Planar elliptic growth

TL;DR

This paper explores the mathematical structure of planar elliptic growth, extending Laplacian growth to elliptic operators, analyzing conservation laws, and presenting exact solutions within a potential theory framework.

Contribution

It introduces a novel elliptic growth model, interprets conservation laws via potential theory, and provides explicit solutions and analysis of well-posedness.

Findings

Conservation laws are linked to potential theory.

Exact solutions of elliptic growth are constructed.

The relation between different forms of elliptic growth is clarified.

Abstract

The planar elliptic extension of the Laplacian growth is, after a proper parametrization, given in a form of a solution to the equation for area-preserving diffeomorphisms. The infinite set of conservation laws associated with such elliptic growth is interpreted in terms of potential theory, and the relations between two major forms of the elliptic growth are analyzed. The constants of integration for closed form solutions are identified as the singularities of the Schwarz function, which are located both inside and outside the moving contour. Well-posedness of the recovery of the elliptic operator governing the process from the continuum of interfaces parametrized by time is addressed and two examples of exact solutions of elliptic growth are presented.