In this short note we prove that the functional I : W1,p(J;R) → R defined by is sequentially weakly lower semicontinuous in W1,p(J,R) if and only if the symmetric part W+ of W is separately convex. We assume that W is real valued, continuous and bounded below by a constant, and that J is an open subinterval of R. We also show that the lower semicontinuous envelope of I cannot in general be obtained by replacing W by its separately convex hull Wsc.

The double-covering map u dc : R 2 → R 2 is given byin cartesian coordinates. This paper examines the conjecture that u dc is the global minimizer of the Dirichlet energy I(u) = B |∇u| 2 dx among all W 1,2 mappings u of the unit ball B ⊂ R 2 satisfying (i) u = u dc on ∂B, and (ii) det ∇u = 1 almost everywhere. Let the class of such admissible maps be A. The chief innovation is to express I(u) in terms of an auxiliary functional G(u − u dc ), using which we show that u dc is a stationary point of I in A, and that u dc is a global minimizer of the Dirichlet energy among members of A whose Fourier decomposition can be controlled in a way made precise in the paper. By constructing variations about u dc in A using ODE techniques, we also show that u dc is a local minimizer among variations whose tangent ψ to A at u dc obeys G(ψ o ) > 0, where ψ o is the odd part of ψ. In addition, a Lagrange multiplier corresponding to the constraint det ∇u = 1 is identified by an analysis which exploits the well-known Fefferman-Stein duality. AbstractLe double-revêtement fonction u dc : R 2 → R 2 est donn paren coordonnes cartesian. Cet article examine la conjecture que u dc est le minimiseur global de l'énergie de Dirichlet I(u) = B |∇u| 2 dx pour les fonctions satisfaisant (i) u ∈ W 1,2 (B), où B est la boule unité de R 2 , (ii) u = u dc sur ∂B, et (iii) det ∇u = 1 presque partout. Soit A la classe admissible de telles fonctions. La principale innovation est ici * 2010 AMS Classification 35A15, 49J40, 49N60 † Department of Mathematics, University of Surrey, Guildford, Surrey, GU2 7XH, UK. tel: +44 (0)1483 682620. email: j.bevan@surrey.ac.uk 1 Double-covering stationary points of a constrained Dirichlet Energy 2 d'exprimer I(u) sous forme d'une fonction auxiliaire G(u − u dc ), avec laquelle nous montrons que u dc est un point stationnaire de I en A, et que u dc est un minimiseur global de l'énergie de Dirichlet parmi les membres de A dont la décomposition de Fourier peutêtre contrôlée d'une manire détaillée dans l'article. En construisant des variations autour de u dc en A par des techniques variationnelles, nous montronségalement que u dc est un minimiseur local parmi les variations dont la tangente ψ de u dc vers A obéissentà G(ψ o ) > 0, où ψ o est la partie impaire de ψ. Additionnellement, un multiplicateur de Lagrange correspondantà la contrainte det ∇u = 1 est identifiée par une analyse qui exploite la dualité de Fefferman-Stein.

The properties of convex representatives of polyconvex functions W : R2×2 → R are studied. The structure of the largest convex representative ϕW : R5 → R is given, and this is used to derive a representation formula for ϕW when W is C1 and satisfies a mild growth condition. The regularity of ϕW is discussed when W is C1; in particular, when W is C1, grows super-quadratically and is strictly polyconvex, it is shown that W has a C1 convex representative.

Abstract. For a class of 2-D elastic energies we show that a radial equilibrium solution is the unique global minimizer in a subclass of all admissible maps. The boundary constraint is a double cover of S 1 ; the minimizer u is C 1 and is such that det ∇u vanishes at one point.Mathematics Subject Classification. 49K15, 49K20, 49J30, 74B20.

Dehydroacetic acid is a common pyrone derivative used commercially as an antibacterial and antifungal agent. Based on the synthesis of dehydroacetic acid (1) from N-hydroxysuccinimdyl acetoacetate, a novel series of enamine-based derivatives were synthesised in order to improve the antibacterial activity of dehydroacetic acid. The antibacterial activities of the synthesised analogues were evaluated against Escherichia coli and Staphylococcus aureus. Derivative 4d (N-Ph) was identified as the most potent inhibitor of S. aureus growth. Overall, derivative 4b (N-Me) showed the best broad-spectrum activity with five-fold greater minimum inhibitory concentration and 11-fold greater minimum biocidal concentration against E. coli when compared to dehydroacetic acid, in addition to improved antibacterial activity against S. aureus.