 Direct method in the calculus of variations

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 $\Vert u_n{\Vert }_X$. Reflexivity supplies a weakly convergent subsequence, and closure puts its limit $u$ in $\mathcal{A}$. Finally,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.^[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.

The Direct Method in the calculus of variations is a powerful approach used to find the extrema (minima or maxima) of functionals, which are mappings from a space of functions to the real numbers. This method primarily involves establishing the existence of a solution to a variational problem and typically uses concepts from analysis, compactness, and weak convergence.