> [!definition] > > Let $U \subset \real^d$ be [[Open Set|open]] and $L$ be a [[Linear Partial Differential Operator|linear partial differential operator]] of order $m$, and suppose that $P_m$ is its principal symbol, then for any $x \in U$, the following are equivalent: > 1. $P_m(x)(\xi) \ne 0$ for all $\xi \in \real^d$ with $\xi \ne 0$. > 2. There exists $C > 0$ such that $|P_m(x)(\xi)| \ge C\abs{\xi}^m$ for all $\xi \in \real^d$. > 3. There exists $C, R > 0$ such that $|P_m(x)(\xi)| \ge C\abs{\xi}^m$ for all $\xi \in \real^d$ with $\abs{\xi} \ge R$. > > If the above holds for every $x \in U$, then $L$ is **elliptic**, being "of $m$-th order in all directions". If $(2)$ holds for the same $C$ for all $x \in U$, then $L$ is **uniformly elliptic**. > > *Proof*. Fix $x \in U$. $(1) \Rightarrow (2)$: Since the unit ball $\partial B(0, 1)$ is compact, $|P_m(x)|$ admits a non-zero minimum, which is our $C$. This extends to all $\xi \in \real^d$ by homogeneity. > > $(2) \Rightarrow (3)$: By the same argument, since $P - P_m$ is of order at most $m - 1$, there exists $C' \ge 0$ such that $\abs{(P - P_m)(x)(\xi)} \le C'\abs{\xi}^{m - 1}$ for all $\xi \in \real^d$. In which case, for sufficiently large $\xi$, the desired result holds. > > $(3) \Rightarrow (1)$: If $P_m(x)(\xi) = 0$ for some $\xi \in \real^d$, then $\limv{\abs{\xi}}|P(x)(\xi)|/|\xi|^m \to 0$, which contradicts $(3)$.