The maximum principle and the Dirichlet problem for Dirac-harmonic maps
type: Item
instance of: scholarly article
title: The maximum principle and the Dirichlet problem for Dirac-harmonic maps
zbMATH Open document ID: 1273.58004
author: Qun Chen
author: Guofang Wang
author: Jürgen Jost
published in: Calculus of Variations and Partial Differential Equations
publication date: 2013-05-07T00:00:00Z
review text: Dirac-harmonic maps are the critical points of a functional that couples the nonlinear sigma model with a spinor field. The functional is motivated by the supersymmetric version of the sigma model. Dirac-harmonic maps have been introduced in [\textit{Q. Chen} et al., Math. Z. 254, No. 2, 409--432 (2006; Zbl 1103.53033)]. While issues like regularity, blow-up analysis and even some partial classification have been studied before, the general existence problem has no satisfying solution yet. A step towards that is made in the current paper, which is concerned with Dirac-harmonic maps from a Riemannian spin manifold with boundary into a regular ball in any Riemannian manifold. The restriction to regular balls allows to generalize a maximum principle for harmonic maps by \textit{W. Jäger} and \textit{H. Kaul} [Manuscr. Math. 28, 269--291 (1979; Zbl 0413.31006)] to the Dirac-harmonic case. The maximum principle helps to derive a priori estimates, which in turn give rise to an existence and uniqueness theorem for Dirac-harmonic maps with small boundary data.
reviewed by: Andreas Gastel
Mathematics Subject Classification ID: 58E20
Mathematics Subject Classification ID: 53C27
zbMATH DE Number: 6161257
zbMATH Keywords: Dirac-harmonic map
zbMATH Keywords: maximum principle
zbMATH Keywords: uniqueness
zbMATH Keywords: existence
zbMATH Keywords: Riemannian spin manifold with boundary
zbMATH Keywords: regular ball
MaRDI profile type: Publication
full work available at URL: s00526-012-0512-5
OpenAlex ID: W1999575410
cites work: Spectral asymmetry and Riemannian Geometry. I
cites work: Spectral asymmetry and Riemannian geometry. II
cites work: Spectral asymmetry and Riemannian geometry. III
cites work: Extrinsic bounds for eigenvalues of the Dirac operator
cites work: Q4274286
cites work: Q3485543
cites work: Q4934552
cites work: Dirac-harmonic maps
cites work: Regularity theorems and energy identities for Dirac-harmonic maps
cites work: Energy identities for Dirac-harmonic maps
cites work: Q5630031
cites work: On the spectrum of the Dirac operator under boundary conditions
cites work: Q5187597
cites work: Eigenvalue boundary problems for the Dirac operator
cites work: Harmonic mappings and minimal submanifolds
cites work: An existence theorem for harmonic mappings of Riemannian manifolds
cites work: Uniqueness of harmonic mappings and of solutions of elliptic equations on Riemannian manifolds
cites work: Uniqueness and stability of harmonic maps and their Jacobi fields
cites work: Q3330158
cites work: Riemannian geometry and geometric analysis
cites work: Geometry and Physics
cites work: Q3998311
cites work: Existence proofs for harmonic mappings with the help of a maximum principle
cites work: Some explicit constructions of Dirac-harmonic maps
cites work: Riemannian center of mass and mollifier smoothing
cites work: Linear and quasilinear elliptic equations
cites work: Hodge decomposition. A method for solving boundary value problems
cites work: Q4847232
cites work: The existence of minimal immersions of 2-spheres
cites work: Q4192407
cites work: Regularity of Dirac-Harmonic Maps
cites work: A structure theorem of Dirac-harmonic maps between spheres
cites work: Regularity for weakly Dirac-harmonic maps to hypersurfaces
DOI: 10.1007/S00526-012-0512-5
Recommended article: Q4293600
Recommended article: Maximum principles, uniqueness and existence for harmonic maps with potential and Landau-Lifshitz equations
Recommended article: A maximum principle for 𝑃-harmonic maps with 𝐿^{𝑞} finite energy
Recommended article: The boundary value problem for Dirac-harmonic maps
Recommended article: A maximum principle for generalizations of harmonic maps in Hermitian, affine, Weyl, and Finsler geometry
Recommended article: Q4806088
Recommended article: Q3763282
Recommended article: Q3070754
Recommended article: Q4516325
Recommended article: On a strong maximum principle for Dirichlet forms
description: scientific article; zbMATH DE number 6161257