Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 4 / 23H / b / iii / Solution

Choose $\varphi \in C_c^{\infty }(\mathbb{R})$ with $\Vert \varphi {\Vert }_2=1$, and for $R>0$ setA change of variables givesThe latter tends to zero as $R\to \infty$. Since the energy is nonnegative,This is vanishing Dirichlet energy by dilation on the line.

The infimum is not attained. If an admissible $u$ had zero energy, then its weak derivative would vanish almost everywhere, so $u$ would be constant almost everywhere. The only constant in $L^2(\mathbb{R})$ is zero, contradicting $\Vert u{\Vert }_2=1$.

Solved by gpt-5.6-sol high.