= Solution
Use the convention $A u_{xx}+2B u_{xy}+C u_{yy}=0$, with discriminant $B^2-AC$. Here $A=1$, $B=0$, $C=\operatorname{sign}(y)$.
For $y>0$, the discriminant is $-1<0$, so the equation is an <elliptic partial differential equation>, already in the canonical <Laplace equation> form $u_{xx}+u_{yy}=0$.
For $y<0$, the discriminant is $1>0$, so it is a <hyperbolic partial differential equation>, the <wave equation> $u_{xx}-u_{yy}=0$. Introduce characteristic coordinates
$$
\xi=x+y,\qquad\eta=x-y.
$$
Then $\partial_x=\partial_\xi+\partial_\eta$ and $\partial_y=\partial_\xi-\partial_\eta$, giving
$$
u_{xx}-u_{yy}=4u_{\xi\eta}=0.
$$
On $y=0$, the discriminant vanishes and the principal part has rank one; the degenerate <parabolic type> is $u_{xx}=0$. This is an interface line, rather than an open parabolic subdomain. Thus
$$
\boxed{\begin{array}{c|c|c}
y>0&\text{elliptic}&u_{xx}+u_{yy}=0\\
y<0&\text{hyperbolic}&u_{\xi\eta}=0\\
y=0&\text{degenerate parabolic}&u_{xx}=0
\end{array}}
$$
This is a <mixed-type partial differential equation>. The classification concerns the principal part and remains meaningful even though the coefficient jumps at the interface.
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