  We generalise results by Lamberti and Lanza de Cristoforis (2005) concerning
the continuity of projections onto eigenspaces of self-adjoint differential
operators with compact inverses as the (spatial) domain of the functions is
perturbed in $\mathbb{R}^2$. Our main case of interest is the Dirichlet
Laplacian. We extend these results from bounds from $H_0^1$ to $H_0^1$ to
bounds from $L^p$ to $L^p$, under the assumption that $(-\Delta^{-1}-z)^{-1}$
is $L^p$ bounded when $z$ lies outside of the spectrum of $-\Delta^{-1}$. We
show that this assumption is met if the initial domain is a square or a
rectangle.
