arXiv subjects

Nassif Ghoussoub

Publications and source records attributed to Nassif Ghoussoub.

At least 37 records · Page 2Linked to original sources

arXiv2017

On Optimal Stochastic Ballistic Transports

For a given Lagrangian $L:[0,T]\times M\times M^\ast\rightarrow \mathbb{R}_+$ and probability measures $μ\in\mathcal{P}(M^\ast)$, $ν\in \mathcal{P}(M)$, we introduce the stochastic ballistic transportation problems \begin{align}\tag{$\star$} \underline{B}(μ,ν):=\inf\left\{\mathbb{E}\left[\langle V,X_0\rangle +\int_0^T L(t,X,β(t,X))\,dt\right]\middle\rvert V\simμ,X_T\sim ν\right\}\\\tag{$\star\star$} \overline{B}(ν,μ):=\sup\left\{\mathbb{E}\left[\langle V,X_T\rangle -\int_0^T L(t,X,β(t,X))\,dt\right]\middle\rvert V\simμ,X_0\sim ν\right\} \end{align} where $X$ is a diffusion process with drift $β$. This cost is based on the stochastic optimal transport problem presented by Mikami and the deterministic ballistic transport introduced by Ghoussoub. We obtain a Kantorovich-style duality result that reformulates this problem in terms of solutions to the Hamilton-Jacobi-Bellman equation \begin{equation*} \frac{\partialϕ}{\partial t}+\frac{1}{2}Δϕ+H(t,x,\nablaϕ)=0, \end{equation*} and show how optimal processes may be thereby attained.

math.AP↗

arXiv2017

Optimal Brownian Stopping between radially symmetric marginals in general dimensions

Given an initial (resp., terminal) probability measure $μ$ (resp., $ν$) on $\mathbb{R}^d$, we characterize those optimal stopping times $τ$ that maximize or minimize the functional $\mathbb{E} |B_0 - B_τ|^α$, $α> 0$, where $(B_t)_t$ is Brownian motion with initial law $B_0\sim μ$ and with final distribution --once stopped at $τ$-- equal to $B_τ\sim ν$. The existence of such stopping times is guaranteed by Skorohod-type embeddings of probability measures in "subharmoic order" into Brownian motion. This problem is equivalent to an optimal mass transport problem with certain constraints, namely the optimal subharmonic martingale transport. Under the assumption of radial symmetry on $μ$ and $ν$, we show that the optimal stopping time is a hitting time of a suitable barrier, hence is non-randomized and is unique.

math.PR↗

arXiv2017

The Hardy-Schrödinger operator with interior singularity: The remaining cases

We consider the remaining unsettled cases in the problem of existence of energy minimizing solutions for the Dirichlet value problem $L_γu-λu=\frac{u^{2^*(s)-1}}{|x|^s}$ on a smooth bounded domain $Ω$ in $\mathbb{R}^n$ ($n\geq 3$) having the singularity $0$ in its interior. Here $γ<\frac{(n-2)^2}{4}$, $0\leq s <2$, $2^*(s):=\frac{2(n-s)}{n-2}$ and $0\leq λ<λ_1(L_γ)$, the latter being the first eigenvalue of the Hardy-Schrödinger operator $L_γ:=-Δ-\fracγ{|x|^2}$. There is a threshold $λ^*(γ, Ω) \geq 0$ beyond which the minimal energy is achieved, but below which, it is not. It is well known that $λ^*(Ω) = 0$ in higher dimensions, for example if $0\leq γ\leq \frac{(n-2)^2}{4}-1$. Our main objective in this paper is to show that this threshold is strictly positive in "lower dimensions" such as when $ \frac{(n-2)^2}{4}-1<γ<\frac{(n-2)^2}{4}$, to identify the critical dimensions (i.e., when the situation changes), and to characterize it in terms of $Ω$ and $γ$. If either $s>0$ or if $γ> 0$, i.e., in {\it the truly singular case}, we show that in low dimensions, a solution is guaranteed by the positivity of the "Hardy-singular internal mass" of $Ω$, a notion that we introduce herein. On the other hand, and just like the case wnen $γ=s=0$ studied by Brezis-Nirenberg and completed by Druet, $n=3$ is the critical dimension, and the classical positive mass theorem is sufficient for the {\it merely singular case}, that is when $s=0$, $γ\leq 0$.

