Past exam of the mathematics course of the University of Cambridge / 2006 / ia / Paper 3 / 11A (Vector Calculus)

Let $u={\varphi }_1-{\varphi }_2$. It is a harmonic function in the bounded region and vanishes on its entire boundary. For smooth solutions and boundary, the divergence theorem applied to $u\nabla u$ gives the energy identityThe integrand is nonnegative, so $\nabla u=0$ in $R$. Thus $u$ is constant on each connected component, and its zero Dirichlet boundary condition makes every constant zero. HenceThis proves the Uniqueness of the Dirichlet problem rather than assuming it.

For the separated expression, write $\varphi =F(x)G(y)$. The hyperbolic factor obeys $F^{\prime \prime }={\lambda }^{2}F$ and the trigonometric factor obeys $G^{\prime \prime }=-{\lambda }^{2}G$, sofor all the constants, including $\lambda =0$. In the half-strip take $a>0$ so the region has positive width. For every nonzero integer $n$, the decaying harmonic sine mode in a half-strip isIts second derivatives cancel, it vanishes at $y=0,a$, and it has the prescribed sine trace at $x=0$. The absolute value in the exponential ensures decay for negative as well as positive $n$. For $n=0$, the zero function is the solution. Thus all integer cases are covered, and the decay is uniform across the strip.