Consider a stationary heat transfer problem as an example, the governing equation is the following second order pde: $$-k\Delta T(x)=\dot{q}(x) $$
In the all text books it said that the strong solution of the equation $T(x)$ which satisfy the above equation needs to be continuous in $\Omega$ and also its first derivative $\frac{d}{dx} T(x)$ should be continuous in $\Omega$ to make sure that the second derivative would exist at every point in $\Omega$
My questions is: is that a sufficient conditions? does continuity in first derivative insure existence of the second derivative?
I think that we could have a situation where we have a continuous first derivative but the second derivative doesn't exist, for example if $$\frac{d}{dx} T(x)=\lvert x\rvert$$ Then the second derivative at $x=0$ doesn't exist
Is there something that I am missing?
Thanks in advance