= Local Sobolev compactness gives lower semicontinuity of a nonnegative integral
If $u_n\rightharpoonup u$ in $H^1(\mathbb R^d)$ and $F:\mathbb R\to[0,\infty)$ is continuous, then $\int F(u)\leq\liminf_n\int F(u_n)$. Start with a subsequence attaining the lower limit. The <Rellich-Kondrachov compactness theorem> gives strong local <L2 space> convergence along a diagonal subsequence, and a further subsequence converges almost everywhere. Its limit is $u$, by uniqueness of the local weak limit. Now apply the <Fatou lemma>. Nonnegativity is essential to this argument; <convexity> of $F$ is unnecessary. In particular $F(s)=1-\cos s$ is allowed despite its failure to be convex.
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