Energy identity for intrinsically biharmonic maps in four dimensions

Hornung, Peter; Moser, Roger

doi:10.2140/apde.2012.5.61

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Energy identity for intrinsically biharmonic maps in four dimensions

Peter Hornung and Roger Moser

Vol. 5 (2012), No. 1, 61–80

DOI: 10.2140/apde.2012.5.61

Abstract

Let $u$ be a mapping from a bounded domain $S \subset ℝ^{4}$ into a compact Riemannian manifold $N$ . Its intrinsic biharmonic energy $E_{2} (u)$ is given by the squared $L^{2}$ -norm of the intrinsic Hessian of $u$ . We consider weakly converging sequences of critical points of $E_{2}$ . Our main result is that the energy dissipation along such a sequence is fully due to energy concentration on a finite set and that the dissipated energy equals a sum over the energies of finitely many entire critical points of $E_{2}$ .

Keywords

biharmonic map, energy identity, bubbling

Mathematical Subject Classification 2000

Primary: 58E20, 35J35

Milestones

Received: 4 November 2009

Revised: 24 November 2010

Accepted: 25 January 2011

Published: 25 June 2012

Authors

Peter Hornung
Max-Planck-Institut für Mathematik in den Naturwissenschaften Inselstraße. 22 04103 Leipzig Germany

Roger Moser
Department of Mathematical Sciences University of Bath Bath BA2 7AY United Kingdom

Vol. 5, No. 1, 2012