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Hilbert's 20th problem concerns the existence of solutions to the fundamental problem in the calculus of variations. I understand that Hilbert, Lesbesgue and Tonelli were pioneers in this area.

In particular, I believe that Hilbert answered his problem soon but there were some gaps. Tonelli pioneered the idea of weak lower semicontinuity but I'm not sure when. I believe that authors did not use the machinery of Sobolev spaces but essentially had such ideas.

Unfortunately, the book by Monna (1975) which would have probably answered this question for me is unavailable in my library.

I would like to know

  1. What was Hilbert's contribution to his 20th problem and when
  2. What was the gap, who filled it (first) and when.
  3. When did Tonelli contribute his ideas of semicontinuity in the calculus of variations.

A link to the papers, if possible would be brilliant or directions to some other book on the history of the Calculus of Variations would be very helpful as well.

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A discussion of Tonelli's contributions and their relation to Hilbert's work can be found in this AMS bulletin. The original work was published in Italian, Fondamenti di Calcolo delle Variazioni (Bologna, 1921 & 1923) --- I have not found an English translation, but an English summary by Tonelli is also in an AMS bulletin. It is noteworthy that Tonelli himself, in this summary but also in other writings, does not mention Hilbert at all. Apparently he considered his contributions to be quite independent.

A more recent discussion of Tonelli's work in the context of Hilbert's 20th (and 19th) problem is given by Francis Clarke (2008):

The decade preceding the formulation of Hilbert’s problems had been marked by a controversy over the Dirichlet principle, which affirms the equivalence between functions $u$ minimizing the Dirichlet functional $J$ and solutions $u$ of Laplace’s equation. As Weierstrass and Hilbert pointed out in response to (notably) Riemann’s assertions, the existence of a minimum here (and the very class in which to seek one) is problematic. Hilbert went on to give the first rigorous treatment of the issue in 1904, in a context which succeeded in limiting the class of functions $u$ involved to Lipschitz ones. But it became clear that a more general type of function space was needed, and finally Sobolev spaces provided a suitable context in which to assert the existence of a solution to the basic problem.

The direct method introduced by Tonelli exploits the weak sequential compactness of a minimizing sequence and the weak lower semicontinuity of the convex functional $J$, to deduce the existence of a solution $u$ in the Sobolev space $W^{1,1}$, as well as the uniqueness (since $J$ is strictly convex), answering Hilbert’s 20th problem. The remaining problem (Hilbert's 19th) was the regularity of the solution $u$, especially since functions in the Sobolev space $W^{1,1}$ are not even continuous necessarily.

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