On the structure of infinity-harmonic maps

Let H ∈ C 2(ℝ N×n ), H ≥ 0. The PDE system arises as the Euler-Lagrange PDE of vectorial variational problems for the functional E ∞(u, Ω) = ‖H(Du)‖ L ∞(Ω) defined on maps u: Ω ⊆ ℝ n → ℝ N . (1) first appeared in the author's recent work. The scalar case though has a long history initiated by Aronsson. Herein we study the solutions of (1) with emphasis on the case of n = 2 ≤ N with H the Euclidean norm on ℝ N×n , which we call the “∞-Laplacian”. By establishing a rigidity theorem for rank-one maps of independent interest, we analyse a phenomenon of separation of the solutions to phases with qualitatively different behaviour. As a corollary, we extend to N ≥ 2 the Aronsson-Evans-Yu theorem regarding non existence of zeros of |Du| and prove a maximum principle. We further characterise all H for which (1) is elliptic and also study the initial value problem for the ODE system arising for n = 1 but with H(·, u, u′) depending on all the arguments.

Location, location, location? Analysing property rents in medieval Gloucester

Although medieval rentals have been extensively studied, few scholars have used them to analyse variations in the rents paid on individual properties within a town. It has been claimed that medieval rents did not reflect economic values or market forces, but were set according to social and political rather than economic criteria, and remained ossified at customary levels. This paper uses hedonic regression methods to test whether property rents in medieval Gloucester were influenced by classic economic factors such as the location and use of a property. It investigates both rents and local rates (landgavel), and explores the relationship between the two. It also examines spatial autocorrelation. It finds significant relationships between urban rents and property characteristics that are similar to those found in modern studies. The findings are consistent with the view that, in Gloucester at least, medieval rents were strongly influenced by classical economic factors working through a competitive urban property market.

Daily thin-ice thickness maps from modis thermal infrared imagery

Accurate knowledge of ice-production rates within the marginal ice zones of the Arctic Ocean requires monitoring of the thin-ice distribution within polynyas. The thickness of the ice layer controls the heat loss and hence the new-ice formation. An established thinice algorithm using high-resolution MODIS data allows deriving the ice-thickness distribution within polynyas. The average uncertainty is ±4.7 cm for ice thicknesses below 0.2 m. In this study, the ice-thickness distributions within the Laptev Sea polynya for the two winter seasons 2007/08 and 2008/09 are calculated. Then, a new method is applied to determine a daily MODIS thin-ice product.