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:
“What we call National-Socialism is the poisonous perversion of ideas which have a long history in German intellectual life.”
—Thomas Mann (1875–1955)
“Social history might be defined negatively as the history of a people with the politics left out.”
—G.M. (George Macaulay)
“Don’t you realize that this is a new empire? Why, folks, there’s never been anything like this since creation. Creation, huh, that took six days, this was done in one. History made in an hour. Why it’s a miracle out of the Old Testament!”
—Howard Estabrook (1884–1978)