Let us consider the problem $$ \min \left\{ \int_0^1 \psi\left(\dot{u}\right) \ dx: u \in \mathcal{C}^1([0,1]), u(0) =0, u(1)=2 \right\} $$ where $\psi \colon \mathbb{R} \longrightarrow \mathbb{R}$ is the function $$\psi (x) = (x^2-1)^2.$$ Hence by my professor is it easy to conjecture that the minimum of the functional is $u_0 (x) = 2x$. Anyway I do not find it "easy to conjecture". To convince myself of this conjecture I tried to work as it follows:
Find with the first variation of the integral a condition that assures the second derivative to be zero
Notice that there is only one linear function that satisfies the requirements of the ambient space of the problem. (By linear I improperly mean $f(x) = mx + q$).
The function $u_0(x)$ is the only minimum of the functional
Let us start with point 1. Let us consider $v \in V = \left \{ v \in \mathcal{C}^1([0,1]) : v(0)=v(1) = 0\right\}$ evaluating the first variation I find the condition: $$ \int_0^1 \left( 4 \dot{u}(x)^3\dot{v}(x) - 2 \dot{u}(x) \dot{v}(x)\right) \ dx $$ which I find pretty strange since given the function $\psi$ I would have expected a condition that will allow me to deduce $\dot{u} = 1$ where the $\inf$ of the function $\psi$ is taken, even tough is not in the ambient space of the problem. Maybe I am doing something wrong, suggestions accepted.
Let us focus on point 2 so that let us consider the convexified function of $\psi$ that coincides with $\psi$ for $|x| \geq 1$ and is zero otherwise. This is a convex function and, for my professor, $u_0(x) = 2x$ is the minimum of the function driven by this convexified function. Why? Anyway if it is true, if I consider the functional $G(u)$ the one driven by the convexified function we should have $F(u) \geq G(u) \geq G(u_0) = F(u_0)$ hence this shall prove that $u_0$ is the minimum of the functional of the problem.
Finally the uniqueness of the minimum is given by the convexity of the functional $G$ hence it is a unique minimum also for $F$.
Can somebody please help me filling in the gaps what I am missing? Thanks in advance.