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This is a transcript from The Calculus of Variations by Jeff Calder

For the given functional $F(x,y,y')$, the sufficient conditions for a weak solution of the Euler-Lagrange equation to be a minimizer is that we require joint convexity in $y$ and $y'$.

As a counter-example, we can take:

$F=\frac{((y')^2-\lambda^2 y^2)} {2}$

which has been claimed to be non-convex. However, I don't know how to verify that. Please help.

S.S
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A twice differentiable function is convex if only if its Hessian matrix is positive semi-definite at every point. We have $$ F''(x,y,y') = \frac{1}{2}\begin{bmatrix} 0 & 0 & 0\\ 0 & -2\lambda^2 & 0\\ 0 & 0 & 2\\ \end{bmatrix}, $$ and so, if $\lambda \neq 0$, the Hessian is never positive semi-definite since it has one eigenvalue $<0$.

Lucas Resende
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