Regularity theory

Summary

Regularity theory is the part of the theory of partial differential equations that shows that solutions are smoother than first assumed. Typically one first obtains a weak solution in a Sobolev space. Then one proves that it has more derivatives, in Sobolev or Hölder spaces, when the coefficients and data are smooth. For elliptic equations the classical results are elliptic regularity, the Schauder estimates, the Calderón–Zygmund estimates and the De Giorgi–Nash–Moser theorem. An important source of the subject is Hilbert's nineteenth problem on the analyticity of minimisers.