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{{Short description|Concept in mathematics}}
'''Regularity''' is a topic of the mathematical study of [[Partial differential equation|partial differential equations]](PDE) such as [[Laplace's equation]], about the integrability and differentiability of [[Weak solution|weak solutions]]. [[Hilbert's nineteenth problem]] was concerned with this concept.<ref name="Elliptic"/>
The motivation for this study is as follows.<ref name="Evans"/> 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|Elliptic boundary value problem}}
Let <math>U</math> be an [[Open set|open]], [[Bounded set|bounded]] subset of <math>\mathbb{R}^n</math>, denote its boundary as <math>\partial U</math> and the variables as <math>x=(x_1,...,x_n)</math>. Representing the PDE as a partial [[differential operator]] <math>L</math> acting on an unknown function <math>
u=u(x)</math> of <math>x\in U</math> results in a BVP of the form <math display="block">\left\{
\begin{align}
L u &= f & &\text{in } U\\
u &=0 & &\text{on } \partial U,
\end{align}\right.
</math> where <math>f: U \rightarrow \mathbb{R}</math> is a given function <math>f=f(x)</math> and <math> u:U\cup \partial U \rightarrow \mathbb{R}</math> and the operator <math>L</math> is of the '''divergence''' '''form''': <math display="block">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),</math>then
* '''Interior regularity''': If ''m'' is a natural number, <math>a^{ij},b^{j},c \in C^{m+1}(U), f\in H^{m}(U)</math> (2) , <math>u\in H_{0}^{1}(U)</math> is a weak solution, then for any open set ''V'' in ''U'' with compact closure, <math>\|u\|_{H^{m+2}(V)}\le C(\|f\|_{H^{m}(U)}+\|u\|_{L^2(U)})</math>(3), where ''C'' depends on ''U, V, L, m'', per se <math>u\in H_{loc}^{m+2}(U)</math>, which also holds if ''m'' is infinity by [[Sobolev inequality|Sobolev embedding theorem]].
* '''Boundary regularity''': (2) together with the assumption that <math>\partial U</math> is <math>C^{m+2}</math> indicates that (3) still holds after replacing ''V'' with ''U,'' i.e. <math>u\in H^{m+2}(U)</math>, 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 law|Conservation laws]], called shock waves.<ref name="Smoller"/>
== References ==
{{reflist
<ref name="Elliptic">{{Cite book |last1=Fernández-Real |first1=Xavier |last2=Ros-Oton |first2=Xavier |date=2022-12-06 |title=Regularity Theory for Elliptic PDE |doi=10.4171/ZLAM/28|arxiv=2301.01564 |isbn=978-3-98547-028-0 |s2cid=254389061 }}</ref>
<ref name="Evans">{{cite book | last=Evans | first=Lawrence C. | title=Partial differential equations | publisher=American mathematical society | publication-place=Providence (R. I.) | date=1998 | isbn=0-8218-0772-2|url= https://math24.wordpress.com/wp-content/uploads/2013/02/partial-differential-equations-by-evans.pdf}}</ref>
<ref name="Smoller">{{cite book | last=Smoller | first=Joel | title=Shock Waves and Reaction—Diffusion Equations | publisher=Springer New York, NY | edition=2 | isbn=978-0-387-94259-9 |doi=10.1007/978-1-4612-0873-0 }}</ref>
}}
[[Category:Elliptic partial differential equations]]
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