Elliptic PDEs have no characteristic surfaces

Question

I’m trying to understand a remark my professor made, that elliptic PDEs are precisely those which do not admit characteristic hyper surfaces. What is the connection between this definition and the usual ones e.g.in terms of the principal symbol?

Answer 1

It's the same (understanding non-existence of characteristics as non-existence of real characteristics ).

If the principal part is $L[u]=au_{xx}+2bu_{xy}+c u_{yy}$, then the equation is said to be elliptic if $b^2-ac<0$. Now, the characteristic form is defined as $Q(\xi,\eta)=a\xi^2+2b\xi\eta+c\eta^2=\eta^2 q(\zeta)$, with $\zeta=-\xi/\eta$ and $q(\zeta)=a\zeta^2-2b\zeta+c$. The two families of curves defined by means of the ODEs

$$ \frac{dy}{dx}=\zeta_{\pm}(x,y), $$

with $\zeta_{\pm}$ the roots of $q(\zeta)=0$, are called characteristic curves. If $b^2-ac<0$ then $\zeta_{\pm}$ are necessarily complex valued and there are no real valued characteristics. On the contrary, if the characteristics are complex it means that necessarily $b^2-ac<0$, so both definitions of ellipticity are equivalent.