= Solution
{c}
Let $w$ be the difference of two solutions. It is a <harmonic function> in the open <rectangle>, continuous on its <closure>, with
$$
w(x,0)=0,\qquad w_y(x,\pi)=0,\qquad
w_x(0,y)=w_x(\pi,y)=0.
$$
Interpret these <Neumann boundary conditions> classically on the open edges, with the corresponding first <derivatives> extending there. We use the <strong maximum principle for harmonic functions> together with <even reflection across a homogeneous Neumann boundary>. This avoids imposing differentiability at the two bottom corners, where the original data fail a <corner compatibility condition for mixed boundary data>.
By continuity on the compact <rectangle>, $w$ attains a maximum $M$. Suppose $M>0$. It cannot occur on the bottom edge, where $w=0$. If it occurs in the interior, the <strong maximum principle for harmonic functions> makes $w$ constant, contradicting its bottom value. If it occurs on an open vertical or top edge, the zero <normal derivative> allows an even <reflection> across that edge. The reflected <harmonic function> then has an interior maximum, giving the same contradiction.
The only remaining possibilities are the two top corners. Reflect evenly across both meeting edges. This gives a bounded <harmonic function> on a punctured neighbourhood of the corner. The <removable singularity for a bounded harmonic function> fills the centre with its continuous value. The corner is now an interior maximum, again impossible. Therefore $w\leq0$. Applying the identical argument to $-w$ gives $w\geq0$.
Thus $\boxed{w=0}$, and \b[the solution is unique] among the continuous classical solutions with the stated edge conditions.
Back to article page
