A linear differential operator \(L\) is called elliptic if
\begin{equation*} \sum_{\lvert \alpha\rvert=m} a_{\alpha}(x)\xi^\alpha\neq 0 \end{equation*}for all \(\xi\neq 0\) and all \(x\in \Omega\).
Remark
- This condition is not strong enough in many applications. Then the uniform ellipticity condition may be imposed.
- analytic coefficients ensure analytic solutions