Here is a question : Consider the Lagrangian $L(x, u, p) = 1/2((p^2 -1)^2) +1/2(u^2)$ and consider the corresponding functional $I(u)$ := $\int_0^1$ $L(x, u(x), u′ (x)) dx.$ Consider $A := \{w\in C^1: w(0)=w(1)=0\}$. Answer the following questions with justification:
• Is it TRUE that $I(u) ≥ 0$ for all $u ∈ A$?
• Consider the ZERO function $v(x) = 0 ∀x ∈ [0, 1].$ Find the value of $I(v)$.
• Does there exist a sequence of function $u_n(x) ∈ A $ such that $\lim_{n\to\infty} I(u_n) = 0$?
• What is the value of $\inf_{u∈A} I(u)$?
My attempt : For the first part it is clear because Lagrangian is sum of two non negative terms and so the integral will also be non negative. Also for second part substituting zero function we shall get $1/2$. I'm stuck in last two parts.
As you can observe that $L$ is always non-negative, $I(u) \ge 0$, Checking that $I(0) = 1/2$ is also trivial. If we suppose that the third part is true then as $0$ is a lower bound then the infimum will be zero. So it remains to show that there is a sequence of functions $u_n \to 0$ for which $I(u_n) \to 0$
here is one example :
$$u_{n}(x)= \begin{cases} n^2x/2 & \text{if } x \in [0,1/n^2]\\ \text{linear function with slope 1} & \text{if } x \in [1/n^2, 1/2n-1/n^2] \\ \text{Again a parabola} & x \in [-1/n^2 +1/2n,1/2n] \end{cases}$$
Make this function symmetric about $1/2$ and periodic of period $1$, then $u_n$ is the required function.