Let \(U\) be a smooth bounded domain and \(f:\overline {U}\to \mathbb {R}\) be a smooth function. Consider the energy
\[ I[v]:=\int _U \left (\sqrt {1+|\nabla v|^2}-fv\right )\,dx \]
on the admissible class
\[ \mathcal {A}=\{v\in C^2(U)\cap C^1(\overline {U}): v=0\text { on }\partial U\}. \]
Suppose that there is \(u\in \mathcal {A}\) satisfying
\[ I[u]=\min _{v\in \mathcal {A}}I[v]. \]
Find and justify the PDE boundary value problem satisfied by \(u\).