On the Best Constant in the Moser-Onofri-Aubin Inequality (2009)
Ghoussoub, Nassif, Lin, Chang-Shou
Let $S^2$ be the 2-dimensional unit sphere and let $J_\alpha $ denote the nonlinear functional on the Sobolev space $H^{1,2}(S^2)$ defined by $$ J_\alpha(u) = \frac{\alpha}{4}\int_{S^2}|\nabla u|^2...
The critical dimension for a 4th order problem with singular nonlinearity (2009)
Cowan, Craig, Esposito, Pierpaolo, Ghoussoub, Nassif, Moradifam, Amir
We study the regularity of the extremal solution of the semilinear biharmonic equation $\bi u=\f{\lambda}{(1-u)^2}$, which models a simple Micro-Electromechanical System (MEMS) device on a ball...
Estimates on Pull-in Distances in MEMS Models and other Nonlinear Eigenvalue Problems (2009)
Ghoussoub, Nassif, Cowan, Craig
Motivated by certain mathematical models for Micro-Electro-Mechanical Systems (MEMS), we give upper and lower $L^\infty$ estimates for the minimal solutions of nonlinear eigenvalue problems of the...
The critical dimension for a fourth order elliptic problem with singular nonlinearity (2008)
Cowan, Craig, Esposito, Pierpaolo, Ghoussoub, Nassif
We study the regularity of the extremal solution of the semilinear biharmonic equation $\bi u=\f{\lambda}{(1-u)^2}$, which models a simple Micro-Electromechanical System (MEMS) device on a ball...
Uniqueness of solutions for an elliptic equation modeling MEMS (2008)
Ghoussoub, Nassif, Esposito, Pierpaolo
We study the effect of the parameter $\lambda$, the dimension $N$, the profile $f$ and the geometry of the domain $\Omega \subset\mathbb{R}^N$, on the question of uniqueness of the solutions to the...
Regularity of the extremal solution in a MEMS model with advection (2008)
Ghoussoub, Nassif, Cowan, Craig
We consider the regularity of the extremal solution of the nonlinear eigenvalue problem (S)_\lambda \qquad {rcr} -\Delta u + c(x) \cdot \nabla u &=& \frac{\lambda}{(1-u)^2} \qquad {in $ \Omega$}, u...
A variational theory for monotone vector fields (2008)
Monotone vector fields were introduced almost 40 years ago as nonlinear extensions of positive definite linear operators, but also as natural extensions of gradients of convex potentials. These...
Simultaneous preconditioning and symmetrization of non-symmetric linear systems (2008)
Ghoussoub, Nassif, Moradifam, Amir
Motivated by the theory of self-duality which provides a variational formulation and resolution for non self-adjoint partial differential equations \cite{G1, G2}, we propose new templates for solving...
Estimates for the quenching time of a parabolic equation modeling electrostatic MEMS (2007)
The singular parabolic problem $u_t=\Delta u -\frac{\lambda f(x)}{(1+u)^2}$ on a bounded domain $\Omega$ of $R^N$ with Dirichlet boundary conditions, models the dynamic deflection of an elastic...
Deformation from symmetry and multiplicity of solutions in nonhomogeneous problems (2007)
Christine Chambers, Christine Chambers, Nassif Ghoussoub, Nassif Ghoussoub
Deformation from symmetry and multiplicity of solutions in non-homogeneous problems
N. Ghoussoub, N. Ghoussoub, L. Tzou, L. Tzou, Nassif Ghoussoub
A variational principle for gradient flows
Bessel potentials and optimal Hardy and Hardy-Rellich inequalities (2007)
Ghoussoub, Nassif, Moradifam, Amir
We give necessary and sufficient conditions on a pair of positive radial functions V and W on a ball B of radius R in R^n,$n \geq 1$, so that the following inequalities hold for all $u \in...
Schrodinger equations and Hamiltonian systems of PDEs with selfdual boundary conditions (2007)
Ghoussoub, Nassif, Moameni, Abbas
Selfdual variational calculus is further refined and used to address questions of existence of local and global solutions for various parabolic semi-linear equations, Hamiltonian systems of PDEs, as...
On the best possible remaining term in the Hardy Inequality (2007)
Ghoussoub, Nassif, Moradifam, Amir
We give a necessary and sufficient condition on a radially symmetric potential $V$ on $\Omega$ that makes it an admissible candidate for an improved Hardy inequality of the following form:...
Ghoussoub, Nassif, Moameni, Abbas
The nonlinear selfdual variational principle established in a preceeding paper [8] -- though good enough to be readily applicable in many stationary nonlinear partial differential equations -- did...
Maximal monotone operators are selfdual vector fields and vice-versa (2006)
If $L$ is a selfdual Lagrangian $L$ on a reflexive phase space $X\times X^*$, then the vector field $x\to \bar\partial L(x):=\{p\in X^*; (p,x)\in \partial L(x,p)\}$ is maximal monotone. Conversely,...
Esposito, Pierpaolo, Ghoussoub, Nassif, Guo, Yujin
We study the branch of semi-stable and unstable solutions (i.e., those whose Morse index is at most one) of the Dirichlet boundary value problem $-\Delta u=\frac{\lambda f(x)}{(1-u)^2}$ on a bounded...
On the Partial Differential Equations of Electrostatic MEMS Devices: Stationary Case (2005)
We analyze the nonlinear elliptic problem $\Delta u=\frac{\lambda f(x)}{(1+u)^2}$ on a bounded domain $\Omega$ of $\R^N$ with Dirichlet boundary conditions. This equation models a simple...
Ghoussoub, Nassif, Moameni, Abbas
Selfdual variational principles are introduced in order to construct solutions for Hamiltonian and other dynamical systems which satisfy a variety of linear and nonlinear boundary conditions...
