A conformal mapping algorithm for the Bernoulli free boundary value problem

International audienceWe propose a new numerical method for the solution of Bernoulli's free boundary valueproblem for harmonic functions in a doubly connected domain $D$ in $\real^2$ where an unknown free boundary $\Gamma_0$ is determined by prescribed Cauchy data on $\Gamma_0$ in addition to a Dirichlet condition on the known boundary $\Gamma_1$.Our main idea is to involve the conformal mapping methodas proposed and analyzed by Akduman, Haddar and Kress~\cite{AkKr,HaKr05}for the solution of a related inverse boundary value problem. For this we interpret the free boundary $\Gamma_0$as the unknown boundary in the inverse problem to construct $\Gamma_0$ from the Dirichlet condition on $\Gamma_0$ and Cauchy data on the known boundary $\Gamma_1$. Our method for the Bernoulli problem iterates on the missing normal derivative on $\Gamma_1$by alternating between the application of the conformal mapping method for the inverse problemand solving a mixed Dirichlet--Neumann boundary value problem in $D$. We present the mathematicalfoundations of our algorithm and prove a convergence result. Some numerical examples will serve as proof of concept of our approach

From rubber bands to rational maps: A research report

This research report outlines work, partially joint with Jeremy Kahn and Kevin Pilgrim, which gives parallel theories of elastic graphs and conformal surfaces with boundary. One one hand, this lets us tell when one rubber band network is looser than another, and on the other hand tell when one conformal surface embeds in another. We apply this to give a new characterization of hyperbolic critically finite rational maps among branched self-coverings of the sphere, by a positive criterion: a branched covering is equivalent to a hyperbolic rational map if and only if there is an elastic graph with a particular "self-embedding" property. This complements the earlier negative criterion of W. Thurston.Comment: 52 pages, numerous figures. v2: New example