Ryan Acosta Babb: Lp continuity of spectral projections for the Laplacian on Euclidean domains

We study the Laplacian on a bounded domain in \(\mathbb{R}^n\). Under appropriate assumptions, the Laplacian admits a complete system of eigenfunctions in \(L^p\), allowing us to define projections onto linear subspaces generated by selected families of these eigenfunctions. Our focus is on how these projection operators behave when the underlying domain in \(\mathbb{R}^n\) is deformed by a diffeomorphism. The main result establishes Lipschitz continuity of the projections in the \(L^p\) operator norm, extending earlier results in \(H_0^1\) due to Lamberti and Lanza de Cristoforis. A crucial ingredient of our analysis is the derivation of \(L^p\) bounds for resolvents of the inverse Laplacian which are also new, to the best of our knowledge. This is joint work with James C. Robinson (Warwick).