arXiv subjects

Nassif Ghoussoub

Publications and source records attributed to Nassif Ghoussoub.

66 records · Page 4Linked to original sources

arXiv2006

Maximal monotone operators are selfdual vector fields and vice-versa

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, any maximal monotone operator $T$ on $X$ is derived from such a potential on phase space, that is there exists a selfdual Lagrangian $L$ on $X\times X^*$ (i.e, $L^*(p, x) =L(x, p)$) such that $T=\bar\partial L$. This solution to problems raised by Fitzpatrick can be seen as an extension of a celebrated result of Rockafellar stating that maximal cyclically monotone operators are actually of the form $T=\partial ϕ$ for some convex lower semi-continuous function on $X$. This representation allows for the application of the selfdual variational theory --recently developed by the author-- to the equations driven by maximal monotone vector fields. Consequently, solutions to equations of the form $Λx\in Tx$ for a given map $Λ: D(Λ)\subset X\to X^*$, can now be obtained by minimizing functionals of the form $I(x)=L(x,Λx)-< x, Λx>$.

math.AP↗

arXiv2005

Compactness along the Branch of Semi-stable and Unstable Solutions for an Elliptic Problem with a Singular Nonlinearity

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 $-Δu=\frac{λf(x)}{(1-u)^2}$ on a bounded domain $Ω\subset \R^N$, which models --among other things-- a simple electrostatic Micro-Electromechanical System (MEMS) device. We extend the results of [11] relating to the minimal branch, by obtaining compactness along unstable branches for $1\leq N \leq 7$ on any domain $Ω$ and for a large class of "permittivity profiles" $f$ . We also show the remarkable fact that power-like profiles $f(x) \simeq |x|^α$ can push back the critical dimension N=7 of this problem, by establishing compactness for the semi-stable branch on the unit ball, also for $N\geq 8$ and as long as $α>α_N=\frac{3N-14-4\sqrt{6}}{4+2\sqrt{6}}$ . As a byproduct, we are able to follow the second branch of the bifurcation diagram and prove the existence of a second solution for $λ$ in a natural range. In all these results, the conditions on the space-dimension and on the power of the profile are essentially sharp.

math.AP↗

arXiv2005

On the Partial Differential Equations of Electrostatic MEMS Devices: Stationary Case

We analyze the nonlinear elliptic problem $Δu=\frac{λf(x)}{(1+u)^2}$ on a bounded domain $Ω$ of $\R^N$ with Dirichlet boundary conditions. This equation models a simple electrostatic Micro-Electromechanical System (MEMS) device consisting of a thin dielectric elastic membrane with boundary supported at 0 above a rigid ground plate located at -1. When a voltage --represented here by $λ$-- is applied, the membrane deflects towards the ground plate and a snap-through may occur when it exceeds a certain critical value $λ^*$ (pull-in voltage). This creates a so-called "pull-in instability" which greatly affects the design of many devices. The mathematical model lends to a nonlinear parabolic problem for the dynamic deflection of the elastic membrane which will be considered in forthcoming papers \cite{GG2} and \cite{GG3}. For now, we focus on the stationary equation where the challenge is to estimate $λ^*$ in terms of material properties of the membrane, which can be fabricated with a spatially varying dielectric permittivity profile $f$. Applying analytical and numerical techniques, the existence of $λ^*$ is established together with rigorous bounds. We show the existence of at least one steady-state when $λ< λ^*$ (and when $λ=λ^*$ in dimension $N< 8$) while none is possible for $λ>λ^*$. More refined properties of steady states --such as regularity, stability, uniqueness, multiplicity, energy estimates and comparison results-- are shown to depend on the dimension of the ambient space and on the permittivity profile.

math.AP↗

arXiv2005

Selfdual Variational Principles for Periodic Solutions of Hamiltonian and Other Dynamical Systems

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 including many of the standard ones. These principles lead to new variational proofs of the existence of parabolic flows with prescribed initial conditions, as well as periodic, anti-periodic and skew-periodic orbits of Hamiltonian systems. They are based on the theory of anti-selfdual Lagrangians introduced and developed recently in [3], [4] and [5].

math.AP↗

arXiv2005

Elliptic Equations with Critical Growth and a Large Set of Boundary Singularities

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 $Ω$ of $\rn$, where ${\mathcal A} $ is a proper linear subspace of $\rn$. Existence problems are related to the question of attainability of the best constant in the following recent inequality of Badiale-Tarantello [1]: $$0<μ_{s,¶}(Ω)=\inf{\int_Ω|\nabla u|^2 dx; u\in \huno \hbox{and}\int_Ω\frac{|u(x)|^{\crit(s)}}{|π(x)|^s} dx=1}$$ where $0<s<2$, $\crit(s)=\frac{2(n-s)}{n-2}$ and where $π$ is the orthogonal projection on a linear space $¶$, where $\hbox{dim}_{\rr}¶\geq 2$. We investigate this question and how it depends on the relative position of the subspace $\Porth$, the orthogonal of $¶$, with respect to the domain $Ω$ as well as on the curvature of the boundary $\partialΩ$ at its points of intersection with $\Porth $.

math.AP↗

arXiv2005

On the existence of Hamiltonian paths connecting Lagrangian submanifolds

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 given Lagrangian submanifolds in $\R^{2N}$. We also consider the case where the Hamiltonian is only semi-convex. A variational principle is also used to establish existence for the corresponding Cauchy problem. The case of periodic solutions will be considered in a forthcoming paper [5].

