Consider for $T>0$ the functional
$$u\mapsto J(u) := \int_0^T (\dot{u}(t)^2-u(t)^2)dt. $$
on the space $W_0^{1,\infty}((0,T),\mathbb{R})$.
(a) Depending on $T$, find the critical points of $J$
(b) Find the corresponding Jacobi-equations (Why are they the same for all critical points here?)
(c) For which values of $T$ can we can conclude that none of the critical points is a minimizer.
For (a) I thought using Euler equations to $F_u-\frac{d}{dt}F_p = -2u-2\ddot{u} =0.$ I would say that $u(t) = c_1\sin(x)+c_2\cos(x)$ are the only solutions, but i'm not sure of this. And in what way do these solutions depend on $T$?
For (b) the Jacobi equations are $\frac{d}{dt}F_{pp}\dot{\eta}+F_{pu}\eta =F_{pu}\dot{\eta}+F_{uu}\eta$. But then I get $2\ddot{\eta}=-2\eta$ which seems like the same equation but now for the unknown $\eta$...but I don't understand the question.
(c) Not sure how to go about this one..?
Thanks for any help or suggestions.
You should read more carefully. For (a), note that we are working on the space $W_0^{1,\infty} ((0,T), \mathbf R)$, hence solutions have boundary values $u(0) = u(T) = 0$.
(Sorry for my short answer, but I seem not be able to add comments yet.)