Linear PDE

Definition

Let $\Omega$ be an open set in $\R^n$ and let $m \ge 1$. A linear partial differential equation of order $m$ in $\Omega$ is a partial differential equation of the following form. $$\sum_{\abs{\alpha} \le m} a_\alpha(x) \, \partial^\alpha u(x) = f(x), \quad x \in \Omega.$$ Here the sum runs over multi-indices $\alpha$. The coefficients $a_\alpha$ and the right-hand side $f$ are given functions on $\Omega$, and $a_\alpha$ is not identically zero for some $\alpha$ with $\abs{\alpha} = m$. The map $P = \sum_{\abs{\alpha} \le m} a_\alpha \partial^\alpha$ is the associated linear differential operator. The equation is homogeneous if $f = 0$ and inhomogeneous otherwise. It has constant coefficients if every $a_\alpha$ is constant.