Microscopical Justification of Solid-State Wetting and Dewetting.

The continuum model related to the , i.e., the problem of determining the equilibrium shape of crystalline drops resting on a substrate, is derived in dimension two by means of a rigorous discrete-to-continuum passage by -convergence of atomistic models taking into consideration the atomic interactions of the drop particles both among themselves and with the fixed substrate atoms.

Magnetoelastic thin films at large strains.

Starting from the three-dimensional setting, we derive a limit model of a thin magnetoelastic film by means of -convergence techniques. As magnetization vectors are defined on the elastically deformed configuration, our model features both Lagrangian and Eulerian terms. This calls for qualifying admissible three-dimensional deformations of planar domains in terms of injectivity.

A Unified Model for Stress-Driven Rearrangement Instabilities.

A variational model to simultaneously treat Stress-Driven Rearrangement Instabilities, such as boundary discontinuities, internal cracks, external filaments, edge delamination, wetting, and brittle fractures, is introduced. The model is characterized by an energy displaying both elastic and surface terms, and allows for a unified treatment of a wide range of settings, from epitaxially-strained thin films to crystalline cavities, and from capillarity problems to fracture models. The existence of minimizing configurations is established by adopting the direct method of the Calculus of Variations.

Law in the Cubic Lattice.

We investigate the Edge-Isoperimetric Problem (EIP) for sets with elements of the cubic lattice by emphasizing its relation with the emergence of the Wulff shape in the crystallization problem. Minimizers of the edge perimeter are shown to deviate from a corresponding cubic Wulff configuration with respect to their symmetric difference by at most elements. The exponent 3 / 4 is optimal.

Sharp [Formula: see text] Law for the Minimizers of the Edge-Isoperimetric Problem on the Triangular Lattice.

We investigate the edge-isoperimetric problem (EIP) for sets of points in the triangular lattice by emphasizing its relation with the emergence of the Wulff shape in the crystallization problem. By introducing a suitable notion of perimeter and area, EIP minimizers are characterized as extremizers of an isoperimetric inequality: they attain maximal area and minimal perimeter among connected configurations. The maximal area and minimal perimeter are explicitly quantified in terms of .