From reflections to elliptic growth

TL;DR

This paper develops a reflection-based method to solve Cauchy problems for harmonic functions and applies it to Hele-Shaw and elliptic growth problems, extending to Helmholtz equations.

Contribution

It introduces a novel reflection technique for harmonic and Helmholtz equations, enabling new solutions to Hele-Shaw and elliptic growth problems.

Findings

Derived explicit solution formulas for harmonic functions using reflections.

Extended the method to Helmholtz equations for elliptic growth.

Provided new insights into solving free boundary problems in fluid dynamics.

Abstract

We use reflections involving analytic Dirichlet and Neumann data on a real-analytic curve in order to find a representation of solutions to Cauchy problems for harmonic functions in the plane. We apply this representation for finding solutions to Hele-Shaw problems. We also generalize the results by deriving the corresponding formulae for Helmholtz equation and applying them to elliptic growth.