A regularity result for polyharmonic maps with higher integrability
type: Item
instance of: scholarly article
title: A regularity result for polyharmonic maps with higher integrability
zbMATH Open document ID: 1172.58003
author: Gilles Angelsberg
author: David Pumberger
published in: Annals of Global Analysis and Geometry
publication date: 2009-07-13T00:00:00Z
full work available at URL: http://hdl.handle.net/20.500.11850/3959
review text: (Extrinsically) \(k\)-polyharmonic maps \(u:\Omega\to N\) from a domain \(\Omega\) in \(\mathbb R^m\) to a compact Riemannian submanifold \(N\subseteq\mathbb R^n\) are the critical points of the higher order energy \(E^k(u):=\int_\Omega|D^ku|^2\,dx\). The natural space in which the direct method gives existence minimizers of \(E^k\) (which are weakly polyharmonic) is the Sobolev space \(W^{k,2}(\Omega,N)\). In the paper, the authors prove some regularity result for weakly polyharmonic maps which holds under two assumptions. The more natural assumption is that of smallness in some presumably natural Morrey spaces. For harmonic and biharmonic maps (i.e., \(k\in\{1,2\}\)), a monotonicity formula shows that stationary weak solutions are in these Morrey spaces, and examples show that without this smallness assumption there can be singularities. However, for \(k\geq3\), the monotonicity formula is still an open problem. The less natural assumption is \(u\in W^{k,2p}\) for some \(p>1\), which does not fit together with the existence results. Since the nonlinearities of the Euler-Lagrange equation are no longer critical in this space (at least under the Morrey space assumption), this second assumption simplifies the regularity proof considerably, the main ingredient being a suitable form of the Gagliardo-Nirenberg inequality. Even though the assumptions therefore must be considered rather strong, it should be noted that by now the paper presents the only available regularity result for polyharmonic maps in the ``supercritical'' case \(m>2k\).
reviewed by: Andreas Gastel
Mathematics Subject Classification ID: 58E20
Mathematics Subject Classification ID: 35J35
zbMATH DE Number: 5577695
zbMATH Keywords: polyharmonic maps
zbMATH Keywords: harmonic maps
zbMATH Keywords: biharmonic maps
zbMATH Keywords: Gagliardo-Nirenberg inequality
zbMATH Keywords: regularity
MaRDI profile type: Publication
OpenAlex ID: W2011935701
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DOI: 10.1007/S10455-008-9122-Z
description: scientific article; zbMATH DE number 5577695