Partial differential equation

Definition

Let $\Omega$ be an open set in $\R^n$ with $n \ge 2$, and let $k \ge 1$ be an integer. For a function $u \colon \Omega \to \R$ of class $C^k$, write $D^j u(x)$ for the collection of all partial derivatives $\partial^\alpha u(x)$ of order $\abs{\alpha} = j$, in multi-index notation. Let $F$ be a given function of these derivatives and of $x$. A partial differential equation of order at most $k$ in $\Omega$ is an equation of the following form. $$F\left(D^k u(x), D^{k-1} u(x), \dots, D u(x), u(x), x\right) = 0 \quad \text{for } x \in \Omega.$$ The function $u$ is the unknown. A classical solution is a function $u \in C^k(\Omega)$ that satisfies the equation at every point of $\Omega$. A system of partial differential equations is defined in the same way with $u$ and $F$ taking values in $\R^N$ and $\R^M$.