Does there exist a (onedimensional) integral functional of calculus of variations (with $f$ finite everywhere)
$$
F(y)=\int\_a^b f(t,y(t),y'(t))\,dt 
$$
such that
$$
\inf\_{y\in Lip([a,b])}F(y)<\inf\_{y\in C^1([a,b])}F(y)
$$
that is, it shows the Lavrentiev phenomenon between $C^1$ and Lipschitz.



Lavrentiev phenomenon between $C^1$ and Lipschitz
Then trivially yes. Just take $F(y)=1+\sum\_{q\in \mathbb Q}\frac{a\_q}{|q-y|^2}$ with $a\_q>0$ such that the series converges a.e. Then change all $+\infty$ values of $F$ to $1$. Now take $[a,b]=[-1,1]$ and define
$$
f(t,y,\xi) =
\begin{cases}
0 &\text{if }y=|t|\\ 
F(y) &\text{otherwise}
\end {cases}.
$$
A Lipschitz function can just stay on the safe path $y=|t|$ and pay $0$ price for the trip, but a $C^1$ function will have to deviate from this path and either have $y\ne |t|$, $y'\ne 0$ at at least one point, which is enough to blow the integral up to $+\infty$, or stay constant, which gives the cost of at least $2$.


