The aim of this note is to study octahedrality in vector-valued Lipschitz-free Banach spaces on a metric space, under topological hypotheses on it, by analysing the weak-star strong diameter 2 property in Lipschitz function spaces. Also, we show an example that proves that our results are optimal and that octahedrality in vector-valued Lipschitz-free Banach spaces actually relies on the underlying metric space as well as on the Banach one.

We describe the structure of the Ricci tensor on a locally homogeneous Lorentzian gradient Ricci soliton. In the non-steady case, we show that the soliton is rigid in dimensions 3 and 4. In the steady case we give a complete classification in dimension 3.

We study some Hardy-type inequalities involving a general norm in ℝn and an anisotropic distance function to the boundary. The case of the optimality of the constants is also addressed.

In this paper we analyse possible extensions of the classical Steklov eigenvalue problem to the fractional setting. In particular, we find a non-local eigenvalue problem of fractional type that approximates, when taking a suitable limit, the classical Steklov eigenvalue problem.

Given a positive, decreasing sequence a, whose sum is L, we consider all the closed subsets of [0, L] such that the lengths of their complementary open intervals are in one-to-one correspondence with the sequence a. The aim of this paper is to investigate the possible values that Assouad-type dimensions can attain for this class of sets. In many cases, the set of attainable values is a closed interval whose endpoints we determine.

We prove that the potential extreme Khovanov cohomology of a link is the cohomology of the independence simplicial complex of its Lando graph. We also provide a family of knots having as many non-trivial extreme Khovanov cohomology modules as desired, that is, examples of H-thick knots that are as far from being H-thin as desired.

A classification of the behaviour of the solutions f(·, a) to the ordinary differential equation (|f′|p-2f′)′ + f - |f′|p-1 = 0 in (0,∞) with initial condition f(0, a) = a and f′(0, a) = 0 is provided, according to the value of the parameter a > 0 when the exponent p takes values in (1, 2). There is a threshold value a* that separates different behaviours of f(·, a): if a > a*, then f(·, a) vanishes at least once in (0,∞) and takes negative values, while f(·, a) is positive in (0,∞) and decays algebraically to zero as r→∞ if a ∊ (0, a*). At the threshold value, f(·, a*) is also positive in (0,∞) but decays exponentially fast to zero as r→∞. The proof of these results relies on a transformation to a first-order ordinary differential equation and a monotonicity property with respect to a > 0. This classification is one step in the description of the dynamics near the extinction time of a diffusive Hamilton–Jacobi equation with critical gradient absorption and fast diffusion.

In parallel to the classical theory of central extensions of groups, we develop a version for extensions that preserve commutativity. It is shown that the Bogomolov multiplier is a universal object parametrizing such extensions of a given group. Maximal and minimal extensions are inspected, and a connection with commuting probability is explored. Such considerations produce bounds for the exponent and rank of the Bogomolov multiplier.

We give a weak-Lp Serrin-type regularity criterion for a weak solution to the three-dimensional magnetohydrodynamics equations in a bounded domain Ω ⊂ ℝ3.

In this paper, we study the singular fourth-order differential equation with a deviating argument:

By using Mawhin's continuation theorem and some analytic techniques, we establish some criteria to guarantee the existence of positive periodic solutions. The significance of this paper is that g has a strong singularity at x = 0 and satisfies a small force condition at x = ∞, which is different from the known ones in the literature.

While the existence of conformal mappings between doubly connected domains is characterized by their conformal moduli, no such characterization is available for harmonic diffeomorphisms. Intuitively, one expects their existence if the domain is not too thick compared to the codomain. We make this intuition precise by showing that for a Dini-smooth doubly connected domain Ω* there exists a ε > 0 such that for every doubly connected domain Ω with ModΩ* < ModΩ < ModΩ* + ε there exists a harmonic diffeomorphism from Ω onto Ω*.

We show how to deform a metric of the form g = gr + dr2 to a metric = Hr + dr2, which is a hyperbolic metric for r less than some fixed λ, and coincides with g for r large. Here by hyperbolic metric we mean a metric of constant sectional curvature equal to -1. We study the extent to which is close to hyperbolic everywhere, if we assume g is close to hyperbolic. A precise definition of the close to hyperbolic concept is given. We also deal with a one-parameter version of this problem. The results in this paper are used in the problem of smoothing Charney–Davis strict hyperbolizations.

We study a weighted maximal weak-type inequality for Haar multipliers that can be regarded as a dual problem of Muckenhoupt and Wheeden. More precisely, if Tε is the Haar multiplier associated with the sequence ε with values in [−1, 1], and is the r-maximal operator, then for any weight w and any function f on [0, 1) we have

for some constant Cr depending only on r. We also show that the analogous result does not hold if we replace by the dyadic maximal operator Md. The proof rests on the Bellman function method; using this technique we establish related weighted Lp estimates for p close to 1, and then deduce the main result by extrapolation arguments.

The convergence and blow-up results are established for the evolution of non-simple closed curves in an area-preserving curvature flow. It is shown that the global solution starting from a locally convex curve converges to an m-fold circle if the enclosed algebraic area A0 is positive, and evolves into a point if A0 = 0.