> [!definition] > > Let $U \subset \real^d$ be open and $L$ be a [[Second Order Linear Partial Differential Operator|second order partial differential operator]] with coefficients $A: U \to L(\real^d, \real^d)$, $b: U \to L(\real^d)$, and $c: U \to \real$, then > 1. $L$ is [[Elliptic Partial Differential Operator|elliptic]] if and only if $A(x)$ is invertible for each $x \in U$. > 2. $L$ is uniformly elliptic if and only if there exists $C \ge 0$ such that $\angles{\xi, A(x)\xi} \ge C\abs{\xi}^2$ for every $x \in U$ and $\xi \in \real^d$.