Expansion of eigenvalues of the perturbed discrete bilaplacian.

We consider the family of discrete Schrödinger-type operators in -dimensional lattice , where is the discrete Laplacian and is of rank-one. We prove that there exist coupling constant thresholds such that for any the discrete spectrum of is empty and for any the discrete spectrum of is a singleton and for and for Moreover, we study the asymptotics of as and as well as The asymptotics highly depends on and

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.