Elliptic Equations with Critical Growth and a Large Set of Boundary Singularities (2005)
Ghoussoub, Nassif, Robert, Frederic
We solve variationally certain equations of stellar dynamics of the form $-\sum_i\partial_{ii} u(x) =\frac{|u|^{p-2}u(x)}{{\rm dist} (x,{\mathcal A} )^s}$ in a domain $\Omega$ of $\rn$, where...
On the existence of Hamiltonian paths connecting Lagrangian submanifolds (2005)
Ghoussoub, Nassif, Moameni, Abbas
We use a new variational method --based on the theory of anti-selfdual Lagrangians developed in [2] and [3]-- to establish the existence of solutions of convex Hamiltonian systems that connect two...
Anti-selfdual Lagrangians on a state space lift to path space provided one adds a suitable selfdual boundary Lagrangian. This process can be iterated by considering the path space as a new state...
We develop the concept and the calculus of anti-self dual (ASD) Lagrangians which seems inherent to many questions in mathematical physics, geometry, and differential equations. They are natural...
Anti-selfdual Lagrangians II: Unbounded non self-adjoint operators and evolution equations (2005)
This paper is a continuation of [13], where new variational principles were introduced based on the concept of anti-selfdual (ASD) Lagrangians. We continue here the program of using these Lagrangians...
The theory of anti-selfdual (ASD) Lagrangians developed in \cite{G2} allows a variational resolution for equations of the form $\Lambda u+Au +\partial \phi (u)+f=0$ where $\phi$ is a convex...
Ghoussoub, Nassif, Robert, Frederic
We establish -among other things- existence and multiplicity of solutions for the Dirichlet problem $\sum_i\partial_{ii}u+\frac{|u|^{\crit-2}u}{|x|^s}=0$ on smooth bounded domains $\Omega$ of $ \rn$...
Nassif Ghoussoub, Frédéric Robert, N. Ghoussoub, F. Robert
Abstract. We establish –among other things – existence and multiplicity of solutions for the Dirichlet problem P i ∂iiu+ |u|2 ⋆ −2 u |x | s = 0 on smooth bounded domains Ω of Rn (n≥3)...
Nassif Ghoussoub, Frédéric Robert, N. Ghoussoub, F. Robert
Abstract. We establish –among other things – existence and multiplicity of solutions for the Dirichlet problem P i ∂iiu+ |u|2 ⋆ −2 u |x | s = 0 on smooth bounded domains Ω of Rn (n ≥ 3)...
A least action principle for steepest descent in a non-convex landscape (2004)
Nassif Ghoussoub, Nassif Ghoussoub, Robert J. Mccann, Robert J. Mccann
A least action principle for steepest descent in a non-convex landscape
Anti-self dual Lagrangians II: Unbounded non self-adjoint operators and evolution equations (2004)
Nassif Ghoussoub, Nassif Ghoussoub, Leo Tzou, Leo Tzou
Anti-selfdual Lagrangians II: Unbounded non self-adjoint operators and evolution equations
A theory of anti-selfdual Lagrangians: Dynamical case (2004)
Nassif Ghoussoub, Nassif Ghoussoub
We develop a concept of anti-self dual Lagrangians that seems inherent to many problems in mathematical physics, Riemannian geometry, and differential equations. On one hand, they represent gradients...
Nassif Ghoussoub, Nassif Ghoussoub
Variational resolutions of non self-adjoint
THE EFFECT OF CURVATURE ON THE BEST CONSTANT IN THE HARDY-SOBOLEV INEQUALITIES (2004)
Nassif Ghoussoub, Frédéric Robert, N. Ghoussoub, F. Robert
The effect of curvature on the best constant in
Anti-self dual Lagrangians II: Unbounded non self-adjoint operators and evolution equations (2004)
and evolution equations
THE EFFECT OF CURVATURE ON THE BEST CONSTANT IN THE HARDY-SOBOLEV INEQUALITIES (2004)
Nassif Ghoussoub, Frédéric Robert, N. Ghoussoub, F. Robert
The effect of curvature on the best constant in
Abstract A Theory of Anti-Selfdual Lagrangians: Dynamical case (2004)
Nassif Ghoussoub, Nassif Ghoussoub
We consider the class of time-dependent anti-selfdual Lagrangians, which –just like the stationary case announced in [5] – enjoys remarkable permamence properties and provides variational...
Nassif Ghoussoub, Robert J. Mccann, Nassif Ghoussoub, Robert J. Mccann
A least action principle for steepest descent in a non-convex landscape
On De Giorgi's Conjecture in Dimension 4 and 5 (2001)
Nassif Ghoussoub, Nassif Ghoussoub, Changfeng Gui, Changfeng Gui
this paper, we develop an approach for establishing in some important cases, a conjecture made by De Giorgi more than 20 years ago. The problem originates in the theory of phase transition and is so...
Operators which factor through Banach lattices not containing c_0 (1990)
Ghoussoub, Nassif, Johnson, William B.
In this supplement to [GJ1], [GJ3], we give an intrinsic characterization of (bounded, linear) operators on Banach lattices which factor through Banach lattices not containing a copy of $c_0$ which...
A refinement of the Riesz decomposition for amarts and semiamarts
Ghoussoub, Nassif, Sucheston, Louis
A real-valued adapted sequence of random variables is an amart if and only if it can be written as a sum of a martingale and a sequence dominated in absolute value by a Doob potential, i.e., a...
On the best possible remaining term in the Hardy inequality
Ghoussoub, Nassif, Moradifam, Amir
We give a necessary and sufficient condition on a radially symmetric potential V on a bounded domain Ω of ℝn that makes it an admissible candidate for an improved Hardy inequality of the following...