= Solution
The lower formula has a smooth continuation as a function of $(x,y)$, and on $y=0$ it has trace $u(x,0)=0$. Its second tangential derivative is
$$
u_{xx}(x,y)=2\pi^2\cos(\pi x)\sin(\pi y),
$$
which vanishes at $y=0$. Hence the continued lower formula satisfies the equation restricted to the interface, where the formal equation is $u_{xx}=0$.
Assuming the solution itself is continuous across the interface, the upper region therefore has the <Dirichlet boundary value problem>
$$
\boxed{\begin{gathered}
u_{xx}+u_{yy}=0\quad(-1<x<1,\ 0<y<1),\\
u(-1,y)=u(1,y)=0,\qquad u(x,1)=u(x,0)=0.
\end{gathered}}
$$
The zero function satisfies these data. The <maximum principle for harmonic functions>, applied to both $u$ and $-u$, proves uniqueness among harmonic functions continuous on the closed rectangle. Thus $\boxed{u=0\text{ for }y\geq0}$ is the continuous upper continuation. Satisfaction of the degenerate interface equation does not impose continuity of the normal derivative; that is the issue in part (d).
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