Boundary Integral Operators: Theory and Application

Time and place

The workshop will take place on the 27th May on Zoom.

Introduction

The theory of integral operators has a long history in developing tools for the study of partial differential equations, including the use of single- and double-layer potentials to treat classical boundary value problems. Since its first developments in the early part of the twentieth century, the use of boundary integral methods has been key in addressing challenging problems in both pure and applied mathematics. Despite being a classical technique, boundary integral operators have experienced a renewed interest owing to their efficiency in tackling topical problems in applied sciences, including the study of plasmonic resonances in nanophotonics, metamaterials, electro-sensing, to name a few. Overall, boundary integrals comprise a fundamental set of mathematical tools that find applications in the fields of inverse problems, wave propagation, optimal design, elasticity, and fluid dynamics.

This MAC-MIGS research afternoon will provide a self-contained exposition of boundary integral methods; from its foundations to some of its latest developments. The goal is to introduce the members of the MAC-MIGS programme, and other researchers at the Maxwell Institute, to the theory of boundary integrals in both its theoretical and numerical aspects. As such, it is aimed at both pure and applied mathematicians. The session will bring three experts in this discipline who will discuss the foundations of the theory and its use in solving exciting problems in science and engineering.

Timetable

Organiser: Matias Ruiz (Edinburgh)

Abstracts

Hyeonbae Kang:
Title: The Neumann-Poincare operator: Theory and Application

Abstract: The Neumann-Poincare operator is a boundary integral operator arising naturally when solving boundary value problems using layer potentials. Its study goes back to C. Neumann and Poincare as the name of the operator suggests. Lately, interest in this operator has been revived in relation to plasmonics, and there has been significant progress in the spectral theory of the operator in the last two decades or so. In this talk I will review such progress in a colloquial way.

Andreas Rosén:
Title: Dirac integral equations for Maxwell scattering

Abstract: I will present a new competitive integral equation reformulation of the Maxwell transmission problem for time-harmonic fields, piecewise constant material parameters and Lipschitz regular interfaces. The equation is based on the hypercomplex Cauchy singular integral for Dirac fields, and consists of an 8/8 matrix of classical operators of double and single layer type.

We demonstrate theoretically and numerically how to tune the six free parameters to avoid false eigenwavenumbers and false essential spectrum, even for plasmonic scattering and for quasi-static scattering. This is joint work with Johan Helsing and Anders Karlsson, Lund.

Bryn Davies:
Title: Discrete approximations of subwavelength resonator systems

Abstract: We will use the theory developed in the previous lectures to conduct rigorous analyses of subwavelength acoustic scattering problems. Our aim is to use these results to establish the mathematical foundations for the design of subwavelength metamaterials. We will present results that characterise a system’s resonance in terms of the eigenpairs of the generalized capacitance matrix. We will then use this theory to explore applications including the design of cochlea-like metamaterials and enhanced sensors based on active metamaterials with exceptional points.