math.AP↗

arXiv2005

Iterations of anti-selfdual Lagrangians and applications to Hamiltonian systems and multiparameter gradient flows

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 space for the newly obtained anti-selfdual Lagrangian. We give here two applications for these remarkable permanence properties. In the first, we establish for certain convex-concave Hamiltonians ${\cal H}$ on a --possibly infinite dimensional--symplectic space $H^2$, the existence of a solution for the Hamiltonian system $-J\dot u (t)=\partial {\cal H} (u(t))$ that connects in a given time T>0, two Lagrangian submanifolds. Another application deals with the construction of a multiparameter gradient flow for a convex potential. Our methods are based on the new variational calculus for anti-selfdual Lagrangians developed in [4], [5] and [7].

math.AP↗

arXiv2005

Anti-selfdual Lagrangians: Variational resolutions of non self-adjoint equations and dissipative evolutions

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 extensions of gradients of convex functions --hence of self-adjoint positive operators-- which usually drive dissipative systems, but also rich enough to provide representations for the superposition of such gradients with skew-symmetric operators which normally generate unitary flows. They yield variational formulations and resolutions for large classes of non-potential boundary value problems and initial-value parabolic equations. Solutions are minima of functionals of the form $I(u)=L(u, Λu)$ (resp. $I(u)=\int_{0}^{T}L(t, u(t), \dot u (t)+Λ_{t}u(t))dt$) where $L$ is an anti-self dual Lagrangian and where $Λ_{t}$ are essentially skew-adjoint operators. However, and just like the self (and antiself) dual equations of quantum field theory (e.g. Yang-Mills) the equations associated to such minima are not derived from the fact they are critical points of the functional $I$, but because they are also zeroes of the Lagrangian $L$ itself.

math.AP↗

arXiv2005

Anti-selfdual Lagrangians II: Unbounded non self-adjoint operators and evolution equations

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 to provide variational formulations and resolutions to various basic equations and evolutions which do not normally fit in the Euler-Lagrange framework. In particular, we consider stationary equations of the form $ -Au\in \partial ϕ(u)$ as well as i dissipative evolutions of the form $-\dot{u}(t)-A_t u(t)+ωu(t) \in \partial ϕ(t, u(t))$ were $ϕ$ is a convex potential on an infinite dimensional space. In this paper, the emphasis is on the cases where the differential operators involved are not necessarily bounded, hence completing the results established in [13] for bounded linear operators. Our main applications deal with various nonlinear boundary value problems and parabolic initial value equations governed by the transport operator with or without a diffusion term.

math.AP↗

arXiv2005

Anti-selfdual Hamiltonians: Variational resolutions for Navier-Stokes and other nonlinear evolutions

The theory of anti-selfdual (ASD) Lagrangians developed in \cite{G2} allows a variational resolution for equations of the form $Λu+Au +\partial ϕ(u)+f=0$ where $ϕ$ is a convex lower-semi-continuous function on a reflexive Banach space $X$, $f\in X^*$, $A: D(A)\subset X\to X^*$ is a positive linear operator and where $Λ: D(Λ)\subset X\to X^{*}$ is a non-linear operator that satisfies suitable continuity and anti-symmetry properties. ASD Lagrangians on path spaces also yield variational resolutions for nonlinear evolution equations of the form $\dot u (t)+Λu(t)+Au(t) +f\in -\partial ϕ(u(t))$ starting at $u(0)=u_{0}$. In both stationary and dynamic cases, the equations associated to the proposed variational principles are not derived from the fact they are critical points of the action functional, but because they are also zeroes of the Lagrangian itself.The approach has many applications, in particular to Navier-Stokes type equations and to the differential systems of hydrodynamics, magnetohydrodynamics and thermohydraulics.

math.AP↗

arXiv2005

Concentration Estimates for Emden-Fowler Equations with Boundary Singularities and Critical Growth

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 $Ω$ of $ \rn$ ($n\geq 3$) involving the critical Hardy-Sobolev exponent $\crit =\frac{2(n-s)}{n-2}$ where $0<s<2$, and in the case where zero (the point of singularity) is on the boundary $\partial Ω$. Just as in the Yamabe-type non-singular framework (i.e., when s=0), there is no nontrivial solution under global convexity assumption (e.g., when $Ω$ is star-shaped around 0). However, in contrast to the non-satisfactory situation of the non-singular case, we show the existence of an infinite number of solutions under an assumption of local strict concavity of $\partial Ω$ at 0 in at least one direction. More precisely, we need the principal curvatures of $\partial Ω$ at 0 to be non-positive but not all vanishing. We also show that the best constant in the Hardy-Sobolev inequality is attained as long as the mean curvature of $\partial Ω$ at 0 is negative, extending the results of [21] and completing our result of [22] to include dimension 3. The key ingredients in our proof are refined concentration estimates which yield compactness for certain Palais-Smale sequences which do not hold in the non-singular case.

math.AP↗

arXiv1990

Operators which factor through Banach lattices not containing c_0

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 complements the characterization of [GJ1], [GJ3] that an operator admits such a factorization if and only if it can be written as the product of two operators neither of which preserves a copy of $c_0$. The intrinsic characterization is that the restriction of the second adjoint of the operator to the ideal generated by the lattice in its bidual does not preserve a copy of $c_0$. This property of an operator was introduced by C. Niculescu [N2] under the name ``strong type B".

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