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For a functional $J[y]=\int_{a}^{b}F(t,y,y')dt$, are there any conditions that ensure extrema over the class of piecewise continuously differentiable functions are all in $C^2[a,b]$?

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I believe that a piecewise smooth extremum would have to satisfy the Weierstrass-Erdmann corner conditions. If these conditions ensure that the extremum is in fact $C^1$, then it solves the Euler-Lagrange equations everywhere. At this point one can appeal to the regularity of solutions of ODEs, which can easily get you $C^2$ or even higher smoothness, depending on the regularity of the functional form of the ODE itself.

For more generality, look up the sufficient conditions for a strong variational extremum (starting at Weak and Strong Extrema here).

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