= Direct method in the calculus of variations
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The <direct method in the calculus of variations> proves existence by passing to a limit of a <minimizing sequence>. A useful abstract version takes a nonempty weakly sequentially closed admissible subset $\mathcal A$ of a reflexive <Banach space> $X$ and a functional $E:X\to\mathbb R\cup\{+\infty\}$. Assume $E$ is finite at some admissible point, bounded below, coercive on $\mathcal A$, and has <weak lower semicontinuity>.
Choose $u_n\in\mathcal A$ with $E(u_n)\to m=\inf_{\mathcal A}E$. <Coercivity> bounds $\|u_n\|_X$. <Reflexivity> supplies a weakly convergent subsequence, and closure puts its limit $u$ in $\mathcal A$. Finally,
$$
m\le E(u)\le\liminf_nE(u_n)=m,
$$
so $u$ attains the infimum. Each hypothesis has a distinct role: compactness must preserve the constraints, and lower semicontinuity must hold for the convergence actually obtained.
For the <Dirichlet energy> with linear forcing on $H_0^1(\Omega)$ over a bounded domain, the <Poincaré inequality> controls the forcing and gives coercivity, while the squared gradient norm is weakly lower semicontinuous. The resulting minimizer solves the weak <Poisson equation> by its first variation.<Partial Differential Equations by John K. Hunter>[Chapter 4, §4.2, variational formulation and the minimizing-sequence approach]{ref} Strict convexity gives uniqueness when the admissible class is convex. The <oscillating minimizing sequence with no minimizer> shows what can fail without suitable lower semicontinuity.
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