Given an open set ?, we consider the problem of providing sharp lower bounds for ? (?), i.e. its second Dirichlet eigenvalue of the p-Laplace operator. After presenting the nonlinear analogue of the Hong-Krahn-Szego inequality, asserting that the disjoint unions of two equal balls minimize ? among open sets of given measure, we improve this spectral inequality by means of a quantitative stability estimate. The extremal cases p = 1 and p = ? are considered as well. © 2012 Springer-Verlag Berlin Heidelberg.

The neuronal functional connectivity is a complex and non-stationary phenomenon creating dynamic networks synchronization determining the brain states and needed to produce tasks. Here, as a measure that quantifies the synchronization between the neuronal electrical activity of two brain regions, we used the normalized compression distance (NCD), which is the length of the compressed file constituted by the concatenated two signals, normalized by the length of the two compressed files including each single signal.

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 investigate the switching dynamics of multistable nematic liquid crystal devices. In particular, we identify a remarkably simple two-dimensional device which exploits hybrid alignment at the surfaces to yield a bistable response. We also consider a three-dimensional tristable nematic device with patterned anchoring, recently implemented in practice, and discuss how the director and disclination patterns change during switching.

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.).

We consider the Dirichlet eigenvalue problem, (the eigenfunction) and ? > 0 (the eigen value), ? is an arbitrary domain in RN with finite measure, 1 < p < ?, 1 < q < p*, p* = Np/(N - p) if 1 < p < N and p* = ? if p >= N. We study several existence and uniqueness results as well as some properties of the solutions. Moreover, we indicate how to extend to the general case some proofs known in the classical case p = q. © 2010 Texas State University - San Marcos.

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.

We construct an open set ? ? ? R on which an eigenvalue problem for the p-Laplacian has no isolated first eigenvalue and the spectrum is not discrete. The same example shows that the usual Lusternik-Schnirelmann minimax construction does not exhaust the whole spectrum of this eigenvalue problem.

In this paper, we consider the isoperimetric problem in the space R with a density. Our result states that, if the density f is lower semi-continuous and converges to a limit a> 0 at infinity, with f<= a far from the origin, then isoperimetric sets exist for all volumes. Several known results or counterexamples show that the present result is essentially sharp. The special case of our result for radial and increasing densities positively answers a conjecture of Morgan and Pratelli (Ann Glob Anal Geom 43(4):331-365, 2013.