Let \(U\) be a bounded domain. Consider the energy
\[ I[v]:=\int _U \frac 12|\nabla v|^2\,dx \]
on the admissible class
\[ \mathcal {A}=\left \{v\in C^2(U)\cap C^1(\overline {U}): v=0\text { on }\partial U,\ \int _U v\,dx=1\right \}. \]
Suppose that there is \(u\in \mathcal {A}\) satisfying
\[ I[u]=\min _{v\in \mathcal {A}}I[v]. \]
Show that \(u\) must solve the PDE boundary value problem \[ \begin {cases} -\Delta u=\lambda & \text {in } U,\\ u=0 & \text {on } \partial U. \end {cases} \]
Here \(\lambda \in \mathbb {R}\) is a Lagrange multiplier associated with the constraint. Find a formula for \(\lambda \) in terms of \(I[u]\).