Let \(U\) be a bounded domain in \(\mathbb {R}^d\) and \(\Gamma \) be a closed subset of \(\partial U\). Consider the energy
\[ I[v]=\frac 12\int _U \left (|Dv|^2+u^2\right )\,dx \]
on the admissible class
\[ \mathcal {A}=\{v\in C^2(\overline {U})\cap C^1(\overline {U}): u|_{\Gamma }=g\},\quad g\in C(\partial U). \]
Suppose that \(u\) is a minimizer for the calculus of variations problem
\[ I[u]=\min _{v\in \mathcal {A}} I[v] \]
find and justify the PDE boundary value problem solved by \(u\). Show that there is at most one solution of this PDE in the admissible class \(\mathcal {A}\).