Conformal Mapping and Invariants – Motivating Numerical PDE Approaches

Overview

The study of conformal mappings and related invariants is a classical topic. Typically, the quantity of interest has been some scalar quantity such as the capacity or modulus of a planar domain. There is a direct connection to PDEs since the standard quantities of interest can be defined as the Dirichlet energy or H1-seminorm of a Laplacian driven by non-homogeneous boundary conditions. The invariants place only very weak restrictions on the types of domains leading to problems with complex geometries where the exact solutions are available at least indirectly. Moreover, new kind of geometric optimization problems can be defined based on the key observation that both the conformal capacity and hyperbolic length are invariant under conformal automorphisms of the unit disk. One model problem is the question: How should a herd of animals group together to minimize the heat loss, i.e., capacity? Alternatively, given some distance constraint, how to maximize the loss. All examples are computed either with hp-FEM or BIE using the PDE approach.