Sign-changing blow-up for scalar curvature type equations

Given $(M,g)$ a compact Riemannian manifold of dimension $n\geq 3$, we are interested in the existence of blowing-up sign-changing families (\ue)_{\eps>0}\in C^{2,\theta}(M), $\theta\in (0,1)$, of solutions to \Delta_g \ue+h\ue=|\ue|^{\frac{4}{n-2}-\eps}\ue\hbox{ in }M\,, where $\Delta_g:=-\hbox{div}_g(\nabla)$ and $h\in C^{0,\theta}(M)$ is a potential. We prove that such families exist in two main cases: in small dimension $n\in \{3,4,5,6\}$ for any potential $h$ or in dimension $3\leq n\leq 9$ when h\equiv\frac{n-2}{4(n-1)}\Scal_g. These examples yield a complete panorama of the compactness/noncompactness of critical elliptic equations of scalar curvature type on compact manifolds. The changing of the sign is necessary due to the compactness results of Druet and Khuri--Marques--Schoen

Sign-changing solutions to elliptic second order equations: glueing a peak to a degenerate critical manifold

We construct blowing-up sign-changing solutions to some nonlinear critical equations by glueing a standard bubble to a degenerate function. We develop a method based on analyticity to perform the glueing when the critical manifold of solutions is degenerate and no Bianchi--Egnell type condition holds.Comment: Final version to appear in "Calculus of Variations and PDEs

Sobolev inequalities for the Hardy-Schr\"odinger operator: Extremals and critical dimensions

In this expository paper, we consider the Hardy-Schr\"odinger operator $-\Delta -\gamma/|x|^2$ on a smooth domain \Omega of R^n with 0\in\bar{\Omega}, and describe how the location of the singularity 0, be it in the interior of \Omega or on its boundary, affects its analytical properties. We compare the two settings by considering the optimal Hardy, Sobolev, and the Caffarelli-Kohn-Nirenberg inequalities. The latter rewrites: $C(\int_{\Omega}\frac{u^{p}}{|x|^s}dx)^{\frac{2}{p}}\leq \int_{\Omega} |\nabla u|^2dx-\gamma \int_{\Omega}\frac{u^2}{|x|^2}dx$ for all $u\in H^1_0(\Omega)$, where \gamma <n^2/4, s\in [0,2) and p:=2(n-s)/(n-2). We address questions regarding the explicit values of the optimal constant C, as well as the existence of non-trivial extremals attached to these inequalities. Scale invariance properties lead to situations where the best constants do not depend on the domain and are not attainable. We consider two different approaches to "break the homogeneity" of the problem: One approach was initiated by Brezis-Nirenberg and by Janelli. It is suitable for the case where 0 is in the interior of \Omega, and consists of considering lower order perturbations of the critical nonlinearity. The other approach was initiated by Ghoussoub-Kang , C.S. Lin et al. and Ghoussoub-Robert. It consists of considering domains where the singularity is on the boundary. Both of these approaches are rich in structure and in challenging problems. If 0\in \Omega, a negative linear perturbation suffices for higher dimensions, while a positive "Hardy-singular interior mass" is required in lower dimensions. If the singularity is on the boundary, then the local geometry around 0 plays a crucial role in high dimensions, while a positive "Hardy-singular boundary mass" is needed for the lower dimensions.Comment: Expository paper. 48 page

Asymptotic analysis for fourth order Paneitz equations with critical growth

We investigate fourth order Paneitz equations of critical growth in the case of $n$-dimensional closed conformally flat manifolds, $n \ge 5$. Such equations arise from conformal geometry and are modelized on the Einstein case of the geometric equation describing the effects of conformal changes of metrics on the $Q$-curvature. We obtain sharp asymptotics for arbitrary bounded energy sequences of solutions of our equations from which we derive stability and compactness properties. In doing so we establish the criticality of the geometric equation with respect to the trace of its second order terms.Comment: 35 pages. To appear in "Advances in the Calculus of Variations

Examples of non-isolated blow-up for perturbations of the scalar curvature equation on non locally conformally flat manifolds

Solutions to scalar curvature equations have the property that all possible blow-up points are isolated, at least in low dimensions. This property is commonly used as the first step in the proofs of compactness. We show that this result becomes false for some arbitrarily small, smooth perturbations of the potential.Comment: Final version to appear in J. of Differential Geometry. References updated, details adde

Off-Shoring of Business Services and De-Industrialization: Threat or Opportunity - and for Whom?

This paper takes a new look at the issue of overseas sourcing of services. In framework in which comparative advantage is endogenous to agglomeration economies and factor mobility, the fragmentation of production made possible by the new communication technologies and low transportation costs allow global firms (multinational corporations or individual firms active in global networks) to simultaneously reap the benefit of agglomeration economies in OECD countries and of low wages prevailing in countries with an ever better educated labour force like India. Thus, the reduction of employment in some routine tasks in rich countries in a general equilibrium helps sustain and reinforces employment in the core competencies in such countries. That is, the loss of some jobs permits to retain the 'core competencies' in the 'core countries'. The welfare implications of this analysis are shown to be not as straightforward as in a neoclassical world.Outsourcing, wage inequality, communication costs