Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 154 / 3 / 1

Letbe the harmonic-oscillator energy space, and denote the minimized quadratic functional by $J(u)$. A minimizing sequence $(u_n)\subset \mathcal{A}(M)$ is bounded in $\Sigma$, because $J$ is its squared Hilbert norm with positive coefficients. After taking a subsequence, $u_n\rightharpoonup u$ weakly in $\Sigma$.

The embedding $\Sigma \hookrightarrow L^4({\mathbb{R}}^2)$ is compact. On any fixed ball this follows from the Rellich-Kondrachov compactness theorem. Outside a large ball, the moment bound makes the $L^2$ tail uniformly small, and interpolation with the uniform $H^1\hookrightarrow L^q$ bounds for some $q>4$ makes the $L^4$ tail uniformly small. Hence $u_n\to u$ strongly in $L^4$.

The constraint passes to the limit, so $\int |u{|}^4=M$. Weak lower semicontinuity gives $J(u)\le \mathrm{liminf}J(u_n)=I(M)$. Therefore $u$ attains the infimum. This is Fixed-L4 minimization in the harmonic-oscillator energy space.