Regularity theory

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.

Elliptic Regularity theory

Main article: Elliptic boundary value problem

Let ${\displaystyle U}$ be an open, bounded subset of ${\displaystyle \mathbb {R} ^{n}}$, denote its boundary as ${\displaystyle \partial U}$ and the variables as ${\displaystyle x=(x_{1},...,x_{n})}$. Representing the PDE as a partial differential operator ${\displaystyle L}$ acting on an unknown function ${\displaystyle u=u(x)}$ of ${\displaystyle x\in U}$ results in a BVP of the form $${\displaystyle \left\{{\begin{aligned}Lu&=f&&{\text{in }}U\\u&=0&&{\text{on }}\partial U,\end{aligned}}\right.}$$ where ${\displaystyle f:U\rightarrow \mathbb {R} }$ is a given function ${\displaystyle f=f(x)}$ and ${\displaystyle u:U\cup \partial U\rightarrow \mathbb {R} }$ and the operator ${\displaystyle L}$ 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

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

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 (PDF). 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.