On the hyperbolicity of Maxwell's equations with a local constitutive law.

Perlick, Volker (2011) On the hyperbolicity of Maxwell's equations with a local constitutive law. Journal of Mathematical Physics, 52. 042903. ISSN 0022-2488

Abstract

Maxwell's equations are considered in metric-free form, with a local but otherwise arbitrary constitutive law. After splitting Maxwell's equations into evolution equations and constraints, we derive the characteristic equation and we discuss its properties in detail. We present several results that are relevant for the question of whether the evolution equations are hyperbolic, strongly hyperbolic, or symmetric hyperbolic. In particular, we give a convenient characterization of all constitutive laws for which the evolution equations are symmetric hyperbolic. The latter property is sufficient, but not necessary, for well-posedness of the initial-value problem. By way of example, we illustrate our results with the constitutive laws of biisotropic media and of Born–Infeld theory.