Hilbert's Nineteenth Problem - History


For C3 solutions Hilbert's problem was answered positively by Sergei Bernstein (1904) in his thesis, who showed that C3 solutions of nonlinear elliptic analytic equations in 2 variables are analytic. Bernstein's result was improved over the years by several authors, such as Petrowsky (1939), who reduced the differentiability requirements on the solution needed to prove that it is analytic. On the other hand direct methods in the calculus of variations showed the existence of solutions with very weak differentiability properties. For many years there was a gap between these results: the solutions that could be constructed were known to have square integrable second derivatives, which was not quite strong enough to feed into the machinery that could prove they were analytic, which needed continuity of first derivatives. This gap was filled independently by Ennio De Giorgi (1956, 1957), and John Forbes Nash (1957, 1958). They were able to show the solutions had first derivatives that were Hölder continuous, which by previous results implied that the solutions are analytic whenever the differential equation has analytic coefficients, thus completing the solution of Hilbert's nineteenth problem.

De Giorgi (1968) gave a counterexample showing that in the case when the solution is vector-valued rather than scalar-valued, the solution need not be analytic.

Read more about this topic:  Hilbert's Nineteenth Problem

Other articles related to "history":

Xia Dynasty - Modern Skepticism
... The Skeptical School of early Chinese history, started by Gu Jiegang in the 1920s, was the first group of scholars within China to seriously question the traditional story of its early history "the ... early Chinese history is a tale told and retold for generations, during which new elements were added to the front end" ...
Casino - History of Gambling Houses
... or another has been seen in almost every society in history ... From the Ancient Greeks and Romans to Napoleon's France and Elizabethan England, much of history is filled with stories of entertainment based on games of chance ... In American history, early gambling establishments were known as saloons ...
Spain - History - Fall of Muslim Rule and Unification
... The breakup of Al-Andalus into the competing taifa kingdoms helped the long embattled Iberian Christian kingdoms gain the initiative ... The capture of the strategically central city of Toledo in 1085 marked a significant shift in the balance of power in favour of the Christian kingdoms ...
Voltaire - Works - Historical
... History of Charles XII, King of Sweden (1731) The Age of Louis XIV (1751) The Age of Louis XV (1746–1752) Annals of the Empire – Charlemagne, A.D ... II (1754) Essay on the Manners of Nations (or 'Universal History') (1756) History of the Russian Empire Under Peter the Great (Vol ... II 1763) History of the Parliament of Paris (1769) ...
History of Computing
... The history of computing is longer than the history of computing hardware and modern computing technology and includes the history of methods intended for pen and paper or for chalk and ...

Famous quotes containing the word history:

    Universal history is the history of a few metaphors.
    Jorge Luis Borges (1899–1986)

    Every library should try to be complete on something, if it were only the history of pinheads.
    Oliver Wendell Holmes, Sr. (1809–1894)

    The visual is sorely undervalued in modern scholarship. Art history has attained only a fraction of the conceptual sophistication of literary criticism.... Drunk with self-love, criticism has hugely overestimated the centrality of language to western culture. It has failed to see the electrifying sign language of images.
    Camille Paglia (b. 1947)