Harmonic maps with fixed singular sets
type: Item
instance of: scholarly article
title: Harmonic maps with fixed singular sets
zbMATH Open document ID: 0777.58013
DOI: 10.1007/BF02921301
author: Libin Mou
author: Robert Hardt
published in: The Journal of Geometric Analysis
publication date: 1993-02-10T00:00:00Z
review text: Let \(\Omega\) be a smooth domain in \(\mathbb{R}^ m\) and \(N\) a compact smooth Riemannian manifold. The authors study the space \({\mathcal H}\) consisting of harmonic maps \(u:\Omega\to N\) that have as singular set a fixed compact subset \(Z\) of \(\Omega\), having finite \((m-3)\)-dimensional Minkowski content. The topology on \({\mathcal H}\) is defined by using Hölder norms on derivatives weighted by powers of distances to \(Z\). The authors study the structure of \({\mathcal H}\) and other properties such as stability and minimality under perturbations. For smooth harmonic maps they prove that \({\mathcal H}\) is a Banach manifold locally diffeomorphic to a submanifold of the product of the boundary data space with a finite dimensional space of Jacobi fields with controlled singular behavior.
Mathematics Subject Classification ID: 58E20
Mathematics Subject Classification ID: 58D15
zbMATH DE Number: 118433
zbMATH Keywords: harmonic maps
zbMATH Keywords: stability
zbMATH Keywords: minimality under perturbations
reviewed by: Vasile Oproiu
MaRDI profile type: Publication
cites work: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II
cites work: Stability and uniqueness properties of the equator map from a ball into an ellipsoid
cites work: Harmonic maps with defects
cites work: Uniqueness in the Cauchy Problem for Partial Differential Equations
cites work: Minimal surfaces with isolated singularities
cites work: Interior estimates for elliptic systems of partial differential equations
cites work: Another Report on Harmonic Maps
cites work: Q5565773
cites work: Q5187597
cites work: Existence and partial regularity of static liquid crystal configurations
cites work: Mappings minimizing theLp norm of the gradient
cites work: Stability of singularities of minimizing harmonic maps
cites work: Area minimizing hypersurfaces with isolated singularities.
cites work: Q3702985
cites work: Q3663717
cites work: Q3330158
cites work: Q3286812
cites work: Q3797914
cites work: Perturbing away singularities of harmonic maps
cites work: Almost every curve in \(R^3\) bounds a unique area minimizing surface
cites work: Measures on spaces of surfaces
cites work: Q3857403
cites work: Q5518064
cites work: Harmonic Maps with Prescribed Finite Singularities
cites work: Q3361581
cites work: A regularity theory for harmonic maps
cites work: Q3706078
cites work: Minimal hypersurfaces with many isolated singularities
cites work: Q3981476
cites work: The Second Variation Formula for Harmonic Mappings
cites work: Q3900552
cites work: Q5289031
full work available at URL: https://doi.org/10.1007/bf02921301
OpenAlex ID: W2003115698
description: scientific article; zbMATH DE number 118433