Let
$L:\Bbb R^N\times\Bbb R^N\rightarrow\Bbb R$
be a Borelian function and
consider the following problems
$$
\inf\left\{F(y)=\int_a^bL(y(t),y'(t))\,{\rm d}t:\,y\in AC([a,b],\Bbb R^N),
y(a)=A,\,y(b)=B\right\} \qquad\quad\! (P)
$$
$$
\inf\left\{F^{**}(y)=\int_a^bL^{**}(y(t),y'(t))\,{\rm d}t:\,y\in AC([a,b],\Bbb R^N),
y(a)=A,\,y(b)=B\right\}\cdot \quad\;\ \! (P^{**})
$$
We give a sufficient condition, weaker then superlinearity, under
which
$\inf F=\inf F^{**}$
if L is just continuous in x. We
then extend a result of Cellina on the Lipschitz regularity of
the minima of (P) when L is not superlinear.