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Description
Starting from 2D lattice spin configurations satisfying the uniform state $\mathbf{e}_3$ outside a bounded domain $\Omega$, we investigate atomistic energies consisting of an exchange Heisenberg term together with an interfacial Dzyaloshinskii--Moriya interaction (DMI). In both discrete and continuum settings, imposing spatial confinement along with a topological constraint ($\text{degree}=1$) is essential to stabilize non-trivial localized skyrmions and prevent their collapse into the uniform state. To treat this constraint, we present two alternative strategies: introducing a non-degenerate discrete topological charge, or imposing an oscillation bound on the spins. The latter ensures that piecewise affine interpolations over a triangulation do not vanish, enabling their projection onto $\mathbb{S}^2$ and the standard prescription of the continuous degree. For sufficiently small DMI strength $\kappa$, both approaches lead to the existence of energy-barrier minimizers (lattice skyrmions) and their variational convergence, up to subsequences, to minimizers of the continuous energy (skyrmions in the continuum), rigorously linking atomistic chiral spin systems to continuum models in ultrathin ferromagnetic films.