Singular Behavior of Harmonic Maps Near Corners

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 $φ$ is the value of the exit angle from the vertex of the reentrant corner for a smooth curve $\mathcal{L}$ and $θ$ 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 $θ$ on $φ$ 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 $θ(φ)$ is described by a linear function.