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Ridha Yacoub

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Bangladesh University of Engineering & Technology

Prof. Dr. Raffaele Pisano, HDR (Habil.)

Université des Sciences et Technologies de Lille (Lille-1)

Cettina Santagati

Università di Catania

Nikos Mantzakouras

National & Kapodistrian University of Athens

Thomas Curtright

University of Miami

California Institute of Technology

Mba I . Michael

Alvan Ikoku Federal College of Education Owerri in Affiliation with the University of Nigeria

University of East London

Gwen Robbins Schug

University of North Carolina at Greensboro

Gabriel Gutierrez-Alonso

University of Salamanca

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Papers by Ridha Yacoub

Topological methods for the prescribed Webster Scalar Curvature problem on CR manifolds

Differential Geometry and its Applications, 2010

We consider the existence of contact forms of prescribed Webster scalar curvature on a (2n+1)-dim... more We consider the existence of contact forms of prescribed Webster scalar curvature on a (2n+1)-dimensional CR compact manifold locally conformally CR equivalent to the standard unit sphere S2n+1 of Cn+1. We give some existence results, using dynamical and topological methods involving the study of the critical points at infinity of the associated noncompact variational problem.

Topological arguments in prescribing the scalar curvature under minimal boundary mean curvature condition on $S^n_+$

Differential and Integral Equations, 2008

ABSTRACT This paper is devoted to the prescribed scalar curvature problem under minimal boundary ... more ABSTRACT This paper is devoted to the prescribed scalar curvature problem under minimal boundary mean curvature condition on the standard n-dimensional half Sphere, with n≥3. Using tools related to the theory of critical points at infinity, we provide some topological conditions, on the level sets of a given positive function on Sn+, under which we prove some existence results.

Existence and Multiplicity results for the prescribed Webster Scalar Curvature Problem on three CR manifolds

This paper is devoted to the existence of contact forms of prescribed Webster scalar curvature on... more This paper is devoted to the existence of contact forms of prescribed Webster scalar curvature on a 3−dimensional CR compact manifold locally conformally CR equivalent to the unit sphere S 3 of C 2. Due to Kazdan-Warner type obstructions, conditions on the function H to be realized as a Webster scalar curvature have to be given. We prove new existence results based on a new type of Euler-Hopf type formula. Our argument gives an upper bound on the Morse index of the obtained solution. We also give a lower bound on the number of conformal contact forms having the same Webster scalar curvature.

$Research paper thumbnail of Concentration phenomena for fractional CR Yamabe type flows$

Concentration phenomena for fractional CR Yamabe type flows

Differential Geometry and its Applications

$Research paper thumbnail of The fractional CR curvature equation on the three-dimensional CR sphere$

The fractional CR curvature equation on the three-dimensional CR sphere

Mathematics Research Reports

In this paper, we address the problem of prescribed fractional Q-curvature on a 3-dimensional sph... more In this paper, we address the problem of prescribed fractional Q-curvature on a 3-dimensional sphere endowed with its standard CR structure. Since the associated variational problem is noncompact, we approach this issue using techniques of Bahri as the theory of critical points at infinity, using topological tools from generalizations of Morse theory. We prove some perturbative existence results.

Existence Results for the Prescribed Webster Scalar Curvature on Higher Dimensional CR Manifolds

Advanced Nonlinear Studies

This paper is about prescribing the Webster scalar curvature on a compact CR manifold of dimensio... more

$Research paper thumbnail of Bahri invariants for fractional Nirenberg-type flows$

Bahri invariants for fractional Nirenberg-type flows

Arabian Journal of Mathematics, 2017

This paper is concerned with prescribing the fractional Q-curvature on the unit sphere S n endowe... more This paper is concerned with prescribing the fractional Q-curvature on the unit sphere S n endowed with its standard conformal structure g 0 , n ≥ 4. Since the associated variational problem is noncompact, we approach this issue with techniques passed by Abbas Bahri, as the well known theory of critical points at infinity, as well as some lesser known topological invariants that appear here as criteria for existence results.

On the Scalar Curvature Equations in high dimension

Advanced Nonlinear Studies, 2002

In order to study the scalar curvature problem on spheres Sn of dimension n > 7, A. Bahri introdu... more In order to study the scalar curvature problem on spheres Sn of dimension n > 7, A. Bahri introduced, in [4], an invariant for Yamabe-type or scalar curvature flows. In this paper, we show that this invariant is not particularly related to Yamabe-type or scalar curvature flows, but it relies on a more general variational principle; and we give a simple proof of it's existence for any variational flow. Through an example, we show how to use this invariant, in order to resolve scalar curvature equations on standard spheres Sn of dimension n 2 5.

