Example 12.1.6 (Elliptic Operators).label

1. (1)

   $\la$ is elliptic everywhere in $\R^{n}$

2. (2)

   $\square$ is nowhere elliptic in $\R^{n}$

3. (3)

   $L=x_{1}\pd_{1}^{2}+\pd_{2}^{2}$ then $\chi_{L}(x,\xi)=x_{1}\xi_{1}^{2}+\xi_{2}^{2}$ so $L$ is elliptic if and only if $x_{1}>0$