arXiv subjects

Theresa M. Simon

Publications and source records attributed to Theresa M. Simon.

9 recordsLinked to original sources

arXiv2026

A weak-strong uniqueness principle for the Mullins-Sekerka equation

We establish a weak-strong uniqueness principle for the two-phase Mullins-Sekerka equation in ambient dimension $d = 2$ and $3$: As long as a classical solution to the evolution problem exists, any weak De Giorgi type varifold solution (see for this notion the recent work of Stinson and the second author, Arch. Ration. Mech. Anal. 248, 8, 2024) must coincide with it. In particular, in the absence of geometric singularities such weak solutions do not introduce a mechanism for (unphysical) non-uniqueness. We also derive a stability estimate with respect to changes in the data. Our method is based on the notion of relative entropies for interface evolution problems, a reduction argument to a perturbative graph setting, and a stability analysis in this perturbative regime relying crucially on the gradient flow structure of the Mullins-Sekerka equation.

math.AP↗

arXiv2025

Euler's elastica functional as a large mass limit of a two-dimensional non-local isoperimetric problem

We consider a large mass limit of the non-local isoperimetric problem with a repulsive Yukawa potential in two space dimensions. In this limit, the non-local term concentrates on the boundary, resulting in the existence of a critical regime in which the perimeter and the non-local terms cancel each other out to leading order. We show that under appropriate scaling assumptions the next-order $Γ$-limit of the energy with respect to the $L^1$ convergence of the rescaled sets is given by a weighted sum of the perimeter and Euler's elastica functional, where the latter is understood via the lower-semicontinuous relaxation and is evaluated on the system of boundary curves. As a consequence, we prove that in the considered regime the energy minimizers always exist and converge to either disks or annuli, depending on the relative strength of the elastica term.

math.AP↗

arXiv2025

Existence of higher degree minimizers in the magnetic skyrmion problem

We demonstrate existence of topologically nontrivial energy minimizing maps of a given positive degree from bounded domains in the plane to $\mathbb S^2$ in a variational model describing magnetizations in ultrathin ferromagnetic films with Dzyaloshinskii-Moriya interaction. Our strategy is to insert tiny truncated Belavin-Polyakov profiles in carefully chosen locations of lower degree objects such that the total energy increase lies strictly below the expected Dirichlet energy contribution, ruling out loss of degree in the limits of minimizing sequences. The argument requires that the domain be either sufficiently large or sufficiently slender to accommodate a prescribed degree. We also show that these higher degree minimizers concentrate on point-like skyrmionic configurations in a suitable parameter regime.

math.AP↗

arXiv2024

Magnetic skyrmions under confinement

We present a variational treatment of confined magnetic skyrmions in a minimal micromagnetic model of ultrathin ferromagnetic films with interfacial Dzylashinksii-Moriya interaction (DMI) in competition with the exchange energy, with a possible addition of perpendicular magnetic anisotropy. Under Dirichlet boundary conditions that are motivated by the asymptotic treatment of the stray field energy in the thin film limit we prove existence of topologically non-trivial energy minimizers that concentrate on points in the domain as the DMI strength parameter tends to zero. Furthermore, we derive the leading order non-trivial term in the $Γ$-expansion of the energy in the limit of vanishing DMI strength that allows us to completely characterize the limiting magnetization profiles and interpret them as particle-like states whose radius and position are determined by minimizing a renormalized energy functional. In particular, we show that in our setting the skyrmions are strongly repelled from the domain boundaries, which imparts them with stability that is highly desirable for applications. We provide explicit calculations of the renormalized energy for a number of basic domain geometries.

math.AP↗

arXiv2023

Theory of magnetic field-stabilized compact skyrmions in thin film ferromagnets

We present a micromagnetic theory of compact magnetic skyrmions under applied magnetic field that accounts for the full dipolar energy and the interfacial Dzyaloshinskii-Moryia interaction (DMI) in the thin film regime. Asymptotic analysis is used to derive analytical formulas for the parametric dependence of the skyrmion size and rotation angle, as well as the energy barriers for collapse and bursting, two processes that lead to a finite skyrmion lifetime. We demonstrate the existence of a new regime at low DMI, in which the skyrmion is stabilized by a combination of non-local dipolar interaction and a magnetic field applied parallel to its core, and discuss the conditions for an experimental realization of such field-stabilized skyrmions.

cond-mat.mtrl-sci↗

arXiv2023

Local minimizers of the interface length functional based on a concept of local paired calibrations

We establish that regular flat partitions are locally minimizing for the interface energy with respect to $L^1$ perturbations of the phases. Regular flat partitions are partitions of open sets in $\mathbb{R}^2$ whose network of interfaces consists of finitely many straight segments with a singular set made up of finitely many triple junctions at which the Herring angle condition is satisfied. This result not only holds for the case of the perimeter functional but for a general class of surface tension matrices. Our proof relies on a localized version of the paired calibration method which was introduced by Lawlor and Morgan (Pac. J. Appl. Math., 166(1), 1994) in conjunction with a relative energy functional that precisely captures the suboptimality of classical calibration estimates. Vice versa, we show that any stationary point of the length functional (in a sense of metric spaces) has to be a regular flat partition.

math.AP↗

arXiv2022

On Rigidity for the Four-Well Problem Arising in the Cubic-to-Trigonal Phase Transformation

We classify all exactly stress-free solutions to the cubic-to-trigonal phase transformation within the geometrically linearized theory of elasticity, showing that only simple laminates and crossing-twin structures can occur. In particular, we prove that although this transformation is closely related to the cubic-to-orthorhombic phase transformation, all its solutions are rigid. The argument relies on a combination of the Saint-Venant compatibility conditions together with the underlying nonlinear relations and non-convexity conditions satisfied by the strain components.

math.AP↗

arXiv2020

A quantitative description of skyrmions in ultrathin ferromagnetic films and rigidity of degree $\pm1$ harmonic maps from $\mathbb{R}^2$ to $\mathbb{S}^2$

We characterize skyrmions in ultrathin ferromagnetic films as local minimizers of a reduced micromagnetic energy appropriate for quasi two-dimensional materials with perpendicular magnetic anisotropy and interfacial Dzyaloshinskii-Moriya interaction. The minimization is carried out in a suitable class of two-dimensional magnetization configurations that prevents the energy from going to negative infinity, while not imposing any restrictions on the spatial scale of the configuration. We first demonstrate existence of minimizers for an explicit range of the model parameters when the energy is dominated by the exchange energy. We then investigate the conformal limit, in which only the exchange energy survives and identify the asymptotic profiles of the skyrmions as degree 1 harmonic maps from the plane to the sphere, together with their radii, angles and energies. A byproduct of our analysis is a quantitative rigidity result for degree $\pm 1$ harmonic maps from the two-dimensional sphere to itself.

math.AP↗

arXiv2020

Convergence rates of the Allen-Cahn equation to mean curvature flow: A short proof based on relative entropies

We give a short and self-contained proof for rates of convergence of the Allen-Cahn equation towards mean curvature flow, assuming that a classical (smooth) solution to the latter exists and starting from well-prepared initial data. Our approach is based on a relative entropy technique. In particular, it does not require a stability analysis for the linearized Allen-Cahn operator. As our analysis also does not rely on the comparison principle, we expect it to be applicable to more complex equations and systems.

math.AP↗