Multiple Solutions for the Non-Abelian Chern--Simons--Higgs Vortex Equations

In this paper we study the existence of multiple solutions for the non-Abelian Chern--Simons--Higgs $(N\times N)$-system: \Delta u_i=\lambda\left(\sum_{j=1}^N\sum_{k=1}^N K_{kj}K_{ji}\re^{u_j}\re^{u_k}-\sum_{j=1}^N K_{ji}\re^{u_j}\right)+4\pi\sum_{j=1}^{n_i}\delta_{p_{ij}},\quad i=1,\dots, N; over a doubly periodic domain $\Omega$, with coupling matrix $K$ given by the Cartan matrix of $SU(N+1),$ (see \eqref{k1} below). Here, $\lambda>0$ is the coupling parameter, $\delta_p$ is the Dirac measure with pole at $p$ and $n_i\in \mathbb{N},$ for $i=1, \dots, N.$ When $N=1, 2$ many results are now available for the periodic solvability of such system and provide the existence of different classes of solutions known as: topological, non-topological, mixed and blow-up type. On the contrary for $N\ge 3,$ only recently in \cite{haya1} the authors managed to obtain the existence of one doubly periodic solution via a minimisation procedure, in the spirit of \cite{nota} . Our main contribution in this paper is to show (as in \cite{nota}) that actually the given system admits a second doubly periodic solutions of "Mountain-pass" type, provided that $3\le N\le 5$. Note that the existence of multiple solutions is relevant from the physical point of view. Indeed, it implies the co-existence of different non-Abelian Chern--Simons condensates sharing the same set (assigned component-wise) of vortex points, energy and fluxes. The main difficulty to overcome is to attain a "compactness" property encompassed by the so called Palais--Smale condition for the corresponding "action" functional, whose validity remains still open for $N\ge 6$.Comment: 34 page

On non-topological solutions for planar Liouville Systems of Toda-type

Motivated by the study of non abelian Chern Simons vortices of non topological type in Gauge Field Theory, we analyse the solvability of planar Liouville systems of Toda type in presence of singular sources. We identify necessary and sufficient conditions on the "flux" pair which ensure the radial solvability of the system. Since the given system includes the (integrable) 2 X 2 Toda system as a particular case, thus we recover the existence result available in this case. Our method relies on a blow-up analysis, which even in the radial setting, takes new turns compared with the single equation case

Radial symmetry and symmetry breaking for some interpolation inequalities

We analyze the radial symmetry of extremals for a class of interpolation inequalities known as Caffarelli-Kohn-Nirenberg inequalities, and for a class of weighted logarithmic Hardy inequalities which appear as limiting cases of the first ones. In both classes we show that there exists a continuous surface that splits the set of admissible parameters into a region where extremals are symmetric and a region where symmetry breaking occurs. In previous results, the symmetry breaking region was identified by showing the linear instability of the radial extremals. Here we prove that symmetry can be broken even within the set of parameters where radial extremals correspond to local minima for the variational problem associated with the inequality. For interpolation inequalities, such a symmetry breaking phenomenon is entirely new

The role of subjective memory complaints in predicting cognitive impairment associated with future Alzheimer’s disease: a community based study

