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.

Hilbert's 20th problem concerns the existence of solutions to the fundamental problem in the calculus of variations. I understand that Hilbert, Lebesgue 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.