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  "body": "\n## Summary\n\nRegularity of minimisers is the study of how smooth the minimisers of a variational problem are.\nThe direct method produces minimisers only in a Sobolev space, so it is a separate question whether they are continuous, differentiable, smooth or analytic.\nIt is the subject of [[Hilbert's nineteenth problem|QPMM2ZPWBW]], solved for scalar problems by the [[De Giorgi–Nash–Moser theorem|32C3D8FG9V]].\nFor vector-valued problems minimisers can be singular, and the theory proves partial regularity: smoothness outside a small closed set.\nIt is a central part of [[regularity theory|N383CX7WKS]] for elliptic equations.\n\n## Explanations\n\n### L1 human\n\nExistence proofs in the calculus of variations give a minimiser that might have kinks, jumps in its derivative, or worse.\nRegularity theory asks whether these defects really occur.\nFor the Dirichlet energy $\\int \\abs{\\nabla u}^2 \\, dx$ the minimisers are harmonic functions, which are smooth, even analytic, inside the domain.\nFor many energies of maps with several components, singular minimisers do exist; then one shows that the bad points form a small set.\n",
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