A threshold phenomenon for embeddings of H0mH^m_0 into Orlicz spaces

We consider a sequence of positive smooth critical points of the Adams-Moser-Trudinger embedding of $H^m_0$ into Orlicz spaces. We study its concentration-compactness behavior and show that if the sequence is not precompact, then the liminf of the $H^m_0$-norms of the functions is greater than or equal to a positive geometric constant.Comment: 14 Page

The Construction of a Partially Regular Solution to the Landau-Lifshitz-Gilbert Equation in R2\mathbb{R}^2

We establish a framework to construct a global solution in the space of finite energy to a general form of the Landau-Lifshitz-Gilbert equation in $\mathbb{R}^2$. Our characterization yields a partially regular solution, smooth away from a 2-dimensional locally finite Hausdorff measure set. This construction relies on approximation by discretization, using the special geometry to express an equivalent system whose highest order terms are linear and the translation of the machinery of linear estimates on the fundamental solution from the continuous setting into the discrete setting. This method is quite general and accommodates more general geometries involving targets that are compact smooth hypersurfaces.Comment: 43 pages, 2 figure

Quantization for an elliptic equation of order 2m with critical exponential non-linearity

On a smoothly bounded domain $\Omega\subset\R{2m}$ we consider a sequence of positive solutions $u_k\stackrel{w}{\rightharpoondown} 0$ in $H^m(\Omega)$ to the equation $(-\Delta)^m u_k=\lambda_k u_k e^{mu_k^2}$ subject to Dirichlet boundary conditions, where $0<\lambda_k\to 0$. Assuming that $$\Lambda:=\lim_{k\to\infty}\int_\Omega u_k(-\Delta)^m u_k dx<\infty,$$ we prove that $\Lambda$ is an integer multiple of \Lambda_1:=(2m-1)!\vol(S^{2m}), the total $Q$-curvature of the standard $2m$-dimensional sphere.Comment: 33 page

Existence of solutions to a higher dimensional mean-field equation on manifolds

For $m\geq 1$ we prove an existence result for the equation $$(-\Delta_g)^m u+\lambda=\lambda\frac{e^{2mu}}{\int_M e^{2mu}d\mu_g}$$ on a closed Riemannian manifold $(M,g)$ of dimension $2m$ for certain values of $\lambda$.Comment: 15 Page

Green's functions for parabolic systems of second order in time-varying domains

We construct Green's functions for divergence form, second order parabolic systems in non-smooth time-varying domains whose boundaries are locally represented as graph of functions that are Lipschitz continuous in the spatial variables and 1/2-H\"older continuous in the time variable, under the assumption that weak solutions of the system satisfy an interior H\"older continuity estimate. We also derive global pointwise estimates for Green's function in such time-varying domains under the assumption that weak solutions of the system vanishing on a portion of the boundary satisfy a certain local boundedness estimate and a local H\"older continuity estimate. In particular, our results apply to complex perturbations of a single real equation.Comment: 25 pages, 0 figur

New improved Moser-Trudinger inequalities and singular Liouville equations on compact surfaces

We consider a singular Liouville equation on a compact surface, arising from the study of Chern-Simons vortices in a self dual regime. Using new improved versions of the Moser-Trudinger inequalities (whose main feature is to be scaling invariant) and a variational scheme, we prove new existence results.Comment: to appear in GAF

Global compactness results for nonlocal problems

International audienceWe obtain a Struwe type global compactness result for a class of nonlinear nonlocal problems involving the fractional $p-$Laplacian operator and nonlinearities at critical growth

Theorems on existence and global dynamics for the Einstein equations

This article is a guide to theorems on existence and global dynamics of solutions of the Einstein equations. It draws attention to open questions in the field. The local-in-time Cauchy problem, which is relatively well understood, is surveyed. Global results for solutions with various types of symmetry are discussed. A selection of results from Newtonian theory and special relativity that offer useful comparisons is presented. Treatments of global results in the case of small data and results on constructing spacetimes with prescribed singularity structure or late-time asymptotics are given. A conjectural picture of the asymptotic behaviour of general cosmological solutions of the Einstein equations is built up. Some miscellaneous topics connected with the main theme are collected in a separate section.Comment: Submitted to Living Reviews in Relativity, major update of Living Rev. Rel. 5 (2002)

Modelling the impact of toxic and disturbance stress on white-tailed eagle (Haliaeetus albicilla) populations

Several studies have related breeding success and survival of sea eagles to toxic or non-toxic stress separately. In the present investigation, we analysed single and combined impacts of both toxic and disturbance stress on populations of white-tailed eagle (Haliaeetus albicilla), using an analytical single-species model. Chemical and eco(toxico)logical data reported from laboratory and field studies were used to parameterise and validate the model. The model was applied to assess the impact of ∑PCB, DDE and disturbance stress on the white-tailed eagle population in The Netherlands. Disturbance stress was incorporated through a 1.6% reduction in survival and a 10–50% reduction in reproduction. ∑PCB contamination from 1950 up to 1987 was found to be too high to allow the return of white-tailed eagle as a breeding species in that period. ∑PCB and population trends simulated for 2006–2050 suggest that future population growth is still reduced. Disturbance stress resulted in a reduced population development. The combination of both toxic and disturbance stress varied from a slower population development to a catastrophical reduction in population size, where the main cause was attributed to the reduction in reproduction of 50%. Application of the model was restricted by the current lack of quantitative dose–response relationships between non-toxic stress and survival and reproduction. Nevertheless, the model provides a first step towards integrating and quantifying the impacts of multiple stressors on white-tailed eagle populations

The Intensity of IUGR-Induced Transcriptome Deregulations Is Inversely Correlated with the Onset of Organ Function in a Rat Model

A low-protein diet applied during pregnancy in the rat results in intrauterine growth restricted (IUGR) fetuses. In humans, IUGR is associated with increased perinatal morbidity, higher incidence of neuro-developmental defects and increased risk of adult metabolic anomalies, such as diabetes and cardiovascular disease. Development and function of many organs are affected by environmental conditions such as those inducing fetal and early postnatal growth restriction. This phenomenon, termed “fetal programming” has been studied unconnectedly in some organs, but very few studies (if any) have investigated at the same time several organs, on a more comparative basis. However, it is quite probable that IUGR affects differentially most organ systems, with possible persistent changes in gene expression. In this study we address transcriptional alterations induced by IUGR in a multi-organ perspective, by systematic analysis of 20-days rat fetuses. We show that (1) expressional alterations are apparently stronger in organs functioning late in foetal or postnatal life than in organs that are functioning early (2) hierarchical classification of the deregulations put together kidney and placenta in one cluster, liver, lungs and heart in another; (3) the epigenetic machinery is set up especially in the placenta, while its alterations are rather mild in other organs; (4) the genes appear deregulated in chromosome clusters; (5) the altered expression cascades varies from organ to organ, with noticeably a very significant modification of the complement and coagulation cascades in the kidney; (6) we found a significant increase in TF binding site for HNF4 proteins specifically for liver genes that are down-regulated in IUGR, suggesting that this decrease is achieved through the action of HNF transcription factors, that are themselves transcriptionnally induced in the liver by IUGR (x 1.84 fold). Altogether, our study suggests that a combination of tissue-specific mechanisms contributes to bring about tissue-driven modifications of gene cascades. The question of these cascades being activated to adapt the organ to harsh environmental condition, or as an endpoint consequence is still raised