Regularity is a topic of the mathematical study of partial differential equations (PDE) such as Laplace's equation, about the integrability and differentiability of weak solutions.
Hilbert's nineteenth problem was concerned with this concept.
[1] The motivation for this study is as follows.
[2] It is often difficult to contrust a classical solution satisfying the PDE in regular sense, so we search for a weak solution at first, and then find out whether the weak solution is smooth enough to be qualified as a classical solution.
Several theorems have been proposed for different types of PDEs.
be an open, bounded subset of
, denote its boundary as
and the variables as
Representing the PDE as a partial differential operator
acting on an unknown function
results in a BVP of the form
{\displaystyle \left\{{\begin{aligned}Lu&=f&&{\text{in }}U\\u&=0&&{\text{on }}\partial U,\end{aligned}}\right.}
is a given function
and the operator
is of the divergence form:
{\displaystyle Lu(x)=-\sum _{i,j=1}^{n}(a_{ij}(x)u_{x_{i}})_{x_{j}}+\sum _{i=1}^{n}b_{i}(x)u_{x_{i}}(x)+c(x)u(x),}
then Not every weak solution is smooth; for example, there may be discontinuities in the weak solutions of conservation laws called shock waves.