A Nonlocal Isoperimetric Problem with Dipolar Repulsion

Abstract

We study a geometric variational problem for sets in the plane in which the perimeter and a regularized dipolar interaction compete under a mass constraint. In contrast to previously studied nonlocal isoperimetric problems, here the nonlocal term asymptotically localizes and contributes to the perimeter term to leading order. We establish existence of generalized minimizers for all values of the dipolar strength, mass and regularization cutoff and give conditions for existence of classical minimizers. For subcritical dipolar strengths we prove that the limiting functional is a renormalized perimeter and that for small cutoff lengths all mass-constrained minimizers are disks. For critical dipolar strength, we identify the next-order Γ-limit when sending the cutoff length to zero and prove that with a slight modification of the dipolar kernel there exist masses for which classical minimizers are not disks.1 |x| 3 . This is the interaction experienced at large distances by spins lying on the plane and oriented perpendicularly to it. Just like the classical Gamow's model, the model under consideration is relevant to the multidomain patterns observed in perpendicularly magnetized thin film ferromagnets [30], as well as ferroelectric films [57], Langmuir monolayers [5] and ferrofluid films subject to a strong perpendicular applied field [31,54] (for an experimental realization involving a magnetic garnet film, see Figure 1). Note, however, that setting G(x) = 1 |x| 3 would result in an ill-defined problem because of the divergence of the integral defining the nonlocal contribution at short scales. Furthermore, redefining the energy in the spirit of nonlocal minimal surfaces [11] to reduce the integral to that over Ω and Ω c , up to an additive constant, would not help, either, since the singularity of the kernel is still too strong. Therefore, a genuine regularization at short scale δ > 0 is necessary to make sense of the energy in (1.1) with this kind of kernel. This is a novel feature of the considered nonlocal isoperimetric problem compared to those studied previously.More specifically, we study the following nonlocal isoperimetric problem, as proposed by . We wish to minimize E(Ω) := αP (Ω) + β 2ˆΩˆΩ g δ (|x − y|) |x − y| 3 dx dy, (1.2) among all finite perimeter sets Ω ⊂ A m of fixed mass m > 0, i.e., with A m := {Ω ⊂ R 2 : P (Ω) < ∞, |Ω| = m}. (1.3) 3 Here, α, β > 0 are fixed parameters, P (Ω) is the perimeter of a measurable set Ω in the sense of De Giorgi [4]: P (Ω) := sup ˆΩ ∇ · φ dx : φ ∈ C 1 c (R 2 ; R 2 ), |φ| ≤ 1 , (1.4)

Second order expansion for the nonlocal perimeter functional

“…We denote the corresponding energies by E (i) for i = 1, 2, 3. When d = 2, E (1) is a sharp interface version of a problem from micromagnetism for thin plates with perpendicular anisotropy [26,29,34,32]. In this setting, {u = 1} and {u = 0} are the magnetic domains where the magnetization points either upward or downward.…”

“…This leads to easier proofs for the Γconvergence for more general kernels compared with the existing literature, e.g. [32,12]. We note that the method of autocorrelation function can also be used for a simpler proof of Dávila's result [18], see Appendix A.…”

“…When u ∈ BV (R 2 ; {0, 1}) and K is defined in (5), the energy has been studied by Muratov and Simon in [32], where the authors show among other results the Γ-convergence of the energy. The Γ-convergence for the energies (6) is studied by Cesaroni and Novaga [12].…”

“…The Γ-convergence for the energies (6) is studied by Cesaroni and Novaga [12]. Our model includes the models considered in both [32] and [12] (in the periodic setting). We generalize some results of these papers, however, with a different strategy of proofs and in a more general setting.…”

“…Moreover, we characterize the minimizers of the limit energy when the volume is small, and show that in 2d stripes have strictly smaller energy than lattice of balls if the volume fraction is close to 1/2. These have not been considered in [32,12].…”