A discussion of Tonelli's contributions and their relation to Hilbert's work can be found in this <A HREF="http://www.ams.org/journals/bull/1926-32-04/S0002-9904-1926-04231-2/S0002-9904-1926-04231-2.pdf">AMS bulletin</A>. 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 <A HREF="http://www.ams.org/journals/bull/1925-31-03/S0002-9904-1925-04002-1/S0002-9904-1925-04002-1.pdf">AMS bulletin.</A> It is noteworthy that Tonelli himself, in this summary but also in other writings I have found from him on the internet, does not mention Hilbert at all. Apparently he considered his contributions to be quite independent.