Weierstrass Erdmann conditions

Question

If I want to minimize the functional $$J[y]=\int_0^{2\pi}(y'^2-y^2)dx$$ over piecewise $C^1$ and $y(0)=y(2\pi)=0$ one solution is obviously $y=0$. However the function $h$ which is $\sin(x)$ on $0<x<\pi$ and then 0 also gives $J[h]=0$. Since it is only piece $C^1$ it should satisfy the WE conditions at $c=\pi$. We have $$\frac{\partial f}{\partial y'}=2y'$$ and so the condition should be that $y'(\pi_-)=y'(\pi_+)$. However $h'(\pi_-)=-1\neq 0=h'(\pi_+)$. How come this minimum does not satisfy the WE conditions?