Analysis & PDE

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

Paul Laurain and Tristan Rivière

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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.

Article information

Anal. PDE, Volume 7, Number 1 (2014), 1-41.

Received: 1 December 2011
Revised: 7 February 2013
Accepted: 3 April 2013
First available in Project Euclid: 20 December 2017

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Zentralblatt MATH identifier

Primary: 35J20: Variational methods for second-order elliptic equations 35J60: Nonlinear elliptic equations 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42] 58E20: Harmonic maps [See also 53C43], etc. 35J47: Second-order elliptic systems
Secondary: 49Q05: Minimal surfaces [See also 53A10, 58E12] 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions [See also 58J60] 32Q65: Pseudoholomorphic curves

analysis of PDEs differential geometry


Laurain, Paul; Rivière, Tristan. Angular energy quantization for linear elliptic systems with antisymmetric potentials and applications. Anal. PDE 7 (2014), no. 1, 1--41. doi:10.2140/apde.2014.7.1.

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