Elliptic, hyperbolic, and parabolic principal symbols

For a real scalar second-order principal symbol

$p_2(x,\xi )={\sum }_{i,j=1}^n{a}_{ij}(x){\xi }_i{\xi }_j,$

assume $a_{ij}=a_{ji}$.

The symbol is elliptic at $x$ when $p_2(x,\xi )\mathrm{\ne }0$ for every $\xi \mathrm{\ne }0$; equivalently, the quadratic form has one definite sign.

Fix a nonzero covector $\tau$. The symbol is strictly hyperbolic at $x$ relative to $\tau$ when $p_2(x,\tau )\mathrm{\ne }0$ and, for every $\zeta$ not proportional to $\tau$, the polynomial $\lambda \mapsto p_2(x,\zeta +\lambda \tau )$ has two distinct real roots. The first condition says that $\tau$ is noncharacteristic and prevents the root condition from becoming vacuous in one dimension.

A space-time operator of the form

$u_t-{\sum }_{i,j=1}^n{a}_{ij}(x,t)u_{x_i{x}_j}+\text{lower-order terms}$

is parabolic at $(x,t)$ when the spatial quadratic form $\sum a_{ij}(x,t){\xi }_i{\xi }_j$ is positive semidefinite and nonzero, and, in the standard time covector, the equation contains a first-order time derivative rather than a second-order one. The nonzero condition ensures that the operator still has a second-order spatial principal part at the point.