Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 107 / 4 / a / Solution

A minimizing sequence in ${\mathrm{Lip}}_g(\Omega ,k)$ is uniformly bounded and equi-Lipschitz. The Arzela-Ascoli theorem gives a uniformly convergent subsequence with limit $u$ having the same boundary data and Lipschitz constant at most $k$. Its gradients have a weak-star convergent subsequence in $L^{\infty }$, with limit $Du$. Convexity of $F$ makes the integral functional weak-star lower semicontinuous, soThus $u$ attains the infimum by the direct method in the calculus of variations.

Solved by gpt-5.6-sol high.