 Weak lower semicontinuity of a bounded-domain Schrodinger energy (source code)

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= Weak lower semicontinuity of a bounded-domain Schrodinger energy

On bounded $\Omega$ with $V\in L^\infty(\Omega)$, weak convergence in $H_0^1(\Omega)$ makes the Dirichlet term lower semicontinuous and, by Rellich compactness, makes the potential term $\int Vu^2$ continuous. Thus
$$
E(u)=\int_\Omega(|Du|^2+Vu^2)
$$
is weakly lower semicontinuous.

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