arXiv subjects

Nassif Ghoussoub

Publications and source records attributed to Nassif Ghoussoub.

At least 55 records · Page 3Linked to original sources

arXiv2012

De Giorgi type results for elliptic systems

We consider the following elliptic system Δu =\nabla H (u) \ \ \text{in}\ \ \mathbf{R}^N, where $u:\mathbf{R}^N\to \mathbf{R}^m$ and $H\in C^2(\mathbf{R}^m)$, and prove, under various conditions on the nonlinearity $H$ that, at least in low dimensions, a solution $u=(u_i)_{i=1}^m$ is necessarily one-dimensional whenever each one of its components $u_i$ is monotone in one direction. Just like in the proofs of the classical De Giorgi's conjecture in dimension 2 (Ghoussoub-Gui) and in dimension 3 (Ambrosio-Cabré), the key step is a Liouville theorem for linear systems. We also give an extension of a geometric Poincaré inequality to systems and use it to establish De Giorgi type results for stable solutions as well as additional rigidity properties stating that the gradients of the various components of the solutions must be parallel. We introduce and exploit the concept of {\it an orientable system}, which seems to be key for dealing with systems of three or more equations. For such systems, the notion of a stable solution in a variational sense coincide with the pointwise (or spectral) concept of stability.

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arXiv2012

Functional Inequalities: New Perspectives and New Applications

This book is not meant to be another compendium of select inequalities, nor does it claim to contain the latest or the slickest ways of proving them. This project is rather an attempt at describing how most functional inequalities are not merely the byproduct of ingenious guess work by a few wizards among us, but are often manifestations of certain natural mathematical structures and physical phenomena. Our main goal here is to show how this point of view leads to "systematic" approaches for not just proving the most basic functional inequalities, but also for understanding and improving them, and for devising new ones - sometimes at will, and often on demand.

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arXiv2011

A Self-dual Polar Factorization for Vector Fields

We show that any non-degenerate vector field $u$ in $ L^{\infty}(Ω, \R^N)$, where $Ω$ is a bounded domain in $\R^N$, can be written as {equation} \hbox{$u(x)= \nabla_1 H(S(x), x)$ for a.e. $x \in Ω$}, {equation} where $S$ is a measure preserving point transformation on $Ω$ such that $S^2=I$ a.e (an involution), and $H: \R^N \times \R^N \to \R$ is a globally Lipschitz anti-symmetric convex-concave Hamiltonian. Moreover, $u$ is a monotone map if and only if $S$ can be taken to be the identity, which suggests that our result is a self-dual version of Brenier's polar decomposition for the vector field $u$ as $u(x)=\nabla ϕ(S(x))$, where $ϕ$ is convex and $S$ is a measure preserving transformation. We also describe how our polar decomposition can be reformulated as a self-dual mass transport problem.

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arXiv2010

Regularity of Extremal Solutions in Fourth Order Nonlinear Eigenvalue Problems on General Domains

We examine the regularity of the extremal solution of the nonlinear eigenvalue problem $Δ^2 u = λf(u)$ on a general bounded domain $Ω$ in $ \IR^N$, with the Navier boundary condition $ u=Δu =0 $ on $ \pOm$. Here $ λ$ is a positive parameter and $f$ is a non-decreasing nonlinearity with $f(0)=1$. We give general pointwise bounds and energy estimates which show that for any convex and superlinear nonlinearity $f$, the extremal solution $ u^*$ is smooth provided $N\leq 5$.

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arXiv2010

Homogenization of maximal monotone vector fields via selfdual variational calculus

We use the theory of selfdual Lagrangians to give a variational approach to the homogenization of equations in divergence form, that are driven by a periodic family of maximal monotone vector fields. The approach has the advantage of using $Γ$-convergence methods for corresponding functionals just as in the classical case of convex potentials, as opposed to the graph convergence methods used in the absence of potentials. A new variational formulation for the homogenized equation is also given.

