Direct method in the calculus of variations

Summary

The direct method proves that a functional has a minimiser without solving any equation. One takes a minimising sequence and uses compactness to extract a subsequence that converges in a weak sense. Lower semicontinuity then shows that the limit is a minimiser. It is the standard existence method of the calculus of variations. It explains why minimisers are first found in Sobolev spaces with their weak topologies. It gave the first rigorous justification of Dirichlet's principle.