On New Difference Sequence Space and Their Generalised Almost Statistical Convergence

The object of the present paper is to introduce the notion of generalised almost statistical (GAS) convergence of bounded real sequences, which generalises the notion of almost convergence as well as statistical convergence of bounded real sequences. We also introduce the concept of Banach statistical limit functional and the notion of GAS convergence mainly depends on the existence of Banach statistical limit functional. We prove the existence of Banach statistical limit functional. Also, the existence GAS convergent sequence, which is neither statistical convergent nor almost convergent. Lastly, some topological properties of the space of all GAS convergent sequences are investigated.

Γ-convergence and relaxations for gradient flows in metric spaces: a minimizing movement approach

We present new abstract results on the interrelation between the minimizing movement scheme for gradient flows along a sequence of Γ-converging functionals and the gradient flow motion for the corresponding limit functional, in a general metric space. We are able to allow a relaxed form of minimization in each step of the scheme, and so we present new relaxation results too.

Variational approximation of size-mass energies for k-dimensional currents

In this paper we produce a Γ-convergence result for a class of energies Fε,ak modeled on the Ambrosio-Tortorelli functional. For the choice k = 1 we show that Fε,a1 Γ-converges to a branched transportation energy whose cost per unit length is a function fan−1 depending on a parameter a > 0 and on the codimension n − 1. The limit cost fa(m) is bounded from below by 1 + m so that the limit functional controls the mass and the length of the limit object. In the limit a ↓ 0 we recover the Steiner energy. We then generalize the approach to any dimension and codimension. The limit objects are now k-currents with prescribed boundary, the limit functional controls both their masses and sizes. In the limit a ↓ 0, we recover the Plateau energy defined on k-currents, k < n. The energies Fε,ak then could be used for the numerical treatment of the k-Plateau problem.

STRIPE PATTERNS IN A MODEL FOR BLOCK COPOLYMERS

We consider a pattern-forming system in two space dimensions defined by an energy [Formula: see text]. The functional [Formula: see text] models strong phase separation in AB diblock copolymer melts, and patterns are represented by {0, 1}-valued functions; the values 0 and 1 correspond to the A and B phases. The parameter ε is the ratio between the intrinsic, material length-scale and the scale of the domain Ω. We show that in the limit ε → 0 any sequence uε of patterns with uniformly bounded energy [Formula: see text] becomes stripe-like: the pattern becomes locally one-dimensional and resembles a periodic stripe pattern of periodicity O(ε). In the limit the stripes become uniform in width and increasingly straight. Our results are formulated as a convergence theorem, which states that the functional [Formula: see text] Gamma-converges to a limit functional [Formula: see text]. This limit functional is defined on fields of rank-one projections, which represent the local direction of the stripe pattern. The functional [Formula: see text] is only finite if the projection field solves a version of the Eikonal equation, and in that case it is the L2-norm of the divergence of the projection field, or equivalently the L2-norm of the curvature of the field. At the level of patterns the converging objects are the jump measures |∇uε| combined with the projection fields corresponding to the tangents to the jump set. The central inequality from Peletier and Röger, Arch. Rational Mech. Anal.193 (2009) 475–537, provides the initial estimate and leads to weak measure-function pair convergence. We obtain strong convergence by exploiting the non-intersection property of the jump set.

A dual method in the calculation of homogenisation and applications

SynopsisWe give a new method for calculating the Γ-limit functional encountered in the problems of homogenisation. We use the Legendre–Lagrange transform in the convex analysis and regularisation method to obtain the explicit expression of the Γ-limit functional. The result can be applied to some nonlocal function spaces.