LEAPS-MPS: The computation of special functions in potential theory in multiply connected domains

Potential theory is the study of functions satisfying Laplace’s equation, an elliptic partial differential equation of widespread scientific importance. A deep understanding of these functions provides valuable insight into many physical processes. Potential theory is very well-understood in simply connected settings, such as in the unit disk or the upper half-plane, and there are many exact solutions of models describing physical phenomena in these classical geometries. However, potential theory is arguably far less well-understood in more complicated multiply connected geometries, i.e. regions with ‘holes’; indeed, many kinds of geometries arising in practical problems involving biological and engineering structures consist of ‘holey’ regions. Therefore, we need to continue to develop new mathematical tools and new mathematical theory to understand how classical exact solutions can be generalized to describe physical phenomena in domains arising in modern practical problems. To this end, this project will enhance our understanding of fundamental potential theory through the computation of several important special functions in multiply connected regions. Broader impacts will include the establishment of a USA branch of the Applied & Computational Complex Analysis Network, and participation in an Industrial Math Clinic which is designed to initiate collaborative projects between teams of undergraduate students and industry on problems which require mathematical expertise.

The main objective of the project is to compute a variety of important special functions arising in potential theory in multiply connected domains, i.e. domains with more than one boundary component. Many functions in potential theory are well-understood in simply connected settings, but far less so in multiply connected settings. The Schottky-Klein prime function is a natural candidate to consider working with in multiply connected domains, and its calculus will underpin much of the proposed work. The principal investigator will devise an optimum numerical scheme for the computation of the prime function in highly multiply connected domains. The PI will then use this numerical scheme when implementing models of schools of swimming fish using potential flow functions combined with the calculus of the prime function. The so-called harmonic-measure distribution function or h-function has a deep connection with planar Brownian motion. The PI will construct analytical formulae for these h-functions, and related functions called g-functions, in several multiply connected domains using the calculus of the prime function. The PI will then extend these computations of h-functions and g-functions onto compact Riemann surfaces such as the sphere and the ring torus.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

Award Number: 2532158
Principal Investigator: Christopher Green
Funds Obligated: $247,390
State: KS