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  "body": "\n## Summary\n\nThe direct method proves that a functional has a minimiser without solving any equation.\nOne takes a minimising sequence and uses compactness to extract a subsequence that converges in a weak sense.\nLower semicontinuity then shows that the limit is a minimiser.\nIt is the standard existence method of the [[calculus of variations|PQ25NHV0KS]].\nIt explains why minimisers are first found in Sobolev spaces with their weak topologies.\nIt gave the first rigorous justification of Dirichlet's principle.\n\n## Explanations\n\n### L1 human\n\nTo show that an energy has a lowest state, take a sequence of states whose energies go down to the lowest possible value.\nIf the energies stay bounded, the states cannot run away, and some part of the sequence settles down to a limit.\nIf the energy cannot jump up in the limit, the limit is a state of lowest energy.\nThe key points are to choose a notion of convergence weak enough for the sequence to converge, yet strong enough for the energy to behave well.\nOn the real line the function $x \\mapsto x^2$ illustrates the steps: any minimising sequence tends to $0$, and $0$ is the minimiser.\n",
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