math.AP↗

arXiv2017

Sign-Changing Solutions for Critical Equations with Hardy Potential

We consider the following perturbed critical Dirichlet problem involving the Hardy-Schrödinger operator on a smooth bounded domain $Ω\subset \mathbb{R}^N$, $N\geq 3$, with $0 \in Ω$: $$ \left\{ \begin{array}{ll}-Δu-γ\frac{u}{|x|^2}-εu=|u|^{\frac{4}{N-2}}u &\hbox{in }Ωu=0 & \hbox{on }\partial Ω, \end{array}\right. $$ when $ε>0$ is small and $γ< {(N-2)^2\over4}$. Setting $ γ_j= \frac{(N-2)^2}{4}\left(1-\frac{j(N-2+j)}{N-1}\right)\in(-\infty,0]$ for $j \in \mathbb{N},$ we show that if $γ\leq \frac{(N-2)^2}{4}-1$ and $γ\neq γ_j$ for any $j$, then for small $ε$, the above equation has a positive --non variational-- solution that develops a bubble at the origin. If moreover $γ<\frac{(N-2)^2}{4}-4,$ then for any integer $k \geq 2$, the equation has for small enough $ε$, a sign-changing solution that develops into a superposition of $k$ bubbles with alternating sign centered at the origin. The above result is optimal in the radial case, where the condition that $γ\neq γ_j$ is not necessary. Indeed, it is known that, if $γ> \frac{(N-2)^2}{4}-1$ and $Ω$ is a ball $B$, then there is no radial positive solution for $ε>0$ small. We complete the picture here by showing that, if $γ\geq \frac{(N-2)^2}{4}-4$, then the above problem has no radial sign-changing solutions for $ε>0$ small. These results recover and improve what is known in the non-singular case, i.e., when $γ=0$.

math.AP↗

arXiv2017

Optimal Ballistic Transport and Hopf-Lax Formulae on Wasserstein Space

We investigate the optimal mass transport problem associated to the following "ballistic" cost functional on phase space $M\times M^*$, $$ b_T(v, x):=\inf\{\langle v, γ(0)\rangle +\int_0^TL(γ(t), {\dot γ}(t))\, dt, γ\in C^1([0, T), M), γ(T)=x\}, $$ where $M=\mathbb{R}^d$, $T>0$, and $L:M\times M \to \mathbb{R}$ is a Lagrangian that is jointly convex in both variables. Under suitable conditions on the initial and final probability measures, we use convex duality à la Bolza and Monge-Kantorovich theory to lift classical Hopf-Lax formulae from state space to Wasserstein space. This allows us to relate optimal transport maps for the ballistic cost to those associated with the fixed-end cost defined on $M\times M$ by $$ c_T(x,y):=\inf\{\int_0^TL(γ(t), {\dot γ}(t))\, dt, γ\in C^1([0, T), M), γ(0)=x, γ(T)=y\}. $$ We also point to links with the theory of mean field games.

math.AP↗

arXiv2017

Mass and Asymptotics associated to Fractional Hardy-Schrödinger Operators in Critical Regimes

We consider linear and non-linear boundary value problems associated to the fractional Hardy-Schrödinger operator $ L_{γ,α}: = ({-}{ Δ})^{\fracα{2}}- \fracγ{|x|^α}$ on domains of $\mathbb{R}^n$ containing the singularity $0$, where $0<α<2$ and $ 0 \le γ< γ_H(α)$, the latter being the best constant in the fractional Hardy inequality on $\mathbb{R}^n$. We tackle the existence of least-energy solutions for the borderline boundary value problem $(L_{γ,α}-λI)u= {\frac{u^{2^\star_α(s)-1}}{|x|^s}}$ on $Ω$, where $0\leq s <α<n$ and $ 2^\star_α(s)={\frac{2(n-s)}{n-α}}$ is the critical fractional Sobolev exponent. We show that if $γ$ is below a certain threshold $γ_{crit}$, then such solutions exist for all $0<λ<λ_1(L_{γ,α})$, the latter being the first eigenvalue of $L_{γ,α}$. On the other hand, for $γ_{crit}<γ<γ_H(α)$, we prove existence of such solutions only for those $λ$ in $(0, λ_1(L_{γ,α}))$ for which the domain $Ω$ has a positive {\it fractional Hardy-Schrödinger mass} $m_{γ, λ}(Ω)$. This latter notion is introduced by way of an invariant of the linear equation $(L_{γ,α}-λI)u=0$ on $Ω$.

math.AP↗

arXiv2016

Structure of optimal martingale transport plans in general dimensions

Given two probability measures $μ$ and $ν$ in "convex order" on $\R^d$, we study the profile of one-step martingale plans $π$ on $\R^d\times \R^d$ that optimize the expected value of the modulus of their increment among all martingales having $μ$ and $ν$ as marginals. While there is a great deal of results for the real line (i.e., when $d=1$), much less is known in the richer and more delicate higher dimensional case that we tackle in this paper. We show that many structural results can be obtained whenever a natural dual optimization problem is attained, provided the initial measure $μ$ is absolutely continuous with respect to the Lebesgue measure. One such a property is that $μ$-almost every $x$ in $\R^d$ is transported by the optimal martingale plan into a probability measure $π_x$ concentrated on the extreme points of the closed convex hull of its support. This will be established in full generality in the 2-dimensional case, and also for any $d\geq 3$ as long as the marginals are in "subharmonic order". In some cases, $π_x$ is supported on the vertices of a $k(x)$-dimensional polytope, such as when the target measure is discrete. Many of the proofs rely on a remarkable decomposition of "martingale supporting" Borel subsets of $\R^d\times \R^d$ into a collection of mutually disjoint components by means of a "convex paving" of the source space. If the martingale is optimal, then each of the components in the decomposition supports a restricted optimal martingale transport for which the dual problem is attained. These decompositions are used to obtain structural results in cases where duality is not attained. On the other hand, they can also be related to higher dimensional Nikodym sets. %On the other hand, they can also lead to natural and intriguing constructions of higher dimensional Nikodym sets.

