Laplacian Growth, Elliptic Growth, and Singularities of the Schwarz Potential

TL;DR

This paper explores the properties of Schwarz functions and potentials in Laplacian and elliptic growth problems, providing new solutions, inverse problem techniques, and extending the theory to higher dimensions.

Contribution

It generalizes the Schwarz potential to higher dimensions and elliptic growth, introduces methods for locating singularities, and constructs new exact solutions.

Findings

Stationary non-physical singularities in the Schwarz function

Generalization of Schwarz potential to elliptic growth

Techniques for locating singularities in higher dimensions

Abstract

The Schwarz function has played an elegant role in understanding and in generating new examples of exact solutions to the Laplacian growth (or "Hele- Shaw") problem in the plane. The guiding principle in this connection is the fact that "non-physical" singularities in the "oil domain" of the Schwarz function are stationary, and the "physical" singularities obey simple dynamics. We give an elementary proof that the same holds in any number of dimensions for the Schwarz potential, introduced by D. Khavinson and H. S. Shapiro [17] (1989). A generalization is also given for the so-called "elliptic growth" problem by defining a generalized Schwarz potential. New exact solutions are constructed, and we solve inverse problems of describing the driving singularities of a given flow. We demonstrate, by example, how \mathbb{C}^n - techniques can be used to locate the singularity set of the Schwarz potential. One of our methods is to prolong available local extension theorems by constructing "globalizing families". We make three conjectures in potential theory relating to our investigation.