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arXiv2009

On the Best Constant in the Moser-Onofri-Aubin Inequality

Let $S^2$ be the 2-dimensional unit sphere and let $J_α$ denote the nonlinear functional on the Sobolev space $H^{1,2}(S^2)$ defined by $$ J_α(u) = \fracα{4}\int_{S^2}|\nabla u|^2 dω+ \int_{S^2} u dω-\ln \int_{S^2} e^{u} dω, $$ where $dω$ denotes Lebesgue measure on $S^2$, normalized so that $\int_{S^2} dω= 1$. Onofri had established that $J_α$ is non-negative on $H^1(S^2)$ provided $α\geq 1$. In this note, we show that if $J_α$ is restricted to those $u\in H^1(S^2)$ that satisfy the Aubin condition: \int_{S^2}e^u x_j dw=0\quad\text{for all}1\leq j\leq 3, then the same inequality continues to hold (i.e., $J_α(u)\geq0$) whenever $α\geq {2/3}-ε_0$ for some $ε_0>0$. The question of Chang-Yang on whether this remains true for all $α\geq {1/2}$ remains open.

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arXiv2009

The critical dimension for a 4th order problem with singular nonlinearity

We study the regularity of the extremal solution of the semilinear biharmonic equation $\bi u=\fλ{(1-u)^2}$, which models a simple Micro-Electromechanical System (MEMS) device on a ball $B\subset\IR^N$, under Dirichlet boundary conditions $u=\partial_νu=0$ on $\partial B$. We complete here the results of F.H. Lin and Y.S. Yang \cite{LY} regarding the identification of a "pull-in voltage" $\la^*>0$ such that a stable classical solution $u_\la$ with $0 \la^*$. Our main result asserts that the extremal solution $u_{λ^*}$ is regular $(\sup_B u_{λ^*} <1)$ provided $ N \le 8$ while $u_{λ^*} $ is singular ($\sup_B u_{λ^*} =1$) for $N \ge 9$, in which case $1-C_0|x|^{4/3}\leq u_{λ^*} (x) \leq 1-|x|^{4/3}$ on the unit ball, where $ C_0:= (\frac{λ^*}{\overlineλ})^{1/3}$ and $ \barλ:= {8/9} (N-{2/3}) (N- {8/3})$.

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arXiv2009

Estimates on Pull-in Distances in MEMS Models and other Nonlinear Eigenvalue Problems

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 form $-Δu = λf(x) F(u)$ on a smooth bounded domain $ Ω$ in $\IR^N$. We are mainly interested in the {\it pull-in distance}, that is the $L^\infty-$norm of the extremal solution $u^*$ and how it depends on the geometry of the domain, the dimension of the space, and the so-called {\it permittivity profile} $f$. In particular, our results provide mathematical proofs for various observed phenomena, as well as rigorous derivations for several estimates obtained numerically by Pelesko \cite{P}, Guo-Pan-Ward \cite{GPW} and others in the case of the MEMS non-linearity $F(u)=\frac{1}{(1-u)^2}$ and for power-law permittivity profiles $f(x)=|x|^α$.

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arXiv2008

The critical dimension for a fourth order elliptic problem with singular nonlinearity

We study the regularity of the extremal solution of the semilinear biharmonic equation $\bi u=\fλ{(1-u)^2}$, which models a simple Micro-Electromechanical System (MEMS) device on a ball $B\subset\IR^N$, under Dirichlet boundary conditions $u=\partial_νu=0$ on $\partial B$. We complete here the results of F.H. Lin and Y.S. Yang \cite{LY} regarding the identification of a "pull-in voltage" $\la^*>0$ such that a stable classical solution $u_\la$ with $0 \la^*$. Our main result asserts that the extremal solution $u_{λ^*}$ is regular $(\sup_B u_{λ^*} <1)$ provided $ N \le 8$ while $u_{λ^*} $ is singular ($\sup_B u_{λ^*} =1$) for $N \ge 17$, in which case $1-C_0|x|^{4/3}\leq u_{λ^*} (x) \leq 1-|x|^{4/3}$ on the unit ball, where $ C_0:= <(\frac{λ^*}{\barλ}>)^{1/3}$ and $ \barλ:= \frac{8 (N-{2/3}) (N- {8/3})}{9}$. The singular character of the extremal solution for the remaining cases (i.e., when $9\leq N\leq 16$) requires a computer assisted proof and will not be addressed in this paper.