In recent years there has been a substantial increase in research examining the role of subjective memory complaints (SMC) in cognitive function and Alzheimer’s disease. These studies have related SMC to many different cognitive outcomes, such as retaining normal cognitive function, a fluctuating cognitive performance and the development of Alzheimer’s disease. Most of these studies have focused on older populations and have employed a limited assessment of cognitive function. This limits the available evidence regarding the clinical utility of SMC. The literature on the role of SMC in younger subjects is scarce. It is not known whether memory complaints are useful in predicting future cases of Alzheimer’s disease in younger community-based subjects. Aims: The main aim of the present study was to determine whether SMC predict the development of cognitive impairment in a younger cohort of subjects, many of whom were under the age of 70 years (73%), based on their risk profile and neuropsychological assessment. A further aim was to ascertain whether the DRS or 7MS are sensitive screening tools for MCI and examine whether the presence of SMC affects the 3-year cognitive outcome of subjects. To address these aims, this study consisted of two parts: a cross-sectional design and a longitudinal follow-up component. Methods: This study was carried out with 86 community-dwelling subjects recruited via advertisement within the catchment area of Central Sydney Area Health Service. The mean age of the subjects was 63.1 years (SD=8.4). Subjective memory complaints were assessed using a single question. Cognitive function was assessed using a comprehensive battery of tests, selected on the basis of their sensitivity to identifying cognitive impairment typically associated with Alzheimer’s disease. After the initial analysis between those with SMC and without SMC, subjects were further classified according to their performance on an episodic memory task (i.e., delayed verbal recall, Rey, 1964) as having normal memory function, SMC or aMCI. Results: Part 1 - Subjective memory complaints (SMC) were reported by 63% of the sample. The initial analysis between subjects with SMC (n=54) and without SMC (n=32) suggested an initial relationship between SMC and cognitive functioning. Subjects with SMC had impaired global cognitive functioning on two brief screening tests (7MS and DRS), working memory, verbal recall and visuomotor speed. However, subsequent screening with the delayed verbal recall test showed that 12 of the 54 subjects with SMC demonstrated significant cognitive impairment, scoring 2 SD below the control group mean. After these subjects were removed to form the aMCI group, the cognitive differences between subjects with SMC and without SMC were no longer apparent. Subjects with aMCI showed evidence of multiple cognitive deficits (below 1 SD of control group mean) with a high percentage of subjects demonstrating impairment on tests of verbal learning, verbal recall, verbal ability and visuomotor speed. Further analysis showed a significant association between age and subjects identified as having SMC (r=-.581, p<.001) and aMCI (r=.692, p<.001). From the age of 60 onwards, both the SMC and aMCI groups demonstrated a more rapid cognitive decline with increasing age in several cognitive domains. Part 2 - After a mean interval of 3.2 years, 43 subjects were followed up. Subjects with aMCI showed evidence of greater decline on both screening tests (7MS; DRS), whilst the SMC group had significantly higher scores. This trend was also apparent with other neuropsychological testing. The analysis of change over time in cognitive function showed that the majority of subjects (both SMC aMCI) either remained stable or improved their cognitive performance. It is likely that the small sample size and short follow-up interval of the present study contributed to the present observation of no change in cognitive function over time. Discussion: The present findings suggest that subjective memory complaints are a poor predictor of cognitive function. In isolation, SMC are unlikely to be useful for identifying cases with significant cognitive impairment. This is particularly relevant for subjects under the age of 70 years. However, for subjects over the age of 70 years, SMC are likely to identify significant cases with neuropsychological assessment (such as animal fluency and delayed recall). Conclusion: The present study showed that SMC are a poor predictor of cognitive function in subjects under the age of 70 years. This study provided evidence that selected and relatively quick to administer formal neuropsychological tests of cognitive function (in particular tests of animal fluency and delayed recall) are better able to identify those at risk of developing cognitive impairment associated with Alzheimer’s disease, at an earlier age. This would thus allow exposure to earlier treatment options, such as donepezil, aricept, vitamin E, and memantine”

Chern--Simons Vortices in the Gudnason Model

We present a series of existence theorems for multiple vortex solutions in the Gudnason model of the ${\cal N}=2$ supersymmetric field theory where non-Abelian gauge fields are governed by the pure Chern--Simons dynamics at dual levels and realized as the solutions of a system of elliptic equations with exponential nonlinearity over two-dimensional domains. In the full plane situation, our method utilizes a minimization approach, and in the doubly periodic situation, we employ an-inequality constrained minimization approach. In the latter case, we also obtain sufficient conditions under which we show that there exist at least two gauge-distinct solutions for any prescribed distribution of vortices. In other words, there are distinct solutions with identical vortex distribution, energy, and electric and magnetic charges.Comment: 39 page

On the symmetry of extremals for the Caffarelli-Kohn-Nirenberg inequalities

In this paper we prove some new symmetry results for the extremals of the Caffarelli-Kohn-Nirenberg inequalities, in any dimension larger or equal than two

Weighted Barycentric Sets and Singular Liouville Equations on Compact Surfaces

Given a closed two dimensional manifold, we prove a general existence result for a class of elliptic PDEs with exponential nonlinearities and negative Dirac deltas on the right-hand side, extending a theory recently obtained for the regular case. This is done by global methods: since the associated Euler functional is in general unbounded from below, we need to define a new model space, generalizing the so-called space of formal barycenters and characterizing (up to homotopy equivalence) its very low sublevels. As a result, the analytic problem is reduced to a topological one concerning the contractibility of this model space. To this aim, we prove a new functional inequality in the spirit of [16] and then we employ a min-max scheme based on a cone-style construction, jointly with the blow-up analysis given in [5] (after [6] and [8]). This study is motivated by abelian Chern- Simons theory in self-dual regime, or from the problem of prescribing the Gaussian curvature in presence of conical singularities (hence generalizing a problem raised by Kazdan and Warner in [26]).Comment: to appear on Journal of Functional Analysis. One proof in Section 3 has been simplified with respect to the previous version, while Section 6 (on open problems) has been substantially improved. At the end on the Introduction, the complete proof of the topological conjecture given in Section 6 (to appear in a forthcoming paper) is announce