The Unified Transform Method: beyond circular or convex domains

TL;DR

This paper introduces a novel transform-based method for solving Laplace's equation in complex planar domains, extending previous techniques to non-convex and Lipschitz domains using properties of the Szegő kernel.

Contribution

The work develops a new transform approach leveraging the Szegő kernel to address boundary value problems in non-convex and Lipschitz domains, expanding the applicability of transform methods.

Findings

Successfully solves boundary value problems in complex domains

Demonstrates numerical implementation of the new transform pairs

Extends transform techniques beyond circular and convex domains

Abstract

A new transform-based approach is presented that can be used to solve mixed boundary value problems for Laplace's equation in non-convex and other planar domains, specifically the so-called Lipschitz domains. This work complements Crowdy (2015, CMFT, 15, 655--687), where new transform-based techniques were developed for boundary value problems for Laplace's equation in circular domains. The key ingredient of the present method is the exploitation of the properties of the Szeg\H{o} kernel and its connection with the Cauchy kernel to obtain transform pairs for analytic functions in such domains. Several examples are solved in detail and are numerically implemented to illustrate the application of the new transform pairs.