Singular behavior of harmonic maps near corners

Abstract

For a harmonic map F : Z harm −→ W transforming the contour of a corner of the boundary ∂Z into a rectilinear segment of the boundary ∂W, the behavior near the vertex of the specified corner is investigated. The behavior of the inverse map F −1 : W −→ 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 L and θ is the value of the exit angle from the vertex image for the image F(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.