Why are elliptic/parabolic/hyperbolic PDEs called "elliptic"/"parabolic"/"hyperbolic"?

I see that the form of a (e.g.) parabolic equation is $$Au_{xx} + 2Bu_{xy} + Cu_{yy} + Du_x + Eu_y + F = 0$$ with $B^2-4AC=0$ whereas the equation of a parabola is $$Ax^2 + 2Bxy + Cuy^2 + Dx + Ey + F = 0$$ with $B^2-4AC=0$.

They are similar, but is there a deeper relationship between these two mathematical concepts than the mere analogy on their notations?

• – KCd
Jan 4 '15 at 1:50
• @KCd: I did not know! I suppose my answer is not as good. Jan 4 '15 at 1:58
It is because the principal symbol of the partial differential operator involved in the process. You can think a typical linear PDE as $$Pu=0$$ where $u$ is in some function space with possible boundary condition given, and $P$ is a linear differential operator like $\frac{\partial^{2}}{\partial y^{2}}+\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial xy}$. The symbol of this operator is given by the polynomial $\xi_{2}^{2}+\xi_1^{2}+\xi_1\xi_2,\xi=(\xi_1,\xi_2)$. So on a purely formal level, (linear) PDEs are classified by their (principal) symbols. For example, an equation is called elliptic if its principal symbol is invertible.