{{Short description|On weak solutions of differential equations}} '''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.<ref name="Elliptic"/>

The motivation for this study is as follows.<ref name="Evans"/> 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|Elliptic boundary value problem}} Let <math>U</math> be an open, 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 elliptic 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 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.

== 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 <math>U\subset \mathbb{R}^n</math>. We assume the boundary of ''U'' to be smooth, and the elliptic operator to be independent of time, with smooth coefficients, i.e.<math display="block">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).</math>In addition, we subscribe the boundary value of ''u'' to be 0.

Then the regularity of the solution is given by the following table, {| class="wikitable" |+ !Equation !<math>u_t+Lu=f</math> (parabolic) !<math>u_{tt}+Lu=f</math> (hyperbolic) |- |Initial Condition |<math>u(0)\in H_{x}^{2m+1}</math> |<math>u(0)\in H_{x}^{m+1},\,(\partial_t u)(0)\in H_{x}^{m}</math> |- |External force |<math>\partial_{t}^k f\in L_t^2 H_{x}^{2(m-k)}\,(k=1,\dots m)</math> |<math>\partial_{t}^k f\in L_t^2 H_{x}^{m-k}\,(k=1,\dots m)</math> |- |Solution |<math>\partial_t^k u\in L_t^2 H_x^{2(m+1-k)},\,(k=1,\dots,m+1)</math> |<math>\partial_t^k u\in L_t^\infty H_x^{m+1-k},\,(k=1,\dots,m+1)</math> |} where ''m'' is a natural number, <math>x\in U</math> denotes the space variable, ''t'' denotes the time variable, ''H<sup>s</sup>'' is a Sobolev space of functions with square-integrable weak derivatives, and ''L<sub>t</sub><sup>p</sup>X'' 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.<ref name="Smoller"/>

== References == <references>

<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. | author-link=Lawrence C. Evans | 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>

</references>

Category:Partial differential equations