Hyperbolic Maxwell variational inequalities of the second kind

Abstract

We analyze a class of hyperbolic Maxwell variational inequalities of the second kind. By means of a local boundedness assumption on the subdifferential of the underlying nonlinearity, we prove a well-posedness result, where the main tools for the proof are the semigroup theory for Maxwell’s equations, the Yosida regularization and the subdifferential calculus. The second part of the paper focuses on a more general case omitting the local boundedness assumption. In this case, taking into account more regular initial data and test functions, we are able to prove a weaker existence result through the use of the minimal section operator associated with the Nemytskii operator of the governing subdifferential. Eventually, we transfer the developed well-posedness results to the case involving Faraday’s law, which in particular allows us to improve the regularity property of the electric field in the weak existence result.