On mappings of strips connected with harmonic functions (Q1324864)

A mapping \(\mathbb{R}^ 3_{(x,y,z)} \to \mathbb{R}^ 3_{(u,v,w)}\) is called a harmonic one in the Lavrent'ev sense if \(\text{grad} u = [\text{grad} v, \text{grad} w]\). A mapping \(\mathbb{R}^ 3_{(x,y,z)} \to \mathbb{R}^ 3_{(u,v,w)}\) is called a quasiharmonic one if \(\text{grad} u = f(u,v) [\text{grad} v, \text{grad} w]\). Let \(\nu (x,y)\) be a bounded, positive, smooth function in \(\mathbb{R}^ n\) and let \({\mathcal D} = \{(x,y,z) : 0 < z < \nu (x,y)\), \((x,y) \in \mathbb{R}^ 2\}\). The author investigates quasiharmonic and Lavrent'ev-sense harmonic mappings of \({\mathcal D}\) with the given function \(u(x,y)\) which is a harmonic one in \({\mathcal D}\) and satisfies the conditions \(u |_{z = 0} = 0\), \(u |_{z = \nu (x,y)} = 1\). The mappings of the domain \(E = \{(x,y,z) : 0 < z < \nu (x,y)\), \((x,y) \in G\}\) with a given function \(w(x,y,z)\), which is a harmonic function satisfying the conditions \(w |_{z = 0} = 0\), \(w |_{z = \nu (x,y,)} = 1\), \({\partial w \over \partial n} |_{(x,y) \in \partial G} = 0\) are investigated too.