Background:
I am still studying This question and I could solve my problems with it if the following question can be answered.
My problem:
Suppose $F:\mathbb{R}\times \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ is $C^{\infty}$.
Suppose $F$ is strictly convex in the third variable but possibly $F_{pp}(t_0,u_0,\beta_0)=0$ for some $(t_0,u_0,\beta_0) \in \mathbb{R}\times \mathbb{R} \times \mathbb{R}$. For example $F(x,z,p)=p^4-e^x\sin(z)$ as in the quoted question.
Can I say that $\frac {\partial}{\partial x}F_p(x,u(x),u'(x))=F_z(x,u(x),u'(x))$ has local solution for every $t_0,u_0,\beta$ such that $u(t_0)=u_0$ and $u'(t_0)=\beta$?
My attempt:
If $F_{pp}$ was strictly positive than I would write $u''=F_{pp}^{-1}(F_z-F_{xp}-F_{zp})$ and I would solve the question with the classical theory about ordinary Cauchy's problems.
But if for some value $F_{pp}=0$?
For the case $F(x,z,p)=p^4-e^x\sin(z)$ we could observe that $F_{pppp}>0$ thus we could reproduce the reasonment as above.
But there are strictly convex function such that does not exists $k \geq 0$ with $\frac{\partial ^k}{\partial p^k}F>0$ as $F(x,z,p)=e^{-\frac{1}{p^2}}$ .