Najoua Gamara - Academia.edu (original) (raw)

Papers by Najoua Gamara

arXiv: Differential Geometry, 2018

We give multiplicity results for the problem of prescribing the scalar curvature on Cauchy- Riema... more We give multiplicity results for the problem of prescribing the scalar curvature on Cauchy- Riemann spheres under Beta-flatness condition. To give a lower bound for the number of solutions, we use Bahri methods based on the theory of critical points at infinity and a Poincare-Hopf type formula.

Partial Differential Equations Arising from Physics and Geometry, 2019

arXiv: Differential Geometry, 2019

In this work, we give a survey on non characteristic domains of Heisenberg groups. We prove that ... more In this work, we give a survey on non characteristic domains of Heisenberg groups. We prove that bounded domains which are diffeomorphic to the solid torus having the center of the group as rotation axis, are non characteristic. Then, we state the following conjecture : The bounded non characteristic domains of the Heisenberg group of dimension 1 are those diffeomorphic to a solid torus having the center of the group as rotation axis.

In this paper we give a new proof for “the CR Pohoz̆aev Identity” and deduce non existence result... more In this paper we give a new proof for “the CR Pohoz̆aev Identity” and deduce non existence results of positive solutions for semi-linear boundary value problems on starshaped domains (P) { −∆Hu = g(u) in Ω u = 0 in ∂Ω, where ∆H is the sublaplacian of the Heisenberg group H , g is a C function on a star-shaped and bounded domain Ω of H. RESUMEN En este art́ıculo presentamos una nueva demostración de la identidad de CR Pohozaev sobre el grupo de Heisenberg y deducimos resultados sobre la no existencia de soluciones positivas para problemas semi-lineales con valores en la frontera sobre dominios estrellados (P) { −∆Hu = g(u) in Ω u = 0 in ∂Ω, donde ∆H es el sublaplaciano del grupo de Heisenberg H , g es una función de clase C sobre un dominio estrellado y acotado Ω de H.

Periodica Mathematica Hungarica

In this work, we study non-characteristic domains of Heisenberg groups. We prove that bounded dom... more In this work, we study non-characteristic domains of Heisenberg groups. We prove that bounded domains which are diffeomorphic to the solid torus having the center of the group as rotation axis, are non-characteristic. Then we state the following conjecture: the bounded non-characteristic domains of the Heisenberg group of dimension 1 are diffeomorphic to a solid torus having the center of the group as revolution axis.

International Journal of Mathematics

We give multiplicity results for the problem of prescribing the scalar curvature on Cauchy–Rieman... more We give multiplicity results for the problem of prescribing the scalar curvature on Cauchy–Riemann spheres under [Formula: see text]-flatness condition. To find a lower bound for the number of solutions, we use Bahri’s methods based on the theory of critical points at infinity and a Poincaré–Hopf-type formula.

Bulletin of the Malaysian Mathematical Sciences Society

We give new estimates on the capacity of a condenser in the first Heisenberg group. As an applica... more We give new estimates on the capacity of a condenser in the first Heisenberg group. As an application, we establish new lower and upper bounds for the first eigenvalue of the Kohn–Laplace operator for a regular bounded open domain of the Heisenberg group.

Arabian Journal of Mathematics

Sub-Riemannian spaces are spaces whose metric structure may be viewed as a constrained geometry, ... more Sub-Riemannian spaces are spaces whose metric structure may be viewed as a constrained geometry, where motion is only possible along a given set of directions, changing from point to point. The simplest example of such spaces is given by the so-called Heisenberg group. The characteristic constrained motion of sub-Riemannian spaces has numerous applications in robotic control in engineering and neurobiology where it arises naturally in the functional magnetic resonance imaging (FMRI). It also arises naturally in other branches of pure mathematics as Cauchy Riemann geometry, complex hyperbolic spaces, and jet spaces. In this paper, we review the use of the relationship between Heisenberg geometry and Cauchy Riemann (CR) geometry. More precisely, we focus on the problem of the prescription of the scalar curvature using techniques related to the theory of critical points at infinity. These techniques were first introduced by Bahri, Bahri and Brezis for the Yamabe conjecture in the Riemannian settings.

Mediterranean Journal of Mathematics, 2017

In this paper, by extending the notions of harmonic transplantation and harmonic radius in the He... more In this paper, by extending the notions of harmonic transplantation and harmonic radius in the Heisenberg group, we give an upper bound for the first eigenvalue for the following Dirichlet problem: (P Ω) −∆ H 1 u = λu in Ω u = 0 on ∂Ω, where Ω is a regular bounded domain of H 1 with smooth boundary and ∆ H 1 is the Kohn-Laplace operator. Using the results of P.Pansu given in [6, 7], which give the relation between the volume of Ω and the perimeter of its boundary. we prove the following result λ 1 (Ω) ≤ C Ω l 2 11 max ξ∈Ω r 2 Ω (ξ) where l 11 is the first strictly positive zero of the Bessel function of first kind and order 1, C Ω is a constant depending of Ω and r Ω (ξ) is the harmonic radius of Ω at a point ξ of Ω.

