Hilbert's 19th problem asks:
Are the solutions of regular problems in the calculus of variations always necessarily analytic?
This was proven to be true (through the work of Sergei Bernstein, Ennio de Giorgi, John Nash, among others).
My question probably stems mostly from my elementary knowledge of the subject, but I am wondering what exactly we gain from this result -- in as-close-to-layman's terms as possible. And from what I gather, it's a good thing that the answer to Hilbert's question is in the affirmative, correct?