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arXiv2008

Uniqueness of solutions for an elliptic equation modeling MEMS

We study the effect of the parameter $λ$, the dimension $N$, the profile $f$ and the geometry of the domain $Ω\subset\mathbb{R}^N$, on the question of uniqueness of the solutions to the following elliptic boundary value problem with a singular nonlinearity: $$ 180pt {{array}{ll} -Δu= \frac{λf(x)}{(1-u)^2} & \hbox{in}Ω0<u<1 &\hbox{in}Ωu=0 &\hbox{on}\partial Ω. {array}. 130pt (S)_{λ, f} $$ This equation has been proposed as a model for a simple electrostatic Micro-Electromechanical System (MEMS) device consisting of a thin dielectric elastic membrane with boundary supported at 0 below a rigid ground plate located at height z = 1.

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arXiv2008

Regularity of the extremal solution in a MEMS model with advection

We consider the regularity of the extremal solution of the nonlinear eigenvalue problem (S)_λ\qquad {rcr} -Δu + c(x) \cdot \nabla u &=& \fracλ{(1-u)^2} \qquad {in $ Ω$}, u &=& 0 \qquad {on $ \pOm$}, where $ Ω$ is a smooth bounded domain in $ \IR^N$ and $ c(x)$ is a smooth bounded vector field on $\bar Ω$. We show that, just like in the advection-free model ($c\equiv 0$), all semi-stable solutions are smooth if (and only if) the dimension $N\leq 7$. The novelty here comes from the lack of a suitable variational characterization for the semi-stability assumption. We overcome this difficulty by using a general version of Hardy's inequality. In a forthcoming paper \cite{CG2}, we indicate how this method applies to many other nonlinear eigenvalue problems involving advection (including the Gelfand problem), showing that they all essentially have the same critical dimension as their advection-free counterparts.

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arXiv2008

A variational theory for monotone vector fields

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 vector fields are not always derived from potentials in the classical sense, and as such they are not always amenable to the standard methods of the calculus of variations. We describe here how the selfdual variational calculus developed recently by the author, provides a variational approach to PDEs and evolution equations driven by maximal monotone operators. To any such a vector field T on a reflexive Banach space X, one can associate a convex selfdual Lagrangian L_T on phase space X x X* that can be seen as a "potential" for T, in the sense that the problem of inverting T reduces to minimizing the convex energy L_T. This variational approach to maximal monotone operators allows their theory to be analyzed with the full range of methods --computational or not-- that are available for variational settings. Standard convex analysis (on phase space) can then be used to establish many old and new results concerned with the identification, superposition, and resolution of such vector fields.

math.AP↗

arXiv2008

Simultaneous preconditioning and symmetrization of non-symmetric linear systems

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 large non-symmetric linear systems. The method consists of combining a new scheme that simultaneously preconditions and symmetrizes the problem, with various well known iterative methods for solving linear and symmetric problems. The approach seems to be efficient when dealing with certain ill-conditioned, and highly non-symmetric systems.

math.NA↗

arXiv2007

Estimates for the quenching time of a parabolic equation modeling electrostatic MEMS

The singular parabolic problem $u_t=Δu -\frac{λf(x)}{(1+u)^2}$ on a bounded domain $Ω$ of $R^N$ with Dirichlet boundary conditions, models the dynamic deflection of an elastic membrane in a simple electrostatic Micro-Electromechanical System (MEMS) device. In this paper, we analyze and estimate the quenching time of the elastic membrane in terms of the applied voltage --represented here by $λ$. As a byproduct, we prove that for sufficiently large $λ$, finite-time quenching must occur near the maximum point of the varying dielectric permittivity profile $f(x)$.

