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
Famous quotes containing the word history:
“The principle that human nature, in its psychological aspects, is nothing more than a product of history and given social relations removes all barriers to coercion and manipulation by the powerful.”
—Noam Chomsky (b. 1928)
“The history of all hitherto existing society is the history of class struggles.”
—Karl Marx (1818–1883)
“You that would judge me do not judge alone
This book or that, come to this hallowed place
Where my friends’ portraits hang and look thereon;
Ireland’s history in their lineaments trace;
Think where man’s glory most begins and ends
And say my glory was I had such friends.”
—William Butler Yeats (1865–1939)