On the Scalar Curvature Equations in high dimension

Advanced Nonlinear Studies, 2002

In order to study the scalar curvature problem on spheres S

Topological arguments in prescribing the scalar curvature under minimal boundary mean curvature condition on Sn+

Differential and Integral Equations

This paper is devoted to the prescribed scalar curvature problem under minimal boundary mean curv... more This paper is devoted to the prescribed scalar curvature problem under minimal boundary mean curvature condition on the standard n-dimensional half Sphere, with n≥3. Using tools related to the theory of critical points at infinity, we provide some topological conditions, on the level sets of a given positive function on Sn+, under which we prove some existence results.

Existence and Multiplicity Results for the Scalar Curvature Problem on the Half-Sphere

Geometry, 2014

In this paper we deal with the scalar curvature problem under minimal boundary mean curvature con... more In this paper we deal with the scalar curvature problem under minimal boundary mean curvature condition on the standard 3-dimensional half-sphere. Using tools related to the theory of critical points at infinity, we give existence results under perturbative and nonperturbative hypothesis, and with the help of some “Morse inequalities at infinity”, we provide multiplicity results for our problem.

On a bi-harmonic equation involving critical exponent: Existence and multiplicity results

Acta Mathematica Scientia, 2011

In this paper, we consider the problem of existence as well as multiplicity results for a bi-harm... more In this paper, we consider the problem of existence as well as multiplicity results for a bi-harmonic equation under the Navier boundary conditions: Δ 2 u = K(x)u p , u > 0 in Ω, Δu = u = 0 on ∂Ω, where Ω is a smooth domain in R n , n 5, and p + 1 = 2n n−4 is the critical Sobolev exponent. We obtain highlightly a new criterion of existence, which provides existence results for a dense subset of positive functions, and generalizes Bahri-Coron type criterion in dimension six. Our argument gives also estimates on the Morse index of the obtained solutions and extends some known results. Moreover, it provides, for generic K, Morse inequalities at infinity, which delivers lower bounds for the number of solutions. As further applications of this Morse theoretical approach, we prove more existence results.

Prescribing the Webster scalar curvature on CR spheres

Comptes Rendus Mathematique, 2011

In this Note, we give two new perturbative results for prescribing the Webster scalar curvature o... more

Existence and Multiplicity results for the prescribed Webster Scalar Curvature Problem on three $C R $ manifolds

by Mohameden O Ahmedou, Ridha Yacoub, and Hichem Chtioui

This paper is devoted to the existence of contact forms of prescribed Webster scalar curvature on... more This paper is devoted to the existence of contact forms of prescribed Webster scalar curvature on a $3-$dimensional CR compact manifold locally conformally CR equivalent to the unit sphere $\mathbb{S}^{3}$ of $\mathbb{C}^{2}$. Due to Kazdan-Warner type obstructions, conditions on the function $H$ to be realized as a Webster scalar curvature have to be given. We prove new existence results based on a new type of Euler-Hopf type formula. Our argument gives an upper bound on the Morse index of the obtained solution. We also give a lower bound on the number of conformal contact forms having the same Webster scalar curvature.

CR Yamabe conjecture – the conformally flat case

Pacific Journal of Mathematics, 2001

We extend the resolution of the CR Yamabe conjecture for one of the two cases left open by D. Jer... more

On a bi-harmonic equation involving critical exponent: Existence and multiplicity results

by Hichem Chtioui and Ridha Yacoub

Acta Mathematica Scientia, 2011

In this paper, we consider the problem of existence as well as multiplicity results for a bi-harm... more In this paper, we consider the problem of existence as well as multiplicity results for a bi-harmonic equation under the Navier boundary conditions: Δ 2 u = K(x)u p , u > 0 in Ω, Δu = u = 0 on ∂Ω, where Ω is a smooth domain in R n , n 5, and p + 1 = 2n n−4 is the critical Sobolev exponent. We obtain highlightly a new criterion of existence, which provides existence results for a dense subset of positive functions, and generalizes Bahri-Coron type criterion in dimension six. Our argument gives also estimates on the Morse index of the obtained solutions and extends some known results. Moreover, it provides, for generic K, Morse inequalities at infinity, which delivers lower bounds for the number of solutions. As further applications of this Morse theoretical approach, we prove more existence results.

Topological methods for the prescribed Webster Scalar Curvature problem on CR manifolds

by Mohameden O Ahmedou, Ridha Yacoub, and Hichem Chtioui

Differential Geometry and Its Applications, 2010

We consider the existence of contact forms of prescribed Webster scalar curvature on a 2n + 1 dim... more We consider the existence of contact forms of prescribed Webster scalar curvature on a 2n + 1 dimensional CR compact manifold locally conformally CR equivalent to the standard unit sphere S 2n+1 of C n+1 . We give some existence results, using dynamical and topological methods involving the study of the critical points at infinity of the associated noncompact variational problem. Mathematics Subject Classification : 53C15, 53C21, 35J65, 18G35.

