Lavrentiev phenomenon between $C^1$ and $C^2$

Question

Does there exist a (onedimensional) functional that exhibits the Lavrentiev phenomenon between $C^1$ and $C^2$ that is

$$ F(y)=\int_a^b f(t,y(t),y'(t))\,dt \quad\text{or possibly}\quad F(y)=\int_a^b f(t,y(t),y'(t),y''(t))\,dt $$

such that

$$ \inf_{y\in C^1([a,b])}F(y)<\inf_{y\in C^2([a,b])}F(y) \;\;?$$

The paper "Mantegazza - Some Elementary Questions in the Calculus of Variations" and the question Lavrentiev phenomenon between $C^1$ and Lipschitz discuss the Lavrentiev phenomenon between $\mathrm{Lip}$ and $C^1$ but do not mention any higher classes.

Having no restrictions of $f$ could maybe give some possibility to construct an example, using some kind of function like $y(t)=t^{5/3}$…?

Answer 1

Does this example work? Take the domain to be $[0, 1]$. Let $H$ be the Cantor function, and $\mathcal H$ its graph. Define $f$ by $$ f (t,x,y) = \begin{cases} 0, &\text{if } (t, y) \in \mathcal H,\\ \max \left\{\dfrac{1}{\text{dist}((t, y), \mathcal H)}, 1\right\},&\text{otherwise} . \end{cases} $$ Then the infimum of $F(g)$ over $g \in C^1$ is $0$, and is realised by taking $g(x) := \int_{0}^{x} H(s) ds$, but I believe the infimum over $C^2$ is $1$, and is realised by any $g$ such that $(t, g’(t))$ stays far away enough from $\mathcal H$.