Abstract: We consider a two-dimensional problem in nonlinear elasticity which corresponds to the cubic-to-tetragonal phase transformation. Our model is frame invariant and the energy density is given by the squared distance from two potential wells. We obtain the quasiconvex envelope of the energy density and therefore the relaxation of the variational problem. Our result includes the constraint of positive determinant.

Abstract: This work is concerned with the derivation of optimal scaling laws, in the sense of matching lower and upper bounds on the energy, for a solid undergoing ductile fracture. The specific problem considered concerns a material sample in the form of an infinite slab of finite thickness subjected to prescribed opening displacements on its two surfaces. The solid is assumed to obey deformation-theory of plasticity and, in order to further simplify the analysis, we assume isotropic rigid-plastic deformations with zero plastic spin. When hardening exponents are given values consistent with observation, the energy is found to exhibit sublinear growth. We regularize the energy through the addition of nonlocal energy terms of the strain-gradient plasticity type. This nonlocal regularization has the effect of introducing an intrinsic length scale into the energy. Under these assumptions, ductile fracture emerges as the net result of two competing effects: whereas the sublinear growth of the local energy promotes localization of deformation to failure planes, the nonlocal regularization stabilizes this process, thus resulting in an orderly progression towards failure and a well-defined specific fracture energy. The optimal scaling laws derived here show that ductile fracture results from localization of deformations to void sheets, and that it requires a well-defined energy per unit fracture area. In particular, fractal modes of fracture are ruled out under the assumptions of the analysis. The optimal scaling laws additionally show that ductile fracture is cohesive in nature, that is, it obeys a well-defined relation between tractions and opening displacements. Finally, the scaling laws supply a link between micromechanical properties and macroscopic fracture properties. In particular, they reveal the relative roles that surface energy and microplasticity play as contributors to the specific fracture energy of the material.

Abstract: The homogeneous bosonic Nordheim equation is a kinetic equation describing the dynamics of the distribution of particles in the space of moments for a homogeneous, weakly interacting, quantum gas of bosons. We show the existence of classical solutions of the homogeneous bosonic Nordheim equation that blow up in finite time. We also prove finite time condensation for a class of weak solutions of the kinetic equation.

Abstract: We consider the minor process of (Hermitian) matrix diffusions with constant diagonal drifts. At any given time, this process is determinantal and we provide an explicit expression for its correlation kernel. This is a measure on the Gelfand–Tsetlin pattern that also appears in a generalization of Warren’s process (Electron. J. Probab. 12:573–590, 2007), in which Brownian motions have level-dependent drifts. Finally, we show that this process arises in a diffusion scaling limit from an interacting particle system in the anisotropic KPZ class in 2+1 dimensions introduced in Borodin and Ferrari (Commun. Math. Phys., 2008). Our results generalize the known results for the zero drift situation.

Abstract: We study couplings q∙ of two equivariant random measures λ∙ and μ∙ on a Riemannian manifold (M,d,m). Given a cost function we ask for minimizers of the mean transportation cost per volume. In case the minimal/optimal cost is finite and λω≪m we prove that there is a unique equivariant coupling minimizing the mean transportation cost per volume. Moreover, the optimal coupling is induced by a transportation map, q∙=(id,T)∗λ∙. We show that the optimal transportation map can be approximated by solutions to classical optimal transportation problems on bounded regions. In case of Lp−cost the optimal transportation cost per volume defines a metric on the space of equivariant random measure with unit intensity.