= Solution
This is \b[<Laplace's equation>]. Zero is \b[not unique]: with $k=\pi/2$,
$$
\boxed{u(x,y)=\sin(kx)\sinh(ky)}
$$
is nonzero and <harmonic>. It vanishes on $y=0$ and $x=0$, while $u_x(1,y)=k\cos(k)\sinh(ky)=0$.
A sufficient additional condition is \b[$u(x,1)=0$] on the top edge. <Green's first identity> then gives
$$
\int_D|\nabla u|^2\,dx\,dy=\int_{\partial D}u\,\partial_nu\,ds-\int_Du\Delta u\,dx\,dy=0.
$$
On the bottom, left and top edges $u=0$; on the right edge the normal derivative is zero. Thus the <gradient> vanishes and $u$ is constant; the bottom value makes that constant zero. For $C^2(D)\cap C^1(\overline D)$, apply the identity on inset rectangles and pass to the boundary using <continuity> of the function and its <gradient>.
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