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I am looking for a function $y(x)$ minimizing an integral $F(x, y, y')$ over a compact interval. $F(\cdot)$ and its partial derivatives of the first and second orders are single valued continuous functions of $(x,y,y')$ in the same compact region.

$y(x)$ is single valued in the interval and admits a derivative $y'(x)$ that is a continuous function of $x$ in the same interval.

I used the Euler-Lagrange condition to determine the optimal function $y(x)$. However, I am struggling with the sufficient conditions to establish that the solution actually is a minimum (I'm new to the calculus of variations).

Application of the Legendre conditions results in $F_{y'y'}(x,y,y') = 0$, as $F_{y'}(x,y,y')$ does not depend on $y'$. Similarly the Weierstrass condition holds with equality. I am looking for another sufficient condition, or a reference. Thank you.

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    Have you considered the possibility that the solution might be one of a family of minima (i.e., a minimum, but not an isolated min)? Perhaps you have, and this is just a silly suggestion. But if, for instance, you asked for the function $y$ joining $y_0$ to $y_1$ and minimizing the integral of $y'$...there'd be a ton of solutions. :) – 2017-02-13
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    Actually I'm looking for conditions to establish that the function is indeed a minimum. And I hoped the Legendre condition would have been enough, but the fact that it holds with equality it makes it inconclusive. @JohnHughes – 2017-02-13
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    Right -- the fact that both tests are inconclusive makes it possible that you're not at an isolated min ... which is why I asked if you're sure that the function really *is* an isolated min. Without that knowledge, you're not likely to find a good test. Over $\Bbb R$: Think of a function $f$ that decreases to 0 at $x = -1$; remains zero until $x = 1$, then rises. Now $0$ is a min for that function, but no "second derivative test" or anything like it will prove that. And there's a good reason: locally, around $0$, $f$ looks exactly like $-f$, which has a global *max* at $x = 0$ (and nearby). – 2017-02-13
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    @JohnHuges You have a really good point, but I was just wondering if there were other conditions, for example a transversality condition, that the mimimand should respect. – 2017-02-14
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    There might be...but you've unfortunately gone beyond what I know of Calc. of Var'ns, so I can't help. Good luck! – 2017-02-14
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    Did you ever make some progress on this? – 2017-08-18

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