The spectral Dirichlet-Neumann map for Laplace's equation in a convex polygon

TL;DR

This paper introduces a novel complex-analytic approach to studying the Dirichlet-Neumann map for Laplace's equation in convex polygons, offering new proofs, numerical methods, and well-posed weak formulations.

Contribution

It develops a new complex analytic framework for the Dirichlet-Neumann map, enabling improved proofs, numerical treatments, and the establishment of well-posed weak problems.

Findings

New proofs of classical results using complex analysis

Development of numerical methods for boundary value problems

Establishment of coercivity estimates for weak formulations

Abstract

We provide a new approach to studying the Dirichlet-Neumann map for Laplace's equation on a convex polygon using Fokas' unified method for boundary value problems. By exploiting the complex analytic structure inherent in the unified method, we provide new proofs of classical results using mainly complex analytic techniques. The analysis takes place in a Banach space of complex valued, analytic functions and the methodology is based on classical results from complex analysis. Our approach gives way to new numerical treatments of the underlying boundary value problem and the associated Dirichlet-Neumann map. Using these new results we provide a family of well-posed weak problems associated with the Dirichlet-Neumann map, and prove relevant coercivity estimates so that standard techniques can be applied.