Narrow Results

We study a thin-shell limit of micromagnetic energy for soft small ferromagnets. The relations between thickness of the magnet t, diameter l and magnetic exchange length w are t/l → 0 and tl/w² ≲ 1. We prove a Γ-convergence of the original 3D problem to a nonlocal 2D problem.

We provide a rigorous derivation of the compressible Reynolds system as a singular limit of the compressible (barotropic) Navier–Stokes system on a thin domain. In particular, the existence of solutions to the Navier–Stokes system with non-homogeneous boundary conditions is shown that may be of independent interest. Our approach is based on new a priori bounds available for the pressure law of hard sphere type. Finally, uniqueness for the limit problem is established in the one-dimensional case.

We present here an analysis of the regularity of minimizers of a variational model for epitaxially strained thin-films. The regularity of energetically-optimal film profiles is studied by extending previous methods and by developing new ideas based on transmission problems. The achieved regularity results relate to both the Stranski-Krastanow and the Volmer-Weber modes, the possibility of different elastic properties between the film and the substrate, and the presence of the surface tensions of all three involved interfaces: film/gas, substrate/gas, and film/substrate. Finally, geometrical conditions are provided for the optimal wetting angle, i.e. the angle formed at the contact point of films with the substrate. In particular, the Young–Dupré law is shown to hold, yielding what appears to be the first analytical validation of such law for a thin-film model in the context of Continuum Mechanics.

The behavior of a thin film made of a ferromagnetic material in the absence of an external magnetic field is described by an energy depending on the magnetization of the film verifying the saturation constraint. The free energy consists of an induced magnetostatic energy and an energy term with density including the exchange energy and the anisotropic energy. We study the behavior of this energy when the thickness of the curved film goes to zero. We show that the minimizers of the free energy converge to the minimizers of a local energy depending on a two-dimensional magnetization by Γ-convergence arguments.