# Singular Behavior of Harmonic Maps Near Corners

## TL;DR

This paper investigates the singular behavior of harmonic maps near corner vertices, revealing a discontinuous relationship between boundary angles that contrasts with the linear behavior seen in conformal maps.

## Contribution

It provides a detailed analysis of how harmonic maps behave near corners, showing a discontinuous dependence of boundary angles, which is a novel insight compared to conformal maps.

## Key findings

- Dependence of boundary angle on corner geometry is discontinuous for harmonic maps.
- Harmonic maps differ qualitatively from conformal maps near corners.
- Behavior of inverse harmonic maps near preimages of vertices is characterized.

## Abstract

For a harmonic map $\mathcal{F}:\mathcal{Z} {\buildrel {\,harm\,} \over\longrightarrow} \mathcal{W}$ transforming the contour of a corner of the boundary $\partial\mathcal{Z}$ into a rectilinear segment of the boundary $\partial\mathcal{W}$, the behavior near the vertex of the specified corner is investigated. The behavior of the inverse map $\mathcal{F}^{-1}:\mathcal{W} \longrightarrow \mathcal{Z}$ near the preimage of the vertex is investigated as well. In particular, we prove that if $\varphi$ is the value of the exit angle from the vertex of the reentrant corner for a smooth curve $\mathcal{L}$ and $\theta$ is the value of the exit angle from the vertex image for the image $\mathcal{F} (\mathcal{L})$ of the specified curve, then the dependence of $\theta$ on $\varphi$ is described by a discontinuous function.Thus, such a behavior of the harmonic map qualitatively differs from the behavior of the corresponding conformal map: for the latter one, the dependence $\theta (\varphi)$ is described by a linear function.