Vol. 4, No. 1, 2011

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ISSN: 1948-206X (e-only)
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Three-term commutator estimates and the regularity of $\half$-harmonic maps into spheres

Francesca Da Lio and Tristan Rivière

Vol. 4 (2011), No. 1, 149–190
Abstract

We prove the regularity of weak 1 2-harmonic maps from the real line into a sphere. A key step is the formulation of the 1 2-harmonic map equation in the form of a nonlocal linear Schrödinger type equation with three-term commutators on the right-hand side. We then establish a sharp estimate for these three-term commutators.

Keywords
harmonic map, nonlinear elliptic PDE, regularity of solutions, commutator estimates
Mathematical Subject Classification 2000
Primary: 58E20, 35B65, 35J20, 35J60, 35S99
Milestones
Received: 23 July 2009
Accepted: 10 March 2010
Published: 27 March 2011
Authors
Francesca Da Lio
Dipartimento di Matematica Pura ed Applicata
Università degli Studi di Padova
Via Trieste 63
35121 Padova
Italy
Department of Mathematics
ETH Zürich
Rämistrasse 101
8092 Zürich
Switzerland
Tristan Rivière
Department of Mathematics
ETH Zürich
Rämistrasse 101
8092 Zürich
Switzerland
http://www.math.ethz.ch/~riviere/