Let $L$ a differential operator of the form $$L(D)=\sum_{|i|=1}^na_i(x)D^i,$$ where $D=(D_1,...,D_n)$ and for $i=(i_1,...,i_n)$, we set $D^i=D_1^{i_1}...D_n^{i_n}$. We set $L'$ the leading part part of $L$ consisting of the termes of highest order. I have problem to see what is exactly $L'$. If $$L(D)=4D_1^3D_2^2+2D_1^2D_2^4,$$
is $L'(D)=4D_1^3+2D_2^4$ ? The thing is an operator is elliptic if $L'(D)=L(D)$, but with this definition, I don't find any elliptic operator, so it should be something else. After I thought that $L(D)=2D_1^2D_2^4+4D_1^3D_1^2=2D^5+2D^6$ and thus $L'(D)=2D^6=2D_1^2D_2^4$, but still not work.
Any idea ?
If $L(D)=\sum_{|i|=1}^m a_i D^i$, then the leading part is $$L'(D)=\sum_{|i|=m}a_i D^i.$$
It's ellptic if $L'(z)\neq 0$ for $z\neq 0$ real (not if $L'=L$).
For example, $L(D)=[D_1,D_2]=D_1D_2-D_2D_1$ is not elliptic, whereas $L(D)=D_1D_2-2D_2D_1$ is elliptic.