Past exam of the algebra, analysis and differential equations 1 course of the University of Oxford / 2013 / Differential Equations / 8 / c

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 iswhich 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 problemThe 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 $\boxed{u=0\text{ for }y\ge 0}$ 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).