Topological methods for the prescribed Webster Scalar Curvature problem on CR manifolds

Differential Geometry and its Applications, 2010

We consider the existence of contact forms of prescribed Webster scalar curvature on a (2n+1)-dim... more We consider the existence of contact forms of prescribed Webster scalar curvature on a (2n+1)-dimensional CR compact manifold locally conformally CR equivalent to the standard unit sphere S2n+1 of Cn+1. We give some existence results, using dynamical and topological methods involving the study of the critical points at infinity of the associated noncompact variational problem.

Topological arguments in prescribing the scalar curvature under minimal boundary mean curvature condition on $S^n_+$

Differential and Integral Equations, 2008

ABSTRACT This paper is devoted to the prescribed scalar curvature problem under minimal boundary ... more ABSTRACT This paper is devoted to the prescribed scalar curvature problem under minimal boundary mean curvature condition on the standard n-dimensional half Sphere, with n≥3. Using tools related to the theory of critical points at infinity, we provide some topological conditions, on the level sets of a given positive function on Sn+, under which we prove some existence results.

Existence and Multiplicity results for the prescribed Webster Scalar Curvature Problem on three CR manifolds

This paper is devoted to the existence of contact forms of prescribed Webster scalar curvature on... more This paper is devoted to the existence of contact forms of prescribed Webster scalar curvature on a 3−dimensional CR compact manifold locally conformally CR equivalent to the unit sphere S 3 of C 2. Due to Kazdan-Warner type obstructions, conditions on the function H to be realized as a Webster scalar curvature have to be given. We prove new existence results based on a new type of Euler-Hopf type formula. Our argument gives an upper bound on the Morse index of the obtained solution. We also give a lower bound on the number of conformal contact forms having the same Webster scalar curvature.

$Research paper thumbnail of Concentration phenomena for fractional CR Yamabe type flows$

Concentration phenomena for fractional CR Yamabe type flows

Differential Geometry and its Applications

$Research paper thumbnail of The fractional CR curvature equation on the three-dimensional CR sphere$

The fractional CR curvature equation on the three-dimensional CR sphere

Mathematics Research Reports

In this paper, we address the problem of prescribed fractional Q-curvature on a 3-dimensional sph... more In this paper, we address the problem of prescribed fractional Q-curvature on a 3-dimensional sphere endowed with its standard CR structure. Since the associated variational problem is noncompact, we approach this issue using techniques of Bahri as the theory of critical points at infinity, using topological tools from generalizations of Morse theory. We prove some perturbative existence results.

Existence Results for the Prescribed Webster Scalar Curvature on Higher Dimensional CR Manifolds

Advanced Nonlinear Studies

This paper is about prescribing the Webster scalar curvature on a compact CR manifold of dimensio... more

$Research paper thumbnail of Bahri invariants for fractional Nirenberg-type flows$

Bahri invariants for fractional Nirenberg-type flows

Arabian Journal of Mathematics, 2017

This paper is concerned with prescribing the fractional Q-curvature on the unit sphere S n endowe... more This paper is concerned with prescribing the fractional Q-curvature on the unit sphere S n endowed with its standard conformal structure g 0 , n ≥ 4. Since the associated variational problem is noncompact, we approach this issue with techniques passed by Abbas Bahri, as the well known theory of critical points at infinity, as well as some lesser known topological invariants that appear here as criteria for existence results.

On the Scalar Curvature Equations in high dimension

Advanced Nonlinear Studies, 2002

In order to study the scalar curvature problem on spheres Sn of dimension n > 7, A. Bahri introdu... more In order to study the scalar curvature problem on spheres Sn of dimension n > 7, A. Bahri introduced, in [4], an invariant for Yamabe-type or scalar curvature flows. In this paper, we show that this invariant is not particularly related to Yamabe-type or scalar curvature flows, but it relies on a more general variational principle; and we give a simple proof of it's existence for any variational flow. Through an example, we show how to use this invariant, in order to resolve scalar curvature equations on standard spheres Sn of dimension n 2 5.

