We present computer simulations of the response of a flexoelectric blue phase network, either in bulk or under confinement, to an applied field. We find a transition in the bulk between the blue phase I disclination network and a parallel array of disclinations along the direction of the applied field. Upon switching off the field, the system is unable to reconstruct the original blue phase but gets stuck in a metastable phase. Blue phase II is comparatively much less affected by the field.

Blue phases are liquid crystals made up by networks of defects, or disclination lines. While existing phase diagrams show a striking variety of competing metastable topologies for these networks, very little is known as to how to kinetically reach a target structure, or how to switch from one to the other, which is of paramount importance for devices. We theoretically identify two confined blue phase I systems in which by applying an appropriate series of electric field it is possible to select one of two bistable defect patterns.

Blue phases are networks of disclination lines, which occur in cholesteric liquid crystals near the transition to the isotropic phase. They have recently been used for the new generation of fast switching liquid crystal displays. Here we study numerically the steady states and switching hydrodynamics of blue phase I (BPI) and blue phase II (BPII) cells subjected to an electric field.

We focus on three different convexity principles for local and nonlocal variational integrals. We prove various generalizations of them, as well as their equivalences. Some applications to nonlinear eigenvalue problems and Hardy-type inequalities are given. We also prove a measure-theoretic minimum principle for nonlocal and non- linear positive eigenfunctions.

The optimal Orlicz target space and the optimal rearrangement- invariant target space are exhibited for embeddings of fractional-order Orlicz-Sobolev spaces. Both the subcritical and the supercritical regimes are considered. In particular, in the latter case the relevant Orlicz-Sobolev spaces are shown to be embedded into the space of bounded continuous functions in Rn. This is a joint work with Andrea Cianchi, Lubos Pick and Lenka Slavikova.

The aim of this talk is to show how de la Vallee Poussin type interpolation based on Chebyshev zeros of rst kind, can be applied to resize an arbitrary color digital image. In fact, using such kind of approximation, we get an image scaling method running for any desired scaling factor or size, in both downscaling and upscaling. The peculiarities and the performance of such method will be discussed.

An optimal embedding theorem for fractional Orlicz-Sobolev spaces into Orlicz spaces will be surveyed. A new embedding for the same fractional spaces into generalized Campanato spaces will be also presented. This is a joint work, in progress, with Andrea Cianchi, Lubos Pick and Lenka Slavikova.

Given (Formula presented.), we discuss the embedding of (Formula presented.) in (Formula presented.). In particular, for (Formula presented.) we deduce its compactness on all open sets (Formula presented.) on which it is continuous. We then relate, for all q up the fractional Sobolev conjugate exponent, the continuity of the embedding to the summability of the function solving the fractional torsion problem in (Formula presented.) in a suitable weak sense, for every open set (Formula presented.).