Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces

We obtain an improved Sobolev inequality in H^s spaces involving Morrey norms. This refinement yields a direct proof of the existence of optimizers and the compactness up to symmetry of optimizing sequences for the usual Sobolev embedding. More generally, it allows to derive an alternative, more transparent proof of the profile decomposition in H^s obtained in [P. Gerard, ESAIM 1998] using the abstract approach of dislocation spaces developed in [K. Tintarev & K. H. Fieseler, Imperial College Press 2007]. We also analyze directly the local defect of compactness of the Sobolev embedding in terms of measures in the spirit of [P. L. Lions, Rev. Mat. Iberoamericana 1985]. As a model application, we study the asymptotic limit of a family of subcritical problems, obtaining concentration results for the corresponding optimizers which are well known when s is an integer ([O. Rey, Manuscripta math. 1989; Z.-C. Han, Ann. Inst. H. Poincare Anal. Non Lineaire 1991], [K. S. Chou & D. Geng, Differential Integral Equations 2000]).Comment: 33 page

Fractional superharmonic functions and the Perron method for nonlinear integro-differential equations

We deal with a class of equations driven by nonlocal, possibly degenerate, integro-differential operators of differentiability order $s\in (0,1)$ and summability growth $p>1$, whose model is the fractional $p$-Laplacian with measurable coefficients. We state and prove several results for the corresponding weak supersolutions, as comparison principles, a priori bounds, lower semicontinuity, and many others. We then discuss the good definition of $(s,p)$-superharmonic functions, by also proving some related properties. We finally introduce the nonlocal counterpart of the celebrated Perron method in nonlinear Potential Theory.Comment: To appear in Math. An

Nonlocal Harnack inequalities

We state and prove a general Harnack inequality for minimizers of nonlocal, possibly degenerate, integro-differential operators, whose model is the fractional p-Laplacian.Comment: To appear in J. Funct. Ana

Local behavior of fractional pp-minimizers

We extend the De Giorgi-Nash-Moser theory to nonlocal, possibly degenerate integro-differential operators.Comment: 26 pages. To appear in Ann. Inst. H. Poincare Anal. Non Lineaire. arXiv admin note: text overlap with arXiv:1405.784

Global estimates for nonlinear parabolic equations

We consider nonlinear parabolic equations of the type $$u_t - div a(x, t, Du)= f(x,t) on \Omega_T = \Omega\times (-T,0),$$ under standard growth conditions on $a$, with $f$ only assumed to be integrable. We prove general decay estimates up to the boundary for level sets of the solutions $u$ and the gradient $Du$ which imply very general estimates in Lebesgue and Lorentz spaces. Assuming only that the involved domains satisfy a mild exterior capacity density condition, we provide global regularity results.Comment: To appear in J. Evol. Equation

A supervised clustering approach for fMRI-based inference of brain states

We propose a method that combines signals from many brain regions observed in functional Magnetic Resonance Imaging (fMRI) to predict the subject's behavior during a scanning session. Such predictions suffer from the huge number of brain regions sampled on the voxel grid of standard fMRI data sets: the curse of dimensionality. Dimensionality reduction is thus needed, but it is often performed using a univariate feature selection procedure, that handles neither the spatial structure of the images, nor the multivariate nature of the signal. By introducing a hierarchical clustering of the brain volume that incorporates connectivity constraints, we reduce the span of the possible spatial configurations to a single tree of nested regions tailored to the signal. We then prune the tree in a supervised setting, hence the name supervised clustering, in order to extract a parcellation (division of the volume) such that parcel-based signal averages best predict the target information. Dimensionality reduction is thus achieved by feature agglomeration, and the constructed features now provide a multi-scale representation of the signal. Comparisons with reference methods on both simulated and real data show that our approach yields higher prediction accuracy than standard voxel-based approaches. Moreover, the method infers an explicit weighting of the regions involved in the regression or classification task

Use of larvae of the wax moth Galleria mellonella as an in vivo model to study the virulence of Helicobacter pylori

BACKGROUND: Helicobacter pylori is the first bacterium formally recognized as a carcinogen and is one of the most successful human pathogens, as over half of the world’s population is colonized by the bacterium. H. pylori-induced gastroduodenal disease depends on the inflammatory response of the host and on the production of specific bacterial virulence factors. The study of Helicobacter pylori pathogenic action would greatly benefit by easy-to-use models of infection. RESULTS: In the present study, we examined the effectiveness of the larvae of the wax moth Galleria mellonella as a new model for H. pylori infection. G. mellonella larvae were inoculated with bacterial suspensions or broth culture filtrates from either different wild-type H. pylori strains or their mutants defective in specific virulence determinants, such as VacA, CagA, CagE, the whole pathogenicity island (PAI) cag, urease, and gamma-glutamyl transpeptidase (GGT). We also tested purified VacA cytotoxin. Survival curves were plotted using the Kaplan-Meier method and LD(50) lethal doses were calculated. Viable bacteria in the hemocoel were counted at different time points post-infection, while apoptosis in larval hemocytes was evaluated by annexin V staining. We found that wild-type and mutant H. pylori strains were able to survive and replicate in G. mellonella larvae which underwent death rapidly after infection. H. pylori mutant strains defective in either VacA, or CagA, or CagE, or cag PAI, or urease, but not GGT-defective mutants, were less virulent than the respective parental strain. Broth culture filtrates from wild-type strains G27 and 60190 and their mutants replicated the effects observed using their respective bacterial suspension. Also, purified VacA cytotoxin was able to kill the larvae. The killing of larvae always correlated with the induction of apoptosis in hemocytes. CONCLUSIONS: G. mellonella larvae are susceptible to H. pylori infection and may represent an easy to use in vivo model to identify virulence factors and pathogenic mechanisms of H. pylori. The experimental model described can be useful to screen a large number of clinical H. pylori strain and to correlate virulence of H. pylori strains with patients’ disease status