= Tonelli existence theorem
{c}
For a scalar fixed-endpoint problem on a bounded interval, suppose the continuous integrand $f(x,y,p)$ is convex in $p$ and satisfies $f(x,y,p)\geq\Theta(|p|)-C$, where $\Theta$ is nonnegative convex and $\Theta(r)/r\to\infty$. If there is one admissible absolutely continuous function of finite energy, a global minimizer exists in $W^{1,1}$.
The lower bound gives uniform integrability of derivatives in a minimizing sequence. Fixed endpoint values then give boundedness and equicontinuity of the functions. Extract uniform convergence of the functions and weak $L^1$ convergence of their derivatives; the limit retains the endpoint values. Convexity in the derivative gives <weak lower semicontinuity of convex gradient energies>, so the limit attains the infimum. Strict convexity of the full integrand on a convex admissible class additionally gives uniqueness.
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