Hello, I have the following problem:

Having a non-linear PDE of second order, there are the initial conditions

$F(x,0)=g(x)$,

$\partial_yF(x,0)=h(x)$.

The question is: Can the second derivatives $\partial^2_{xy}F(x,0)$ and $\partial^2_{xx}F(x,0)$ be calculated by taking derivatives on the initial conditions directly?

Thanks beforehand.