Mixed-type equation

Remarks

Example: the Tricomi equation $u_{yy} + y \, u_{xx} = 0$ on $\Omega = \R^2$, written as $A u_{xx} + 2B u_{xy} + C u_{yy}$, has $A = y$, $B = 0$ and $C = 1$, so $AC - B^2 = y$. It is elliptic on the open set $y > 0$ and hyperbolic on the open set $y < 0$, so it is of mixed type. Non-example: the equation $u_{xx} + (1 + y^2) u_{yy} = 0$ has $AC - B^2 = 1 + y^2 > 0$ everywhere, so it is elliptic on all of $\R^2$ and not of mixed type. If the coefficients are continuous, the elliptic and hyperbolic regions are open, and in a connected $\Omega$ they are separated by a closed set where the coefficient matrix is degenerate.