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. The compactness of energy-equibounded sequences and energy lower semicontinuity are shown with respect to a proper selected topology in a class of admissible configurations that extends the classes previously considered in the literature. In particular, graph-like constraints previously considered for the setting of thin films and crystalline cavities are substituted by the more general assumption that the free crystalline interface is the boundary, consisting of an at most fixed finite number m of connected components, of sets of finite perimeter. Finally, it is shown that, as m → ∞, the energy of minimal admissible configurations tends to the minimum energy in the general class of configurations without the bound on the number of connected components for the free interface. Contents

Under suitable assumptions on the family of anisotropies, we prove the existence of a weak global 1 n+1 -Hölder continuous in time mean curvature flow with mobilities of a bounded anisotropic partition in any dimension using the method of minimizing movements. The result is extended to the case when suitable driving forces are present. We improve the Hölder exponent to 1 2 in the case of partitions with the same anisotropy and the same mobility and provide a weak comparision result in this setting for a weak anisotropic mean curvature flow of a partition and an anisotropic mean curvature two-phase flow.

We study the discrete spectrum of the two-particle Schrödinger operator H ̂ μ λ ( K ) , K ∈ T 2 , associated to the Bose–Hubbard Hamiltonian H ̂ μ λ of a system of two identical bosons interacting on site and nearest-neighbor sites in the two dimensional lattice Z 2 with interaction magnitudes μ ∈ R and λ ∈ R , respectively. We completely describe the spectrum of H ̂ μ λ ( 0 ) and establish the optimal lower bound for the number of eigenvalues of H ̂ μ λ ( K ) outside its essential spectrum for all values of K ∈ T 2 . Namely, we partition the (μ, λ)-plane such that in each connected component of the partition the number of bound states of H ̂ μ λ ( K ) below or above its essential spectrum cannot be less than the corresponding number of bound states of H ̂ μ λ ( 0 ) below or above its essential spectrum.

We study the mean curvature motion of a droplet flowing by mean curvature on a horizontal hyperplane with a possibly nonconstant prescribed contact angle. Using the minimizing movements method we show the existence of a weak evolution, and its compatibility with a distributional solution. We also prove various comparison results.Date: October 2, 2018. Key words and phrases. Mean curvature flow with prescribed contact angle, sets of finite perimeter, capillary functional, minimizing movements.