Non-Convex Planar Harmonic Maps

by   Shahar Z. Kovalsky, et al.

We formulate a novel characterization of a family of invertible maps between two-dimensional domains. Our work follows two classic results: The Radó-Kneser-Choquet (RKC) theorem, which establishes the invertibility of harmonic maps into a convex planer domain; and Tutte's embedding theorem for planar graphs - RKC's discrete counterpart - which proves the invertibility of piecewise linear maps of triangulated domains satisfying a discrete-harmonic principle, into a convex planar polygon. In both theorems, the convexity of the target domain is essential for ensuring invertibility. We extend these characterizations, in both the continuous and discrete cases, by replacing convexity with a less restrictive condition. In the continuous case, Alessandrini and Nesi provide a characterization of invertible harmonic maps into non-convex domains with a smooth boundary by adding additional conditions on orientation preservation along the boundary. We extend their results by defining a condition on the normal derivatives along the boundary, which we call the cone condition; this condition is tractable and geometrically intuitive, encoding a weak notion of local invertibility. The cone condition enables us to extend Alessandrini and Nesi to the case of harmonic maps into non-convex domains with a piecewise-smooth boundary. In the discrete case, we use an analog of the cone condition to characterize invertible discrete-harmonic piecewise-linear maps of triangulations. This gives an analog of our continuous results and characterizes invertible discrete-harmonic maps in terms of the orientation of triangles incident on the boundary.


page 2

page 4

page 5

page 7

page 8

page 17


The Harmonic GBC Function Map is a Bijection if the Target Domain is Convex

Harmonic generalized barycentric coordinates (GBC) functions have been u...

PDE-Based Parameterisation Techniques for Planar Multipatch Domains

This paper presents a PDE-based parameterisation framework for addressin...

An L^p-comparison, p∈ (1,∞), on the finite differences of a discrete harmonic function at the boundary of a discrete box

It is well-known that for a harmonic function u defined on the unit ball...

Weak convergence of Monge-Ampere measures for discrete convex mesh functions

For mesh functions which satisfy a convexity condition at the discrete l...

Methods and applications of PDMP samplers with boundary conditions

We extend Monte Carlo samplers based on piecewise deterministic Markov p...

Inverse Scale Space Iterations for Non-Convex Variational Problems: The Continuous and Discrete Case

Non-linear filtering approaches allow to obtain decompositions of images...

A Liouville principle for the random conductance model under degenerate conditions

We consider a random conductance model on the d-dimensional lattice, d∈[...

Please sign up or login with your details

Forgot password? Click here to reset