Singular Behavior of Harmonic Maps Near Corners
| Autor: | Bezrodnykh, S. I., Vlasov, V. I. |
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| Rok vydání: | 2018 |
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| Druh dokumentu: | Working Paper |
| DOI: | 10.1080/17476933.2018.1536705 |
| Popis: | 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. Comment: This manusctript is accepted for publication in Journal "Complex Variables and Elliptic Equations" |
| Databáze: | arXiv |
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