PDE Seminar Abstracts

On the eigenvalues of the Robin Laplacian with a complex parameter

James Kennedy
University of Lisbon, Portugal
Mon 2nd Sep 2019, 12-1pm, Carslaw Room 829 (AGR)


We are interested in the eigenvalues of the Laplacian on a bounded domain with boundary conditions of the form u ν + αu = 0, where ν is the outer unit normal to the boundary and α should be considered a parameter on which the eigenvalues depend.

For positive α this operator, and in particular its eigenvalues, interpolate in a strong sense between those of the Neumann (α = 0) and Dirichlet (formally α = ) Laplacians. In recent years, however, the case of large negative α has been studied intensively, and in particular the asymptotics of the eigenvalues in the singular limit α - is well understood: there is a sequence of eigenvalues which diverges like - α2, independently of the geometry of the domain, while any non-divergent eigenvalues converge to points in the spectrum of the Dirichlet Laplacian.

Here, after giving a brief overview of what is known for real α, we will present a number of new results for the corresponding problem when α is a (usually large) complex parameter. This is based on ongoing joing work with Sabine Bögli (Imperial College London) and Robin Lang (University of Stuttgart).