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.
Categories: math.CV