Scalar partial differential equations, order, and classical solutions

Let $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$ be open, let $m\ge 1$, and let

$F:\mathrm{\Omega }\times {\mathbb{R}}^{N(m,n)}\to \mathbb{R}$

be a function of the point $x$ and of the jet coordinates $z_{\alpha }(\mathrm{\mid }\alpha \mathrm{\mid }\le m)$, where $N(m,n)$ is the number of multi-indices of length at most $m$. Assume that $F$ depends nontrivially on at least one jet coordinate. A scalar partial differential equation of order at most $m$ is an equation

$F(x,(D^{\alpha }u(x){)}_{\mathrm{\mid }\alpha \mathrm{\mid }\le m})=0$

for an unknown scalar field $u:\mathrm{\Omega }\to \mathbb{R}$.

Its order is the largest $\mathrm{\mid }\alpha \mathrm{\mid }$ for which $F$ depends nontrivially on $z_{\alpha }$. A classical solution of an equation of order at most $m$ is a function $u\in C^m(\mathrm{\Omega })$ satisfying the displayed equation at every $x\in \mathrm{\Omega }$.

If $\mathrm{\Sigma }\subseteq \mathrm{\Omega }$ is a codimension-one surface, prescribing values of $u$ and possibly of some derivatives on $\mathrm{\Sigma }$ is Cauchy data. If $\mathrm{\partial }\mathrm{\Omega }$ is used instead, the prescribed values are boundary data.