math.AP↗

arXiv2007

Bessel potentials and optimal Hardy and Hardy-Rellich inequalities

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 $u \in C_{0}^{\infty}(B)$: $\int_{B}V(x)|\nabla u |^{2}dx \geq \int_{B} W(x)u^2dx$, and $\int_{B}V(x)|Δu |^{2}dx \geq \int_{B} W(x)|\nabla u|^{2}dx+(n-1)\int_{B}(\frac{V(x)}{|x|^2}-\frac{V_r(|x|)}{|x|})|\nabla u|^2dx$. This characterization makes a very useful connection between Hardy-type inequalities and the oscillatory behaviour of certain ordinary differential equations, and helps in the identification of a large number of such couples (V, W) - that we call Bessel pairs -as well as the best constants in the corresponding inequalities. This allows us to improve, extend, and unify many results -old and new- about Hardy and Hardy-Rellich type inequalities, such as those obtained by Caffarelli-Kohn-Nirenberg, Brezis-Vazquez, Wang-Willem, Adimurthi-Chaudhuri-Ramaswamy, Filippas-Tertikas, Adimurthi-Grossi -Santra, Tertikas-Zographopoulos, and Blanchet-Bonforte-Dolbeault-Grillo-Vasquez.

math.AP↗

arXiv2007

Schrodinger equations and Hamiltonian systems of PDEs with selfdual boundary conditions

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 well as certain nonlinear Schrodinger evolutions. This allows for the resolution of such equations under general time boundary conditions which include the more traditional ones such as initial value problems, periodic and anti-periodic orbits, but also yield new ones such as "periodic orbits up to an isometry" for evolution equations that may not have periodic solutions. In the process, we introduce a method for perturbing selfdual functionals in order to induce coercivity and compactness, while keeping the system selfdual.

math.AP↗

arXiv2007

On the best possible remaining term in the Hardy Inequality

We give a necessary and sufficient condition on a radially symmetric potential $V$ on $Ω$ that makes it an admissible candidate for an improved Hardy inequality of the following form: \begin{equation}\label{gen-hardy.0} \hbox{$\int_Ω|\nabla u |^{2}dx - (\frac{n-2}{2})^{2} \int_Ω\frac{|u|^{2}}{|x|^{2}}dx\geq c\int_Ω V(|x|)|u|^{2}dx$ \quad for all $u \in H^{1}_{0}(Ω)$.} \end{equation}

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arXiv2007

Anti-symmetric Hamiltonians (II): Variational resolutions for Navier-Stokes and other nonlinear evolutions

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 not however cover the case of nonlinear evolutions such as the Navier-Stokes equations. One of the reasons is the prohibitive coercivity condition that is not satisfied by the corresponding selfdual functional on the relevant path space. We show here that such a principle still hold for functionals of the form I(u)= \int_0^T \Big [ L (t, u(t),\dot {u}(t)+Λu(t)) +< Λu(t), u(t) > \Big ] dt +\ell (u(0)- u(T), \frac {u(T)+ u(0)}{2}) where $L$ (resp., $\ell$) is an anti-selfdual Lagrangian on state space (resp., boundary space), and $Λ$ is an appropriate nonlinear operator on path space. As a consequence, we provide a variational formulation and resolution to evolution equations involving nonlinear operators such as the Navier-Stokes equation (in dimensions 2 and 3) with various boundary conditions. In dimension 2, we recover the well known solutions for the corresponding initial-value problem as well as periodic and anti-periodic ones, while in dimension 3 we get Leray solutions for the initial-value problems, but also solutions satisfying $u(0)=αu(T)$ for any given $α$ in $(-1,1)$. Our approach is quite general and does apply to many other situations.

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