Regularity of polyharmonic maps in the critical dimension
type: Item
instance of: scholarly article
title: Regularity of polyharmonic maps in the critical dimension
zbMATH Open document ID: 1183.58010
author: Andreas Gastel
author: Christoph Scheven
published in: Communications in Analysis and Geometry
publication date: 2009-09-22T00:00:00Z
review text: The authors consider the intrinsic and extrinsic \(m\)-polyenergy functional on a domain \(\Omega\) of \(\mathbb{R}^{n}\), \(n\geq 2m\), action on functions \(u:\Omega\to N\), \(u\in W^{m,2}(\Omega,N)\), where \(N\) is a Riemannian manifold immersed in \(\mathbb{R}^K\) \[ E_m(u)=\frac{1}{2}\int_{\Omega}|D^mu|^2dx \quad \quad \tilde{E}_m(u)=\frac{1}{2}\int_{\Omega}|\nabla^{m-1}Du|^2dx, \] and study the regularity of weakly extrinsic and intrinsic \(m\)-ployharmonic maps, resp., that is, of the critical points of \(E_m\) and of \(\tilde{E}_m\), resp., for all variations \(u_t=\pi_N(u+tW)\), with \(\pi_N\) the nearest-point retraction onto \(N\). Their first main theorem states that if \(n=2m\) and \(N\) is compact, such critical points are smooth, generalizing a result of \textit{C.\ Wang} [Math. Z. 247, No.~1, 65--87 (2004; Zbl 1064.58016)] for the case \(m=2\). For the case \(n\geq 2m\), the authors establish a second main theorem that obtains smoothness of a weakly extrinsic or intrinsic \(m\)-polyharmonic map that is locally Hölder continuous. The proof of the first theorem consists in showing that \(u\) is Hölder continuous for the case of critical dimension \(n=2m\). To prove their results they use Lorentz spaces \(L^{p,q}(\Omega)\), and recall Sobolev-Lorentz embedding theorems, Poincaré-type inequalities, and derive a fundamental elliptic inequality for the \(L^{p,q}(\Omega)\)-norm of the Hessian of \(f\) on a ball of radius \(r<1\) in terms of the norms of \(\Delta f\), \(Df\) and \(f\) on the unit ball. By interpolation methods and using Hodge decompositions, the norm of higher order derivatives of \(f\) is estimated. Reformulating an orthogonal condition that extrinsically polyharmonic maps satisfy, the authors derive a new form of the Euler-Lagrangian equation, and using a Uhlenbeck's gauge theorem and arguments due to Schoen an Uhlenbeck, leads to the construction of Coulomb frames satisfying appropriate estimates, and consequent decay and growth estimate…
reviewed by: Isabel Salavessa
Mathematics Subject Classification ID: 58E20
Mathematics Subject Classification ID: 35G20
zbMATH DE Number: 5606319
zbMATH Keywords: polyharmonic maps
zbMATH Keywords: regularity theorem
zbMATH Keywords: higher order nonlinear systems
MaRDI profile type: Publication
DOI: 10.4310/CAG.2009.V17.N2.A2
description: scientific article; zbMATH DE number 5606319