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Mohameden Ould Ahmedou

Publications and source records attributed to Mohameden Ould Ahmedou.

13 recordsLinked to original sources
arXiv2014

Morse theory and the resonant Q-curvature problem

In this paper, we study the prescribed $Q$-curvature problem on closed four-dimensional Riemannian manifolds when the total integral of the $Q$-curvature is a positive integer multiple of the one of the four-dimensional round sphere. This problem has a variational structure with a lack of compactness. Using some topological tools of the theory of "critical points at infinity" combined with a refined blow-up analysis and some dynamical arguments, we identify the accumulations points of all noncompact flow lines of a pseudogradient flow, the so called critical points at infinity of the associated variational problem, and associate to them a natural Morse index. We then prove strong Morse type inequalities, extending the full Morse theory to this noncompact variational problem. Finally, we derive from our results Poincaré-Hopf index type criteria for existence, extending known results in the literature and deriving new ones.

math.DG↗
arXiv2013

On the supercritical mean field equation on pierced domains

We consider the problem $$(P_\eps)\qquad Δu +λ{e^{u}\over\int\limits_{Ω\setminus B(ξ,\eps)}e^u}=0\ \hbox{in}\ Ω\setminus B(ξ,\eps),\quad u =0\ \hbox{on}\ \partial\(Ω\setminus B(ξ,\eps)\), $$ where $Ω$ is a smooth bounded open domain in $\rr^2$ which contains the point $ξ.$ We prove that if $λ>8π,$ problem $(P_\eps)$ has a solutions $u_\eps$ such that $$u_\eps(x)\to {8π+ λ\over2} G(x,ξ) \ \hbox{uniformly on compact sets of $Ω\setminus\{ξ\}$}$$ as $\eps$ goes to zero. Here $G$ denotes Green's function of Dirichlet Laplacian in $Ω.$ If $λ\not\in 8π\mathbb N$ we will not make any symmetry assumptions on $Ω,$ while if $λ\in 8π\mathbb N$ we will assume that $Ω$ is invariant under a rotation through an angle ${8π^2\over \la} $ around the point $ξ.$

math.AP↗
arXiv2009

Conformal metrics of prescribed scalar curvature on $4-$manifolds: The degree zero case

In this paper, we consider the problem of existence and multiplicity of conformal metrics on a riemannian compact $4-$dimensional manifold $(M^4,g_0)$ with positive scalar curvature. We prove new exitence criterium which provides existence results for a dense subset of positive functions and generalizes Bahri-Coron and Chang-Gursky-Yang Euler-Hopf type criterium. Our argument gives estimates on the Morse index of the solutions and has the advantage to extend known existence results. Moreover it provides, for generic $K$ {\it Morse Inequalities at Infinity}, which give a lower bound on the number of metrics with prescribed scalar curvature in terms of the topological contribution of its "critical points at Infinity" to the difference of topology between the level sets of the associated Euler-Lagrange functional.

math.DG↗
arXiv2005

An obstruction for the mean curvature of a conformal immersion S^{n}-> R^{n+1}

We prove a Pohozaev type identity for non-linear eigenvalue equations of the Dirac operator on Riemannian spin manifolds with boundary. As an application, we obtain that the mean curvature H of a conformal immersion S^{n}-> R^{n+1} satisfies $\int \partial_X H=0$ where X is a conformal vector field on S^{n} and where the integration is carried out with respect to the Euclidean volume measure of the image. This identity is analogous to the Kazdan-Warner obstruction that appears in the problem of prescribing the scalar curvature on S^{n} inside the standard conformal class.

math.DG↗
arXiv2004

On a Yamabe Type Problem on Three Dimensional Thin Annulus

We consider a Yamabe type problem on a family $A_ε$ of annulus shaped domains of $\R^3$ which becomes "thin" as $ε$ goes to zero. We show that, for any given positive constant $C$, there exists $ε_0$ such that for any $ε< ε_0$, the problem has no solution $u_ε$ whose energy is less than $C$. Such a result extends to dimension three a result previously known in higher dimensions. Although the strategy to prove this result is the same as in higher dimensions, we need a more careful and delicate blow up analysis of asymptotic profiles of solutions $u_ε$ when $ε$ goes to zero.

math.AP↗
arXiv2002

On a geometric equation with critical nonlinearity on the boundary

A theorem of Escobar asserts that, on a positive three dimensional smooth compact Riemannian manifold with boundary which is not conformally equivalent to the standard three dimensional ball, a necessary and sufficient condition for a $C^2$ function $H$ to be the mean curvature of some conformal flat metric is that $H$ is positive somewhere. We show that, when the boundary is umbilic and the function $H$ is positive everywhere, all such metrics stay in a compact set with respect to the $C^2$ norm and the total degree of all solutions is equal to -1.

math.AP↗
arXiv2001

Prescribing a fourth order conformal invariant on the standard sphere, Part II: blow-up analysis and applications

In this paper we perform a fine blow-up analysis for a fourth order elliptic equation involving critical Sobolev exponent, related to the prescription of some conformal invariant on the standard sphere. We derive from this analysis some a priori estimates in dimension 5 and 6. On the five dimensionl sphere these a priori estimates combined with the perturbation result of Part I allow us to obtain some existence results. On the six dimensional sphere we prove the existence of at least one solution when an associated index is different from zero.

math.AP↗
arXiv2001

Compactness results in conformal deformations of Riemannian metrics on manifolds with boundaries

This paper is devoted to the study of a problem arising from a geometric context, namely the conformal deformation of a Riemannian metric to a scalar flat one having constant mean curvature on the boundary. By means of blow-up analysis techniques and the Positive Mass Theorem, we show that on locally conformally flat manifolds with umbilic boundary all metrics stay in a compact set with respect to the $C^2$-norm and the total Leray-Schauder degree of all solutions is equal to -1. Then we deduce from this compactness result the existence of at least one solution to our problem.

math.AP↗
arXiv2001

Prescribing a fourth order conformal invariant on the standard sphere - Part I: a perturbation result

In this paper we study some fourth order elliptic equation involving the critical Sobolev exponent, related to the prescription of a fourth order conformal invariant on the standard sphere. We use a topological method to prove the existence of at least a solution when the function to be prescribed is close to a constant and a finite dimensional map associated to it has non-zero degree.

math.AP↗