1,720,960 research outputs found
Hardy-Sobolev inequalities with singularities on non smooth boundary. Part 2: Influence of the global geometry in small dimensions
No full textWe consider Hardy-Sobolev nonlinear equations on domains with singularities. We introduced this problem in Cheikh-Ali [4]. Under a local geometric hypothesis, namely that the generalized mean curvature is negative (see (7) below), we proved the existence of extremals for the relevant Hardy-Sobolev inequality for large dimensions. In the present paper, we tackle the question of small dimensions that was left open. We introduce a “mass”, that is a global quantity, the positivity of which ensures the existence of extremals in small dimensions. As a byproduct, we prove the existence of solutions to a perturbation of the initial equation via the Mountain-Pass Lemma.SCOPUS: ar.jinfo:eu-repo/semantics/inPres
{A}nalyse asymptotique des équations de {H}ardy-{S}obolev dans des espaces singuliers
No full textIn this manuscript, divided into 3 parts, we study the existence of extremal for Hardy-Sobolev inequalities.Part 1: We obtain the (non-)existence of singulars solutions for the perturbative Hardy-Schrödinger equation on a non-smooth domain with the singular point 0 on the boundary of the domain. In particular, we introduce a geometric quantity G which generalizes the mean curvature for ”Large dimensions” and the new notion of the mass in ”Small dimensions”. Our main result is that, in the case of a subcritical perturbation, an interaction appears between the perturbation and G at 0 (resp. m) for large dimensions (resp. small dimensions). In addition, the negativity of the curvature G (resp. the positivity of the mass m) for the large dimensions (resp. small dimensions) is sufficient when the perturbation has no effect.Part 2: In this part, we perform a blow-up analysis of solutions for the Hardy-Sobolev equation of minimizing type. First, we obtain an optimal control of the family of solutions. After, we get specific informations about the blowup point using a Pohozaev identity.Part 3: We consider the best constant in a critical Sobolev inequality of second order. We show non-rigidity for the optimizers above a certain threshold, namely, we prove that the best constant is achieved by a nonconstant solution of the associated fourth order elliptic problem under Neumann boundary conditions. Our arguments rely on asymptotic estimates of the Rayleigh quotient. We also show rigidity below another threshold.Dans ce manuscrit, divisé en 3 parties, nous étudions des extrémales d’inégalités de Hardy-Sobolev. Partie 1 : Nous obtenons l’existence de solutions singulières pour l’équation de Hardy-Schrödinger perturbée ou non sur un domaine non régulier avec le point singulier 0 de l’équation se trouvant sur le bord du domaine. En particulier, nous introduisons une courbure géométrique G qui généralise la courbure moyenne pour les ”grandes dimensions” et une notion nouvelle de masse m pour les ”petites dimensions”. Notre résultat principal est que dans le cas d’un potentiel variable du terme perturbatif sous-critique, une interaction entre perturbation et G en 0 (resp. m) dans le cas grandes dimensions (resp. petites dimensions) apparait. En plus, la négativité de la courbure G (resp. la positivité de la masse m) pour les grandes dimensions (resp. petites dimensions) est suffisant lorsque la perturbation n’a aucun effet. .Partie 2 : Dans cette partie, nous travaillons sur l’analyse asymptotique des sous-extrémales explosives. Nous effectuons une analyse de blow-up pour une équation de Hardy-Sobolev. Dans un premier temps, nous obtenons un contrôle ponctuel optimal de la suite de solutions. Dans un second temps, nous obtenons des informations précises sur le point d’explosion en utilisant une identité de Pohozaev. Partie 3 : Nous considérons la meilleure constante dans une inégalité critique de second ordre de Sobolev. Nous montrons la non-rigidité pour les optimiseurs au-dessus d’un certain seuil, à savoir nous prouvons que la meilleure constante est atteinte par une solution non constante du problème elliptique de quatrième ordre sous des conditions limites de type Neumann. Nos arguments reposent sur des estimations asymptotiques du quotient de Rayleigh. Nous montrons également la rigidité en dessous d’un autre seuil pour les solutions de moindre énergie
Hardy–Sobolev inequalities with singularities on non smooth boundary: Hardy constant and extremals. Part I: Influence of local geometry
No full textLet Ω be a domain of Rn, n≥3. The classical Caffarelli–Kohn–Nirenberg inequality rewrites as the following inequality: for any s∈[0,2] and any γ0 such that (HS)∫Ω[Formula presented]dx[Formula presented]≤K(Ω,γ,s)∫ΩSCOPUS: ar.jDecretOANoAutActifinfo:eu-repo/semantics/publishe
Ground-state blowing-up solutions for a Hardy-Sobolev equations on a manifold
No full textVersion à paraître au "Journal of Geometric Analysis"International audienceWe prove the existence of blowing-up families of solutions to an equation of Hardy-Sobolev type in high dimensions. These families are of minimal type. The sole condition is that the potential of the linear operator touches a critical potential at the singular point. This condition is sharp as shown by the first author in [Cheikh-Ali, Pacific J. of Math. 2022]
