On double-covering stationary points of a constrained Dirichlet energy
The author deals with the problem of minimizing the Dirichlet integral \(I(\mathbf{u})=\int_B|\nabla \mathbf{u}|^2\,d\mathbf{x}\) among mappings belonging to the class \[ \mathcal{A}=\{\mathbf{u}\in W^{1,2}(B,\mathbb{R}^2):\;\mathbf{u}(\mathbf{x})= \mathbf{u}_{\text{dc}}(\mathbf{x})\;\text{if}\,x\in \partial B,\;\text{det}\,\nabla \mathbf{u}=1\,\text{a.e.}\}, \] where \(B\) is a unit ball in \(\mathbb{R}^2\) and \[ \mathbf{u}_{\text{dc}}(\mathbf{x})=\frac 1{\sqrt 2|\mathbf{x}|}\left(\begin{matrix} x_2^2-x_1^2\\ 2x_1x_2\end{matrix}\right) \] is the double-covering map in cartesian coordinates \(\mathbf{x}=(x_1,x_2)^T.\) The main result of the paper is that the function \(\mathbf{u}_{\text{dc}}\) is the global minimizer of the Dirichlet energy \(I(\mathbf{u})\) over \(\mathcal{A}\).
scientific article; zbMATH DE number 6347282
scientific article