Relaxed Micromagnetism using Young Measures

Abstract

Equilibrium magnetization states in ferromagnetic materials are characterized by minima of the energy ε in (I.1) (for α = 0); as the set A in (I.3) is non-convex, it is not weakly closed in L ²(ω; ℝ^d), and a solution to (I.4) does not have to exist for uniaxial materials; cf. for instance [71]. An implication is that highly oscillatory minimizing sequences of ε do not have weak limits in A. One way to overcome this problem is to convexify the anisotropy energy, see Chapter 2. In this chapter, we consider another scheme which leaves ε in (I.1) (for α = 0); S ø ^** is not known or where it is not desirable to compute it; moreover, the microstructure is computed directly in terms of Young measure-valued solutions — rather than from certain averages in a post-processing step.