On double-covering stationary points of a constrained Dirichlet energy
type: Item
instance of: scholarly article
title: On double-covering stationary points of a constrained Dirichlet energy
zbMATH Open document ID: 1311.49009
published in: Annales de l'Institut Henri Poincaré. Analyse Non Linéaire
publication date: 2014-09-24T00:00:00Z
review text: 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 \(\mathcal{A}\).
reviewed by: Igor Bock
Mathematics Subject Classification ID: 49J20
Mathematics Subject Classification ID: 49N60
zbMATH DE Number: 6347282
zbMATH Keywords: Dirichlet integral
zbMATH Keywords: minimizer
zbMATH Keywords: double-covering map
author: Jonathan J. Bevan
MaRDI profile type: Publication
full work available at URL: j.anihpc.2013.04.001
OpenAlex ID: W2066199079
cites work: Convexity conditions and existence theorems in nonlinear elasticity
cites work: Discontinuous equilibrium solutions and cavitation in nonlinear elasticity
cites work: Some Open Problems in Elasticity
cites work: Maximal smoothness of solutions to certain Euler–Lagrange equations from nonlinear elasticity
cites work: Minimizers with topological singularities in two dimensional elasticity
cites work: Compensated compactness and Hardy spaces
cites work: Direct methods in the calculus of variations
cites work: Weak local minimizers in finite elasticity
cites work: Q3134551
cites work: On the partial regularity of energy-minimizing, area-preserving maps
cites work: \(H^p\) spaces of several variables
cites work: Q4838489
cites work: Sufficient conditions for regularity of area-preserving deformations
cites work: Explicit energy-minimizers of incompressible elastic brittle bars under uniaxial extension
cites work: Non-uniqueness of minimizers for strictly polyconvex functionals
cites work: Q3550716
cites work: On the symmetry of energy-minimising deformations in nonlinear elasticity. I: Incompressible materials
cites work: On the symmetry of energy-minimising deformations in nonlinear elasticity. II: Compressible materials
cites work: On the global stability of two-dimensional, incompressible, elastic bars in uniaxial extension
DOI: 10.1016/J.ANIHPC.2013.04.001
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description: scientific article; zbMATH DE number 6347282