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 <A HREF="http://archive.numdam.org/ARCHIVE/ASNSP/ASNSP_1940_2_9_3-4/ASNSP_1940_2_9_3-4_289_0/ASNSP_1940_2_9_3-4_289_0.pdf">other writings</A>, 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 <A HREF="http://math.univ-lyon1.fr/~clarke/Clarke_Regularity.pdf">Francis Clarke</A> (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.