Advanced Nonlinear Studies, 2017

This work is an adaptation of one of the methods based on the variational critical points at infi... more This work is an adaptation of one of the methods based on the variational critical points at infinity theory of Abbas Bahri [

In this paper we give a new proof for “the CR Poho˘zaev Identity” and deduce non existence result... more In this paper we give a new proof for “the CR Poho˘zaev Identity” and deduce non existence results of positive solutions for semi-linear boundary value problems on starshaped domains (P) � −�\delta u = g(u) in \omega u = 0 in boudary of \omega where $ �\delta$ is the sublaplacian of the Heisenberg group H^n, g is a C^1 function on a star-shaped and bounded domain of H^n.

We review upper and lower bound isoperimetric properties of the fundamental eigenfunction of the ... more We review upper and lower bound isoperimetric properties of the fundamental eigenfunction of the Dirichlet Laplacian and announce new reverse H¨older type inequalities for norms of this function in the case of a wedgelike membrane.

Nonlinear Analysis: Theory, Methods & Applications, 2015

Abstract In this work, we give new existence and multiplicity results for the solutions of the pr... more Abstract In this work, we give new existence and multiplicity results for the solutions of the prescription problem for the Webster scalar curvature on a 3-dimensional Pseudo Hermitian Manifold. The critical points of prescribed functions verify mixed conditions. We establish some Morse Inequalities at Infinity and a Poincare–Hopf type formula to give a lower bound on the number of solutions as well as an upper bound for the Morse index of such solutions.

Advanced Nonlinear Studies

This paper is a preliminary work on Heisenberg group domains, devoted to the study of the Green’s... more This paper is a preliminary work on Heisenberg group domains, devoted to the study of the Green’s function for the Kohn Laplacian on domains far away from the set of characteristic points. We give some estimates of the Green’s function, its regular part and their derivatives analogous to those proved by A. Bahri, Y. Li and O. Rey [Calc. Var. Partial Differ. Equ. 3, No. 1, 67–93 (1995; Zbl 0814.35032)] and O. Rey in [J. Funct. Anal. 89, No. 1, 1–52 (1990; Zbl 0786.35059)] for Euclidean domains, while the study of such functions on the set of characteristic points of the given domain will be discussed in a forthcoming paper.

We extend to the CR framework an equivalent problem to the well known Kazdan- Warner problem on R... more We extend to the CR framework an equivalent problem to the well known Kazdan- Warner problem on Riemannian manifolds. Let (M, θ) be a three-dimensional pseudo Hermitian CR compact manifold, locally conformally CR equivalent to the sphere S

Arabian Journal of Mathematics, 2013

In this paper, we give existence and multiplicity results for the problem of prescribing the Webs... more In this paper, we give existence and multiplicity results for the problem of prescribing the Webster scalar curvature on the three CR sphere of C 2 under mixed conditions: non-degenerancy and flatness.

Advances in Pure and Applied Mathematics, 2012

Abstract. In this paper, we obtain existence result for prescribed curvature satisfying a CR “fla... more Abstract. In this paper, we obtain existence result for prescribed curvature satisfying a CR “flatness condition” by using topological methods results: the theory of critical points at infinity.

Contemporary Mathematics, 2011

We review upper and lower bound isoperimetric properties of the fundamental eigenfunction of the ... more We review upper and lower bound isoperimetric properties of the fundamental eigenfunction of the Dirichlet Laplacian and announce new reverse Hölder type inequalities for norms of this function in the case of wedgelike membrane.

Mediterranean Journal of Mathematics, 2012

Journal of The European Mathematical Society - J EUR MATH SOC, 2001

Let (M, θ) be a compact CR manifold of dimension 2n + 1 with a contact form θ, and L = (2 + 2/n) ... more Let (M, θ) be a compact CR manifold of dimension 2n + 1 with a contact form θ, and L = (2 + 2/n) b + R its associated CR conformal laplacien. The CR Yamabe conjecture states that there is a contact formθ on M conformal to θ which has a constant Webster curvature. This problem is equivalent to the existence of a function u such that      Lu = u 1+2/n on M u > 0. D. Jerison and J.M. Lee solved the CR Yamabe problem in the case where n ≥ 2 and (M, θ) is not locally CR equivalent to the sphere S 2n+1 of C n. In a join work with R. Yacoub, the CR Yamabe problem was solved for the case where (M, θ) is locally CR equivalent to the sphere S 2n+1 for all n. In the present paper, we study the case n = 1, left by D. Jerison and J.M. Lee, which completes the resolution of the CR Yamabe conjecture for all dimensions.