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?
Best Answer
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.
But this leaves the question that why the symbol matters at all in the classification process. Intuitively, it seems "clear" that the highest derivative should matter most. However to show this rigorously is in fact quite deep; a proof of elliptic operators satisfies some basic nice properties takes at least half a page and usually require one knew pseudo-differential operators. I suppose others might be better at answering this. If you can follow, I also think Klainerman's article on PDE might be helpful, in which he discussed thoroughly the classification of PDEs.