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<p>We investigate the formation of singularities in a baby Skyrme type energy model, which describes magnetic solitons in two-dimensional ferromagnetic systems. In presence of a diverging anisotropy term, which enforces a preferred background state of the magnetization, we establish a weak compactness of its topological charge density, which converges to an atomic measure with quantized weights. We characterize the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\Gamma \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation>-limit of the energies as the total variation of this measure. In the case of lattice type energies, we first need to carefully define a notion of discrete topological charge for <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathbb {S}^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">S</mi> </mrow> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>-valued maps. We then prove a corresponding compactness and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\Gamma \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation>-convergence result, thereby bridging the discrete and continuum theories.</p>

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Energy Concentration in a Two-Dimensional Magnetic Skyrmion Model: Variational Analysis of Lattice and Continuum Theories

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  • 发表日期：

    2026年1月14日

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  • L. Briani,
  • M. Cicalese,
  • L. Kreutz

摘要

We investigate the formation of singularities in a baby Skyrme type energy model, which describes magnetic solitons in two-dimensional ferromagnetic systems. In presence of a diverging anisotropy term, which enforces a preferred background state of the magnetization, we establish a weak compactness of its topological charge density, which converges to an atomic measure with quantized weights. We characterize the \(\Gamma \) Γ -limit of the energies as the total variation of this measure. In the case of lattice type energies, we first need to carefully define a notion of discrete topological charge for \(\mathbb {S}^2\) S 2 -valued maps. We then prove a corresponding compactness and \(\Gamma \) Γ -convergence result, thereby bridging the discrete and continuum theories.