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 construct 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.

Elliptic regularity theory

Main article: Elliptic boundary value problem

Let U be an open, bounded subset of \mathbb{R}^n, denote its boundary as \partial U and the variables as x=(x_1,...,x_n). Representing the PDE as a partial differential operator L acting on an unknown function u=u(x) of x\in U results in a BVP of the form

\left\{
\begin{align}
L u &= f & &\text{in } U\\
u &=0 & &\text{on } \partial U,
\end{align}\right.

where f: U \rightarrow \mathbb{R} is a given function f=f(x) and u:U\cup \partial U \rightarrow \mathbb{R} and the elliptic operator L is of the divergence form:

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

  • Interior regularity: If m is a natural number, a^{ij},b^{j},c \in C^{m+1}(U), f\in H^{m}(U) (2), u\in H_{0}^{1}(U) is a weak solution, then for any open set V in U with compact closure, \|u\|_{H^{m+2}(V)}\le C(\|f\|_{H^{m}(U)}+\|u\|_{L^2(U)})(3), where C depends on U, V, L, m, per se u\in H_{loc}^{m+2}(U), which also holds if m is infinity by Sobolev embedding theorem.
  • Boundary regularity: (2) together with the assumption that \partial U is C^{m+2} indicates that (3) still holds after replacing V with U, i.e. u\in H^{m+2}(U), which also holds if m is infinity.

Parabolic and Hyperbolic regularity theory

Parabolic and hyperbolic PDEs describe the time evolution of a quantity u governed by an elliptic operator L and an external force f over a space U\subset \mathbb{R}^n. We assume the boundary of U to be smooth, and the elliptic operator to be independent of time, with smooth coefficients, i.e.

Lu(t,x)= - \sum_{i,j=1}^n \big(a_{ij} (x) u_{x_i}(t,x)\big)_{x_j} + \sum_{i=1}^n b_i(x) u_{x_i}(t,x) + c(x) u(t,x).

In addition, we subscribe the boundary value of u to be 0.

Then the regularity of the solution is given by the following table,

Equationu_t+Lu=f (parabolic)u_{tt}+Lu=f (hyperbolic)
Initial Conditionu(0)\in H_{x}^{2m+1}u(0)\in H_{x}^{m+1},\,(\partial_t u)(0)\in H_{x}^{m}
External force\partial_{t}^k f\in L_t^2 H_{x}^{2(m-k)}\,(k=1,\dots m)\partial_{t}^k f\in L_t^2 H_{x}^{m-k}\,(k=1,\dots m)
Solution\partial_t^k u\in L_t^2 H_x^{2(m+1-k)},\,(k=1,\dots,m+1)\partial_t^k u\in L_t^\infty H_x^{m+1-k},\,(k=1,\dots,m+1)

where m is a natural number, x\in U denotes the space variable, t denotes the time variable, Hs is a Sobolev space of functions with square-integrable weak derivatives, and LtpX is the Bochner space of integrable X-valued functions.

Counterexamples

Not every weak solution is smooth; for example, there may be discontinuities in the weak solutions of conservation laws called shock waves.[3]

References

  1. ^ Fernández-Real, Xavier & Ros-Oton, Xavier (2022-12-06). Regularity Theory for Elliptic PDE. arXiv:2301.01564. doi:10.4171/ZLAM/28. ISBN 978-3-98547-028-0. S2CID 254389061
  2. ^ Evans, Lawrence C. (1998). Partial differential equations. Providence (R. I.): American mathematical society. ISBN 0-8218-0772-2.
  3. ^ Smoller, Joel. Shock Waves and Reaction—Diffusion Equations. 2 ed. Springer New York, NY. doi:10.1007/978-1-4612-0873-0. ISBN 978-0-387-94259-9.