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Abstract. We investigate time harmonic Maxwell equations in heterogeneous media, where the permeability µ and the permittivity ε are piecewise constant. The associated boundary value problem can be interpreted as a transmission problem. In a very natural way the interfaces can have edges and corners. We give a detailed description of the edge and corner singularities of the electromagnetic fields.

In this paper we consider the wave equation on 1-d networks with a delay term in the boundary and/or transmission conditions. We first show the well posedness of the problem and the decay of an appropriate energy. We give a necessary and sufficient condition that guarantees the decay to zero of the energy. We further give sufficient conditions that lead to exponential or polynomial stability of the solution. Some examples are also given.

Exponential stability analysis via Lyapunov method is extended to the one-dimensional heat and wave equations with time-varying delay in the boundary conditions. The delay function is admitted to be timevarying with an a priori given upper bound on its derivative, which is less than 1. Sufficient and explicit conditions are derived that guarantee the exponential stability. Moreover the decay rate can be explicitly computed if the data are given.

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