> [!definition] > > Let $\Omega \subset \real^d$ be [[Open Set|open]], $L$ be a [[Linear Partial Differential Operator|linear partial differential operator]] on $\Omega$, and $x \in \Omega$, then $L$ is **elliptic** at $x$ if the [[Characteristic Form|characteristic form]] $\chi_L(x, \cdot)$ has no zeroes in $\mathbb R^d \setminus \bracs{0}$.