculus of Variations:integral equations. Hilbert did so because he recognized that in this branch by its flexibility he was coming closer to his goal of an unifying methodological approach to analysis.