Direct method in the calculus of variations

A proof strategy for establishing that a variational problem has a minimizer. It combines compactness of minimizing sequences with lower semicontinuity of the objective functional.

Minimizing sequence: A sequence of admissible objects whose objective values approach the infimum of a minimization problem. The method begins with a sequence whose values approach the infimum.
Calculus of variations: The mathematical study of extrema of functionals, often defined on spaces of functions. The direct method addresses minimization problems formulated in this broader subject.
Dirichlet principle: The principle that a harmonic function minimizes the integral of the squared gradient among functions with the same boundary values. Its existence proof illustrates minimizing energy over a constrained function class.
Euler–Lagrange equation: A differential equation satisfied by stationary points of a sufficiently smooth variational functional. It characterizes candidates, while the direct method proves a minimizer exists without first solving that equation.
Weak convergence: Convergence defined by testing against continuous linear functionals, weaker than convergence in norm. Bounded minimizing sequences often yield weakly convergent subsequences.
Functional: A map whose inputs are functions or other elements of an infinite-dimensional space and whose outputs are usually scalars. The quantity minimized by the method is a functional.
Calculus of variations for minimal surfaces: The variational study of surfaces minimizing area under prescribed boundary or geometric constraints. Area-minimization problems use compactness and lower semicontinuity to establish minimizers.
Calculus of variations, indirect methods: Methods for studying variational extrema through necessary conditions such as differential equations and first variations. Indirect methods derive extremal conditions rather than obtaining existence from compactness.
Weak lower semicontinuity: A functional is weakly lower semicontinuous if its value at a weak limit does not exceed the limit inferior of sequence values. It ensures the limit's objective value is no greater than the minimizing sequence's limiting value.
Sobolev space: A function space whose elements and weak derivatives satisfy specified integrability conditions. Sobolev spaces provide common admissible spaces with useful compactness and semicontinuity properties.

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