= Solution
For $u^{(m,n)}(x,y)=\sin(m\pi x)\sin(n\pi y)$, direct differentiation yields
$$
\partial_{xx}u^{(m,n)}=-(m\pi)^2u^{(m,n)},\qquad
\partial_{yy}u^{(m,n)}=-(n\pi)^2u^{(m,n)}.
$$
Since the planar <Laplacian> is $\Delta=\partial_{xx}+\partial_{yy}$, the required <eigenvalue> of $-\Delta+I$ is
$$
\boxed{\lambda_{m,n}=1+\pi^2(m^2+n^2)>0.}
$$
On the sides $x=0,1$ the first <sine> factor vanishes, and on $y=0,1$ the second <sine> factor vanishes. Thus the homogeneous <Dirichlet boundary condition> holds on the whole boundary. The <eigenfunction> is nonzero, for instance at $(x,y)=(1/(2m),1/(2n))$, where its value is one.
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