= Separated Laplace mode with a Robin edge
For $0<x<L$, $0<y<H$, choose $k=m\pi/H>0$ and seek a <harmonic function> with zero values on the horizontal edges, boundary value $\sin(ky)$ at $x=L$, and $\phi+\alpha\partial_n\phi=0$ at $x=0$, with constant $\alpha\ge0$. <Separation of variables> gives $\phi=[\alpha k\cosh(kx)+\sinh(kx)]\sin(ky)/[\alpha k\cosh(kL)+\sinh(kL)]$. The outward derivative at the left edge is $-\partial_x$, giving $F(0)-\alpha F'(0)=0$. The denominator is positive and <Poisson uniqueness with a nonnegative Robin normal coefficient> proves uniqueness in the regular solution class.
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