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

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 $\left\{{\begin{aligned}Lu&=f&&{\text{in }}U\\u&=0&&{\text{on }}\partial U,\end{aligned}}\right.$ where $f:U\rightarrow \mathbb {R}$ is a given function $f=f(x)$ and $u:U\cup \partial U\rightarrow \mathbb {R}$ and the 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, $\|u\|_{H^{m+2}(V)}\leq C(\|f\|_{H^{m}(U)}+\|u\|_{L^{2}(U)})$ (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.

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

^ 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.
^ Evans, Lawrence C. (1998). Partial differential equations (PDF). Providence (R. I.): American mathematical society. ISBN 0-8218-0772-2.
^ 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.

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[Elliptic-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.

[Evans-2] Evans, Lawrence C. (1998). Partial differential equations (PDF). Providence (R. I.): American mathematical society. ISBN 0-8218-0772-2.

[Smoller-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.

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