Eigenfunctions for $2$-dimensional tori and for rectangles with Neumann boundary conditions

Consider the eigenfunctions u for a 2D-torus, respectively, the free rectangular membrane, so that $-\Delta u=\lambda u$ on $\mathcal R(c,d)=(0,c)\times(0,d)$ with periodic, respectively, Neumann boundary conditions. In this note we show that if $u>0$ on $\partial\mathcal R(c,d)$ then $u\equiv C>0$ in $\mathcal R(c,d)$.