Quadrature domains for the Helmholtz equation

In this talk, which is based on joint work with Stephen J. Gardiner (UCD Dublin), I will present some recent progress on the existence of quadrature domains with respect to the Helmholtz operator 𝛥+𝑘² for 𝑘>0. The study of classical quadrature domains, i.e. when 𝑘=0, can be motivated in several ways, one being its connection to so called Hele-Shaw flow and flows in porous media. For 𝑘>0 they turn out to come up naturally in connection with scattering theory, in the way that they turn out to be so called non-scattering domains. We will give a brief introduction to the concepts of quadrature domains, discuss what is similar and what is different for the classical case 𝑘=0 and the case 𝑘>0, and as far as time permits try to give some details on how some of the results are proved.