Regularity of weak solution to Maxwell's equations and applications to microwave heating (Q1880752)

The regularity of a weak solution of the time-harmonic Maxwell equations is considered in a bounded anisotropic medium \(\Omega\), and it is shown that the weak solution of the linearly-degenerate system \(\nabla\times(\gamma(x)\nabla\times{\mathbf E})+ \xi(x){\mathbf E}={\mathbf J}(x)\), \(x\in\Omega\subset\mathbb{R}^3\) is Hölder continuous under regularity conditions for \(\gamma(x)\) and \(\xi(x)\). A coupled electric and temperature microwave heating system is then examined, and it is shown that under certain conditions the system has a weak solution. Then, finally, the existence of a global classical solution is shown to exist for the special case where the electric wave only propagates in one direction.