Let $X=\{u\in C^1[0,1]|u(0)=0\}$ and let $I:X\to\mathbb{R}$ be defined as $I(u)=\int_0^1 (u'^2-u^2)$. Which of the following is correct?

Question

Let $X=\{u\in C^1[0,1]|u(0)=0\}$ and let $I:X\to\mathbb{R}$ be defined as $I(u)=\int_0^1 (u'^2-u^2)$. Which of the following is correct?

$(a)$ I is bounded below

$(b)$ I is not bounded below

$(c)$ I attains its infimum

$(d)$ I does not attain its infimum

Attempt:

Using Euler-Lagrange equations,

$2u+2u'=0$

Therefore $u(x)=c_1\sin(x)+c_2 \cos(x)$

It is given that $u(0)=0$ so it implies that $c_2=0$.

$\therefore u(x)=c_1\sin(x)$

What should be the next step? Please give some hints.

Answer 1

$u(x) =\int_0^{x} u'(t) \, dt$ so $u(x)^{2} \leq x\int_0^{x} u'(t) ^{2} \, dt \leq \int_0^{1} u'(t) ^{2} \, dt$ which shows (after integration) that $I \geq 0$. Hence a) is true, b) is false. The infimum is attained when $u \equiv 0$ so c) is true and d) is false.