Going Beyond Counting First Authors in Author Co-citation Analysis
Get PDFThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Variations on the Author
Get PDF“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
Asymptotic Analysis of Hardy-Sobolev equations in singular spaces
No full textDans ce manuscrit, divisé en 3 parties, nous étudions des extrémales d’inégalités de Hardy-Sobolev. Partie 1 : Nous obtenons l’existence de solutions singulières pour l’équation de Hardy-Schrödinger perturbée ou non sur un domaine non régulier avec le point singulier 0 de l’équation se trouvant sur le bord du domaine. En particulier, nous introduisons une courbure géométrique G qui généralise la courbure moyenne pour les ”grandes dimensions” et une notion nouvelle de masse m pour les ”petites dimensions”. Notre résultat principal est que dans le cas d’un potentiel variable du terme perturbatif sous-critique, une interaction entre perturbation et G en 0 (resp. m) dans le cas grandes dimensions (resp. petites dimensions) apparait. En plus, la négativité de la courbure G (resp. la positivité de la masse m) pour les grandes dimensions (resp. petites dimensions) est suffisant lorsque la perturbation n’a aucun effet. Partie 2 : Dans cette partie, nous travaillons sur l’analyse asymptotique des sous-extrémales explosives. Nous effectuons une analyse de blow-up pour une équation de Hardy-Sobolev. Dans un premier temps, nous obtenons un contrôle ponctuel optimal de la suite de solutions. Dans un second temps, nous obtenons des informations précises sur le point d’explosion en utilisant une identité de Pohozaev. Partie 3 : Nous considérons la meilleure constante dans une inégalité critique de second ordre de Sobolev. Nous montrons la non-rigidité pour les optimiseurs au-dessus d’un certain seuil, à savoir nous prouvons que la meilleure constante est atteinte par une solution non constante du problème elliptique de quatrième ordre sous des conditions limites de type Neumann. Nos arguments reposent sur des estimations asymptotiques du quotient de Rayleigh. Nous montrons également la rigidité en dessous d’un autre seuil pour les solutions de moindre énergie.In this manuscript, divided into 3 parts, we study the existence of extremal for Hardy-Sobolev inequalities. Part 1: We obtain the (non-)existence of singulars solutions for the perturbative Hardy-Schrödinger equation on a non-smooth domain with the singular point 0 on the boundary of the domain. In particular, we introduce a geometric quantity G which generalizes the mean curvature for ”Large dimensions” and the new notion of the mass in ”Small dimensions”. Our main result is that, in the case of a subcritical perturbation, an interaction appears between the perturbation and G at 0 (resp. m) for large dimensions (resp. small dimensions). In addition, the negativity of the curvature G (resp. the positivity of the mass m) for the large dimensions (resp. small dimensions) is sufficient when the perturbation has no effect. Part 2: In this part, we perform a blow-up analysis of solutions for the Hardy-Sobolev equation of minimizing type. First, we obtain an optimal control of the family of solutions. After, we get specific informations about the blowup point using a Pohozaev identity. Part 3: We consider the best constant in a critical Sobolev inequality of second order. We show non-rigidity for the optimizers above a certain threshold, namely, we prove that the best constant is achieved by a nonconstant solution of the associated fourth order elliptic problem under Neumann boundary conditions. Our arguments rely on asymptotic estimates of the Rayleigh quotient. We also show rigidity below another threshold
Appropriate Similarity Measures for Author Cocitation Analysis
Get PDFWe provide a number of new insights into the methodological discussion about author cocitation analysis. We first argue that the use of the Pearson correlation for measuring the similarity between authors’ cocitation profiles is not very satisfactory. We then discuss what kind of similarity measures may be used as an alternative to the Pearson correlation. We consider three similarity measures in particular. One is the well-known cosine. The other two similarity measures have not been used before in the bibliometric literature. Finally, we show by means of an example that our findings have a high practical relevance.information science;Pearson correlation;cosine;similarity measure;author cocitation analysis
THE SECOND BEST CONSTANT FOR THE HARDY-SOBOLEV INEQUALITY ON MANIFOLDS
No full textWe consider the second best constant in the Hardy-Sobolev inequality on a Riemannian manifold. More precisely, we are interested in the existence of extremal functions for this inequality. This problem was tackled by Djadli and Druet (Calc. Var. Partial Differential Equations 12 (2001), 59-84) for Sobolev inequalities. Here, we establish the corresponding result for the singular case. In addition, we perform a blow-up analysis of solutions to Hardy-Sobolev equations of minimizing type. This yields information on the value of the second best constant in the related Riemannian functional inequality.SCOPUS: ar.jDecretOANoAutActifinfo:eu-repo/semantics/publishe
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