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January & February  2009, 23(1&2): 435-454. doi: 10.3934/dcds.2009.23.435

Non-linear electromagnetism and special relativity

Denis Serre 1,

1. 

École Normale Supérieure de Lyon, UMPA (UMR 5669 CNRS), ENS de Lyon, 46, allée d’Italie, 69364 Lyon, cedex 07, France

Received  September 2007 Revised  July 2008 Published  September 2008

We continue the study of nonlinear Maxwell equations for electromagnetism in the formalism of B. D. Coleman & E. H. Dill. We exploit here the assumption of Lorentz invariance, following I. Białinicki-Barula. In particular, we show that nonlinearity forbids the convexity of the electromagnetic energy density. This justifies the study of rank-one convex and of polyconvex densities, begun in [8, 16]. We also show the alternative that either electrodynamics is linear, or dispersion is lost as the electromagnetic field becomes intense.
Keywords: Born--Infeld model, quasiconvexity, Maxwell equations.
Mathematics Subject Classification: Primary: 78A25, 83A05; Secondary: 35L45.
Citation: Denis Serre. Non-linear electromagnetism and special relativity. Discrete & Continuous Dynamical Systems, 2009, 23 (1&2) : 435-454. doi: 10.3934/dcds.2009.23.435
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