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 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.

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 elliptic 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.

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 ${\displaystyle U\subset \mathbb {R} ^{n}}$. We assume the boundary of U to be smooth, and the elliptic operator to be independent of time, with smooth coefficients, i.e.$${\displaystyle Lu(t,x)=-\sum _{i,j=1}^{n}{\big (}a_{ij}(x)u_{x_{i}}(t,x){\big )}_{x_{j}}+\sum _{i=1}^{n}b_{i}(x)u_{x_{i}}(t,x)+c(x)u(t,x).}$$In addition, we subscribe the boundary value of u to be 0.

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

Equation	${\displaystyle u_{t}+Lu=f}$ (parabolic)	${\displaystyle u_{tt}+Lu=f}$ (hyperbolic)
Initial Condition	${\displaystyle u(0)\in H_{x}^{2m+1}}$	${\displaystyle u(0)\in H_{x}^{m+1},\,(\partial _{t}u)(0)\in H_{x}^{m}}$
External force	${\displaystyle \partial _{t}^{k}f\in L_{t}^{2}H_{x}^{2(m-k)}\,(k=1,\dots m)}$	${\displaystyle \partial _{t}^{k}f\in L_{t}^{2}H_{x}^{m-k}\,(k=1,\dots m)}$
Solution	${\displaystyle \partial _{t}^{k}u\in L_{t}^{2}H_{x}^{2(m+1-k)},\,(k=1,\dots ,m+1)}$	${\displaystyle \partial _{t}^{k}u\in L_{t}^{\infty }H_{x}^{m+1-k},\,(k=1,\dots ,m+1)}$

where m is a natural number, ${\displaystyle x\in U}$ denotes the space variable, t denotes the time variable, H^s is a Sobolev space of functions with square-integrable weak derivatives, and L_t^pX is the Bochner space of integrable X-valued functions.

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