Regularity of minimizers

Summary

Regularity of minimisers is the study of how smooth the minimisers of a variational problem are. The direct method produces minimisers only in a Sobolev space, so it is a separate question whether they are continuous, differentiable, smooth or analytic. It is the subject of Hilbert's nineteenth problem, solved for scalar problems by the De Giorgi–Nash–Moser theorem. For vector-valued problems minimisers can be singular, and the theory proves partial regularity: smoothness outside a small closed set. It is a central part of regularity theory for elliptic equations.