Angular energy quantization for linear elliptic systems with antisymmetric potentials and applications

Laurain, Paul; Rivière, Tristan

doi:10.2140/apde.2014.7.1

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Angular energy quantization for linear elliptic systems with antisymmetric potentials and applications

Paul Laurain and Tristan Rivière

Vol. 7 (2014), No. 1, 1–41

DOI: 10.2140/apde.2014.7.1

Abstract

We establish a quantization result for the angular part of the energy of solutions to elliptic linear systems of Schrödinger type with antisymmetric potentials in two dimensions. This quantization is a consequence of uniform Lorentz–Wente type estimates in degenerating annuli. Moreover this result is optimal in the sense that we exhibit a sequence of functions satisfying our hypothesis whose radial part of the energy is not quantized. We derive from this angular quantization the full energy quantization for general critical points to functionals which are conformally invariant or also for pseudoholomorphic curves on degenerating Riemann surfaces.

Keywords

analysis of PDEs, differential geometry

Mathematical Subject Classification 2010

Primary: 35J20, 35J60, 53C42, 58E20, 35J47

Secondary: 49Q05, 53C21, 32Q65

Milestones

Received: 1 December 2011

Revised: 7 February 2013

Accepted: 3 April 2013

Published: 7 May 2014

Authors

Paul Laurain
Department of Mathematics IMJ-Université Paris 7 75013 Paris France
http://www.math.jussieu.fr/~laurainp/
Tristan Rivière
Department of Mathematics ETH Zentrum CH-8093 Zürich Switzerland
http://www.math.ethz.ch/~riviere/

Vol. 7, No. 1, 2014