 Solution

Let $u_1,u_2$ be two solutions with the same data, and write $w=u_1-u_2$ on the rectangle $\Omega =(0,\pi )\times (-1,1)$. By linearity,The normal derivative is $-w_x$ on the left edge and $w_x$ on the right edge, so it vanishes on both. The energy method applies Green's first identity, which follows by integratingIt givesThe boundary term vanishes on each horizontal edge because $w=0$, and on each vertical edge because ${\partial }_{n}w=0$. The interior term vanishes because $w$ is a harmonic function. Since the continuous integrand $|\nabla w{|}^2$ is nonnegative, a zero integral forces $\nabla w=0$. The rectangle is connected, so $w$ is constant; its zero value on either horizontal edge makes that constant zero. HenceThis proof assumes enough boundary regularity for Green's first identity. It also works in the finite Dirichlet energy class using the weak identity with test function $w$, so it remains applicable when corner derivatives are not continuous. It proves uniqueness conditional on existence and does not impose a compatibility condition for arbitrary $f,g$.