On a Class of Degenerate Abstract Parabolic Problems and Applications to Some Eddy Current Models

Abstract

We present an abstract framework for parabolic type equations which possibly degenerate on certain spatial regions. The degeneracies are such that the equations under investigation may admit a type change ranging from parabolic to elliptic type problems. The approach is an adaptation of the concept of so-called evolutionary equations in Hilbert spaces and is eventually applied to a degenerate eddy current type model. The functional analytic setting requires minimal assumptions on the boundary and interface regularity. The degenerate eddy current model is justified as a limit model of non-degenerate hyperbolic models of Maxwell's equations.

Nonlocal $H$-convergence for topologically nontrivial domains

“…Remark 3.4 (On range closed). In[12, Remark 4.7] one can find conditions for the closedness of ran(curl) (or, equivalently, ran( c url)). A sufficient condition…”