$L^p$ continuity of eigenprojections for 2-d Dirichlet Laplacians under perturbations of the domain

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 $(-{\mathrm{\Delta }}^{-1}-z{)}^{-1}$ is $L^p$ bounded when $z$ lies outside of the spectrum of $-{\mathrm{\Delta }}^{-1}$. We show that this assumption is met if the initial domain is a square or a rectangle.