The Effects of Differential Constraints and Surface Energies on Scaling Laws for Singular Perturbation Problems

Abstract

Motivated by mathematical models for phase transformations, arising, for example, in shape-memory alloys, and for micromagnetism in ferromagnetic materials, we study a class of A-free differential inclusions quantitatively. For this reason, we consider singular perturbation models of differential inclusions under an A-free constraint and determine the scaling in the singular perturbation parameter. The focus of this thesis is to characterize the influence of the order of the operator on the possible scaling laws and to show that the model is robust under changes of the surface energy. The first is achieved by establishing lower bounds for the compatible and incompatible two-well problem. It turns out that the scaling is determined by the maximal vanishing order on the sphere of the symbol of the differential operator applied to the compatible direction. These lower bounds are proven to be optimal for the divergence operator and for a higher order generalization of the curl and curl curl operators as an annihilator of symmetrized derivatives. The influence of the surface energy is studied by comparing sharp and diffuse interface models, as well as suitable interpolations of these. We deduce the lower bound for the diffuse model by estimating the energy from below through the sharp interface model and complement them with matching upper bounds for a model class of wells in the case of the curl operator. Furthermore, an <em>N</em>-well setting giving rise to higher order laminates for the curl operator is studied. If the energy penalizes only oscillations in a certain direction, we observe that for almost all directions this anisotropic energy scales like the full isotropic energy. The exceptional directions are those, where the anisotropic energy does not penalize the oscillations of the "inner-most" laminate. The scaling for these matches the ones of a lower order laminate. Furthermore, a non-algebraic scaling law for a divergence-free <em>T₃</em> structure is discussed. This result quantifies the dichotomy between the rigidity of exact solutions and the flexibility of approximate solutions for the associated inclusion.