Analysis and Probability seminar

Abstract: Consider the Laplacian on a bounded domain in Rⁿ. Under suitable conditions, this Laplacian will have a complete set of eigenfunctions in Lp, and we can consider projections onto linear spaces spanned by some collections of these eigenfunctions. We are interested in the behaviour of these projections as we deform the domain in Rⁿ. Our main result is the Lipschitz continuity in the Lp operator norm of these projection operators, generalising existing results for H¹₀ by Lamberti and Lanza de Cristoforis. A key tool needed in our setting is Lp bounds on resolvents of the inverse Laplacian, which are also new, to the best of our knowledge.

This is joint work with James C. Robinson (Warwick).