We generalize the ‘sup+C inf’ inequality obtained by Shafrir to the solutions of

with Ω ⊂ ℝ2 open and bounded, α ∈ (0,1) and V any measurable function which satisfies 0 < a ≤ V ≤ b < + ∞.

We describe the numerical invariants and the curvature ellipse attached to the second fundamental form of a spacelike surface in a four-dimensional Minkowski space. We then study the configuration of the V-principal curvature lines on a spacelike surface when the normal field V is lightlike (the lightcone configuration). We end with some observations on the mean directionally curved lines and on the asymptotic lines on spacelike surfaces.

We study the thermodynamic consistency of phase-field models, which include gradient terms of the density ρ in the free-energy functional such as the van der Waals–Cahn–Hilliard model. It is well known that the entropy inequality admits gradient and higher-order gradient terms of ρ in the free energy only if either the energy flux or the entropy flux is represented by a non-classical form. We identify a non-classical entropy flux, which is not restricted to isothermal processes, so that gradient contributions are possible.

We then investigate equilibrium conditions for the van der Waals–Cahn–Hilliard phase-field model in the sharp interface limit. For a single substance thermodynamics provides two jump conditions at the sharp interface, namely the continuity of the Gibbs free energies of the adjacent phases and the discontinuity of the corresponding pressures, which is balanced by the mean curvature. We show that these conditions can be also extracted from the van der Waals–Cahn–Hilliard phase-field model in the sharp interface limit. To this end we prove an asymptotic expansion of the density up to the first order. The results are based on local energy estimates and uniform convergence results for the density.

We study the existence, multiplicity and non-existence of positive solutions for the singular two-point boundary-value problems

where λ is a non-negative real parameter and f ∈ C((0, 1) × [0,∞), (0,∞)). Here, f(t, u) may be singular at t = 0 and/or 1. To obtain the main results we use the global continuation theorem and fixed-point index theory.

We use methods of bifurcation theory to study properties of solution curves for a class of quasilinear two-point problems. Unlike semilinear equations, here, solution curves may stop at some point, or solution curves may turn the ‘wrong way’ (compared with semilinear equations) as in a paper by Habets and Omari, where the prescribed mean curvature equation was considered. This class of equations will be our main example. Another difference from semilinear equations is that the bifurcation diagram may depend on the length of the interval, as was discovered recently by Pan, who considered the prescribed mean curvature equation and f(u) = eu. We generalize this result to convex f(u), with f(0) > 0, and to more general quasilinear equations. We also give formulae which allow us to compute all possible turning points and the direction of the turn, generalizing similar formulae in Korman et al. We also present a numerical computation of the bifurcation curves.

We study the asymptotic behaviour of a Hele-Shaw flow produced by the injection of fluid from a finite number of points at different speeds. We prove that, as time tends to infinity, the boundary of the fluid domain approaches the circle centred at the barycentre of the injection points with weights proportional to the injection rates. The distances from the barycentre to the boundary points are estimated from both above and below.

The strong* topology s*(X) of a Banach space X is defined as the locally convex topology generated by the seminorms x ↦ ‖Sx‖ for bounded linear maps S from X into Hilbert spaces. The w-right topology for X, ρ(X), is a stronger locally convex topology, which may be analogously characterized by taking reflexive Banach spaces in place of Hilbert spaces. For any Banach space Y , a linear map T : X → Y is known to be weakly compact precisely when T is continuous from the w-right topology to the norm topology of Y. The main results deal with conditions for, and consequences of, the coincidence of these two topologies on norm bounded sets. A large class of Banach spaces, including all C*-algebras and, more generally, all JB*-triples, exhibit this behaviour.

We consider weak solutions of parabolic systems of the type

where the structure function b is differentiable with respect to x and satisfies standard ellipticity and growth properties with polynomial growth rate p ∊ (2n/(n + 2), 2). We investigate regularity properties of the solution, including the existence of second-order spatial derivatives, the existence of the time derivative and the higher integrability of the spatial gradient. As an application, we derive dimension estimates for the singular set of solutions of homogeneous parabolic systems. More precisely, we establish the bound

provided the structure function depends Höolder continuously on the space variable with Höolder exponent β ∊(0, 1].

In general, population systems are often subject to environmental noise. To examine whether the presence of such noise affects these systems significantly, we perturb the functional Kolmogorov-type system

into the stochastic functional differential equation

We show that different environmental noise structures have different effects on the population system with unbounded delay. Under two classes of different environmental noise perturbations, we establish existence theorems of the global positive solution to the unbounded delay stochastic functional Kolmogorov-type system. As the desired results for population dynamics, we also examine asymptotic boundedness, including the moment boundedness, stochastically ultimate boundedness and the moment average boundedness in time. To illustrate our idea more clearly, as a special case we also discuss a Lotka–Volterra system with unbounded delay.