Optimal Shape of Isolated Ferromagnetic Domains

Abstract

We investigate the energy of an isolated magnetized domain Ω ⊂ R n for n = 2, 3. In non-dimensionalized variables, the energy given bypenalizes the interfacial area of the domain as well as the energy of the corresponding magnetostatic field. Here, the magnetostatic potential Φ is determined by ∆Φ = ∂ 1 χ Ω , corresponding to uniform magnetization within the domain. We consider the macroscopic regime |Ω| → ∞, in which we derive compactness and Γlimit for the cross-section of the anisotropically rescaled configuration. The limit energy is local and the limiting shape of minimizers can be calculated.

Asymptotic shape of isolated magnetic domains

“…However, a proof for this statement seems quite technical and is ongoing work. We note that ([18], Thm. 1.2) gives a characterization of an integral version of the length and radius of configuration.…”

“…In [18], it has been shown that minimizers of our energy functional (1.2) with volume constraint (1.3) exist in all dimensions n and for all prescribed masses μ≥0. For 2≤n≤7, for any local minimizer of (1.2), there is a regular representative normalΩ for its positivity set which is open, bounded and has a smooth boundary.…”

“…and the regular representative is obtained after modification by a zero set. Also, the scaling of the ground state energy and some qualitative properties of the shape have been investigated in [18] for n=3 and in [19] for n≥4. We note that for small volume μ, the energy (1.2) is a perturbation of the perimeter energy and minimizers have approximately the shape and energy of a ball with mass μ.…”

Nucleation barriers for the cubic‐to‐tetragonal phase transformation in the absence of self‐accommodation

“…Note that this scaling of the macroscpic energy corresponds to an elastic field of strength |∇ | ∼ −1 on a region of volume 3 . The estimates (19) and (20) together with the assumption 0 2 = yield…”

“…In this situation, the elastic energy is even more degenerate allowing for a much larger class of configurations. Models related to the optimal shape of isolated nuclei in the framework of nonlocal isoperimetric problem have also been investigated in the context of Diblock copolymer models, [1,6,[14][15][16][17]23,24] copolymer-homopolymer blends, [7,8] ferromagnetics [19,22] and quantum systems, [28] these references are certainly not exhaustive.…”