A nonlinear hyperbolic Maxwell system using measure-valued functions (Q641607)

Maxwell's equations with a power law conductivity term (Ohm's law) and suitably matching non-linear dielectricity and permeability are studied in a bounded Lipschitz domain with a totally reflecting boundary. Due to the choice of power law for the constitutive equations, the electromagnetic field \((E,H)\) is naturally situated in \(L^p(\Omega)\times L^{p'}(\Omega)\), \(p, p'\in]1,\infty[\), \(\frac{1}{p}+\frac{1}{p'}=1\). Existence is approached by a Galerkin approximation argument using measure-valued functions. Technical difficulties in obtaining the needed a priori estimates in this \(L^p\)-\(L^{p'}\) duality setting are handled by choosing an adapted basis representation for the finite-dimensional approximations ensuring that \(\nabla\times\) leaves the approximating subspaces invariant.