On the Scalar Curvature Equations in high dimension

Advanced Nonlinear Studies, 2002

In order to study the scalar curvature problem on spheres S

Topological arguments in prescribing the scalar curvature under minimal boundary mean curvature condition on Sn+

Differential and Integral Equations

This paper is devoted to the prescribed scalar curvature problem under minimal boundary mean curv... more This paper is devoted to the prescribed scalar curvature problem under minimal boundary mean curvature condition on the standard n-dimensional half Sphere, with n≥3. Using tools related to the theory of critical points at infinity, we provide some topological conditions, on the level sets of a given positive function on Sn+, under which we prove some existence results.

Existence and Multiplicity Results for the Scalar Curvature Problem on the Half-Sphere

Geometry, 2014

In this paper we deal with the scalar curvature problem under minimal boundary mean curvature con... more In this paper we deal with the scalar curvature problem under minimal boundary mean curvature condition on the standard 3-dimensional half-sphere. Using tools related to the theory of critical points at infinity, we give existence results under perturbative and nonperturbative hypothesis, and with the help of some “Morse inequalities at infinity”, we provide multiplicity results for our problem.

On a bi-harmonic equation involving critical exponent: Existence and multiplicity results

Acta Mathematica Scientia, 2011

In this paper, we consider the problem of existence as well as multiplicity results for a bi-harm... more In this paper, we consider the problem of existence as well as multiplicity results for a bi-harmonic equation under the Navier boundary conditions: Δ 2 u = K(x)u p , u > 0 in Ω, Δu = u = 0 on ∂Ω, where Ω is a smooth domain in R n , n 5, and p + 1 = 2n n−4 is the critical Sobolev exponent. We obtain highlightly a new criterion of existence, which provides existence results for a dense subset of positive functions, and generalizes Bahri-Coron type criterion in dimension six. Our argument gives also estimates on the Morse index of the obtained solutions and extends some known results. Moreover, it provides, for generic K, Morse inequalities at infinity, which delivers lower bounds for the number of solutions. As further applications of this Morse theoretical approach, we prove more existence results.

Prescribing the Webster scalar curvature on CR spheres

Comptes Rendus Mathematique, 2011

In this Note, we give two new perturbative results for prescribing the Webster scalar curvature o... more

Existence and Multiplicity results for the prescribed Webster Scalar Curvature Problem on three $C R $ manifolds

by Mohameden O Ahmedou, Ridha Yacoub, and Hichem Chtioui

This paper is devoted to the existence of contact forms of prescribed Webster scalar curvature on... more This paper is devoted to the existence of contact forms of prescribed Webster scalar curvature on a $3-$dimensional CR compact manifold locally conformally CR equivalent to the unit sphere $\mathbb{S}^{3}$ of $\mathbb{C}^{2}$. Due to Kazdan-Warner type obstructions, conditions on the function $H$ to be realized as a Webster scalar curvature have to be given. We prove new existence results based on a new type of Euler-Hopf type formula. Our argument gives an upper bound on the Morse index of the obtained solution. We also give a lower bound on the number of conformal contact forms having the same Webster scalar curvature.

CR Yamabe conjecture – the conformally flat case

Pacific Journal of Mathematics, 2001

We extend the resolution of the CR Yamabe conjecture for one of the two cases left open by D. Jer... more

On a bi-harmonic equation involving critical exponent: Existence and multiplicity results

by Hichem Chtioui and Ridha Yacoub

Acta Mathematica Scientia, 2011

In this paper, we consider the problem of existence as well as multiplicity results for a bi-harm... more In this paper, we consider the problem of existence as well as multiplicity results for a bi-harmonic equation under the Navier boundary conditions: Δ 2 u = K(x)u p , u > 0 in Ω, Δu = u = 0 on ∂Ω, where Ω is a smooth domain in R n , n 5, and p + 1 = 2n n−4 is the critical Sobolev exponent. We obtain highlightly a new criterion of existence, which provides existence results for a dense subset of positive functions, and generalizes Bahri-Coron type criterion in dimension six. Our argument gives also estimates on the Morse index of the obtained solutions and extends some known results. Moreover, it provides, for generic K, Morse inequalities at infinity, which delivers lower bounds for the number of solutions. As further applications of this Morse theoretical approach, we prove more existence results.

Topological methods for the prescribed Webster Scalar Curvature problem on CR manifolds

by Mohameden O Ahmedou, Ridha Yacoub, and Hichem Chtioui

Differential Geometry and Its Applications, 2010

We consider the existence of contact forms of prescribed Webster scalar curvature on a 2n + 1 dim... more We consider the existence of contact forms of prescribed Webster scalar curvature on a 2n + 1 dimensional CR compact manifold locally conformally CR equivalent to the standard unit sphere S 2n+1 of C n+1 . We give some existence results, using dynamical and topological methods involving the study of the critical points at infinity of the associated noncompact variational problem. Mathematics Subject Classification : 53C15, 53C21, 35J65, 18G35.