Is the minimum of this functional $C^{infty}$?

The problem:

Let us define
$$
mathscr{F}(u)=int_0^1(u'(x))^4-e^xsin (u(x)), mathrm{d}x
$$
for $u in W^{1,1}([0,1])$ such that $u(0)=A$ and $u(1)=B$.
It don’t matters what $A$ or $B$ are, because I am interested in the reasoning.

I am asked to study if the minimum $u$ is such that $u in C^{infty}([0,1])$ or eventually $u in C^{infty}([0,1]-E)$ where $E$ is a closed and negligible subset of $[0,1]$, i.e. it is closed and $mu(E)=0$.

An attempt:

First of all, thanks to Ioffe’s theorem, we know that $mathscr{F}$ is sequentially weakly lower semi continuous in $W^{1,1}([0,1])$. Further more, because $(u'(x))^4-e^xsin (u(x)) geq (u'(x))^4-e$, a minimum $u in W^{1,1}$ exists.

But what can be said about the regularity of $u$? I know the Tonelli’s partial regularity theorem but I cannot directly apply it here because $F_{pp}$ is not defined positive but we only have $F_{pp} geq 0$.
I know that if $u'(x) neq 0$ then $u$ is $C^{infty}$ in a neighborhood of $x$.

My question is: $u in C^{infty}([0,1])$?

Further more I would like to understand when can we apply Tonelli’s theorem if $F_{pp} geq 0$ but not $F_{pp} > 0$ i.e. if $F_{pp}$ is positive semidefinite but not positive definite.

Remark: I tried first solving the related Question.