A new div-curl result. Applications to the homogenization of elliptic systems and to the weak continuity of the Jacobian

In this paper a new div-curl result is established in an open set $\Omega$ of $\mathbb{R}^N$, $N\geq 2$, for the product of two sequences of vector-valued functions which are bounded respectively in $L^p(\Omega)^N$ and $L^q(\Omega)^N$, with ${1/p}+{1/q}=1+{1/(N-1)}$, and whose respectively divergence and curl are compact in suitable spaces. We also assume that the product converges weakly in $W^{-1,1}(\Omega)$. The key ingredient of the proof is a compactness result for bounded sequences in $W^{1,q}(\Omega)$, based on the imbedding of $W^{1,q}(S\_{N-1})$ into $L^{p'}(S\_{N-1})$ ($S\_{N-1}$ the unit sphere of $\mathbb{R}^N$) through a suitable selection of annuli on which the gradients are not too high, in the spirit of De Giorgi and Manfredi. The div-curl result is applied to the homogenization of equi-coercive systems whose coefficients are equi-bounded in $L^\rho(\Omega)$ for some \rho\textgreater{}{N-1\over 2} if N\textgreater{}2, or in $L^1(\Omega)$ if $N=2$. It also allows us to prove a weak continuity result for the Jacobian for bounded sequences in $W^{1,N-1}(\Omega)$ satisfying an alternative assumption to the $L^\infty$-strong estimate of Brezis and Nguyen. Two examples show the sharpness of the results

Analysis of tourism competitiveness and the key influencers

The aim of this research is measure and analyse the tourism competitiveness, which has become one of the great challenges of tourism researchers in recent years. We analyze the relationship between tourism competitiveness and major tourist magnitudes, relations with most tourist destinations, and some pillars that measure competitiveness such as sustainability. A methodology for calculating an indicator of tourism competitiveness based on a double reference point is proposed, taking into account a level of aspiration and reserve level for each pillar of competitiveness. They are considered different degrees of offset between the pillars so that a series of synthetic indices are calculated. Later the rankings obtained for each country are analyzed and conducted a series of analyses. This tool provides managers with a useful and accurate tool, an interactive flexible and easy to use multi-criteria decision on their part. Allows simulation of situations, monitoring and control of various indicators. Later statistical tools for analyzing and comparing tourism competitiveness and other quantities are used.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

Asymptotic behavior of an elastic beam fixed on a small part of one of its extremities

We study the asymptotic behavior of the solution of an anisotropic, heterogeneous, linearized elasticity problem in a cylinder whose diameter $\epsilon$ tends to zero. The cylinder is assumed to be fixed (homogeneous Dirichlet boundary condition) on the whole of one of its extremities, but only on a small part (of size $\epsilon r^\epsilon$) of the second one; the Neumann boundary condition is assumed on the remainder of the boundary. We show that the result depends on $r^\epsilon$, and that there are 3 critical sizes, namely $r^\epsilon=\epsilon^3$, $r^\epsilon=\epsilon$, and $r^\epsilon=\epsilon^{1/3}$, and in total 7 different regimes. We also prove a corrector result for each behavior of $r^\epsilon$.Comment: Preliminary version of a Note to be published in a slightly abbreviated form in C. R. Acad. Sci. Paris, Ser. I, 338 (2004), pp. 975-98