Quantitative analysis of passive intermodulation and surface roughness

We explore the relationship between rough surface conductors and the phenomenon of passive intermodulation. The underlying surface is taken to be the boundary of a Lipschitz domain, and a characteristic angle of the domain is used to track boundary dependence on the various fields. To model electro-thermal passive intermodulation in particular, we consider a specific type of temperature-dependent conductivity and determine conditions on the conductivity under which one can use fixed point arguments to solve an induction heating and Joule heating problem on a Lipschitz domain. In the latter problem, we also consider a time-dependent permittivity function $\epsilon$. Finally, weak solutions to a magneto-quasi-static problem are obtained when the permeability µ is temperature dependent and is allowed to degenerate in a certain way. An interesting effect of the rough surface is the inherently limited Sobolev regularity of the electric field, which can be improved if one assumes additional constraints on the boundary.