math.AP↗

arXiv2015

Metric Selfduality and Monotone Vector Fields on Manifolds

We develop a "metrically selfdual" variational calculus for $c$-monotone vector fields between general manifolds $X$ and $Y$, where $c$ is a coupling on $X\times Y$. Remarkably, many of the key properties of classical monotone operators known to hold in a linear context, extend to this non-linear setting. This includes an integral representation of $c$-monotone vector fields in terms of $c$-convex selfdual Lagrangians, their characterization as a partial $c$-gradients of antisymmetric Hamiltonians, as well as the property that these vector fields are generically single-valued. We also use a symmetric Monge-Kantorovich transport to associate to any measurable map its closest possible $c$-monotone "rearrangement". We also explore how this metrically selfdual representation can lead to a global variational approach to the problem of inverting $c$-monotone maps, an approach that has proved efficient for resolving non-linear equations and evolutions driven by monotone vector fields in a Hilbertian setting.

math.AP↗

arXiv2015

Sobolev inequalities for the Hardy-Schrödinger operator: Extremals and critical dimensions

In this expository paper, we consider the Hardy-Schrödinger operator $-Δ-γ/|x|^2$ on a smooth domain Ωof R^n with 0\in\barΩ, and describe how the location of the singularity 0, be it in the interior of Ωor on its boundary, affects its analytical properties. We compare the two settings by considering the optimal Hardy, Sobolev, and the Caffarelli-Kohn-Nirenberg inequalities. The latter rewrites: $C(\int_Ω\frac{u^{p}}{|x|^s}dx)^{\frac{2}{p}}\leq \int_Ω |\nabla u|^2dx-γ\int_Ω\frac{u^2}{|x|^2}dx$ for all $u\in H^1_0(Ω)$, where γ<n^2/4, s\in [0,2) and p:=2(n-s)/(n-2). We address questions regarding the explicit values of the optimal constant C, as well as the existence of non-trivial extremals attached to these inequalities. Scale invariance properties lead to situations where the best constants do not depend on the domain and are not attainable. We consider two different approaches to "break the homogeneity" of the problem: One approach was initiated by Brezis-Nirenberg and by Janelli. It is suitable for the case where 0 is in the interior of Ω, and consists of considering lower order perturbations of the critical nonlinearity. The other approach was initiated by Ghoussoub-Kang , C.S. Lin et al. and Ghoussoub-Robert. It consists of considering domains where the singularity is on the boundary. Both of these approaches are rich in structure and in challenging problems. If 0\in Ω, a negative linear perturbation suffices for higher dimensions, while a positive "Hardy-singular interior mass" is required in lower dimensions. If the singularity is on the boundary, then the local geometry around 0 plays a crucial role in high dimensions, while a positive "Hardy-singular boundary mass" is needed for the lower dimensions.

math.AP↗

arXiv2015

Borderline variational problems involving fractional Laplacians and critical singularities

We consider the problem of attainability of the best constant in the following critical fractional Hardy-Sobolev inequality: \begin{equation*} μ_{γ,s}(\R^n):= \inf\limits_{u \in H^{\fracα{2}} (\R^n)\setminus \{0\}} \frac{ \int_{\R^n} |({-}{ Δ})^{\fracα{4}}u|^2 dx - γ\int_{\R^n} \frac{|u|^2}{|x|^α}dx }{(\int_{\R^n} \frac{|u|^{2_α^*(s)}}{|x|^{s}}dx)^\frac{2}{2_α^*(s)}}, \end{equation*} where $0\leq s<α<2$, $n>α$, ${2_α^*(s)}:=\frac{2(n-s)}{n-α},$ and $γ\in \mathbb{R}$. This allows us to establish the existence of nontrivial weak solutions for the following doubly critical problem on $\R^n$, \begin{equation*} \left\{\begin{array}{lll} ({-}{ Δ})^{\fracα{2}}u- γ\frac{u}{|x|^α}&= |u|^{2_α^*-2} u + {\frac{|u|^{2_α^*(s)-2}u}{|x|^s}} & \text{in } {\R^n}\\ \hfill u&>0 & \text{in } \R^n, \end{array}\right. \end{equation*} where $2_α^*:=\frac{2 n}{n-α}$ is the critical $α$-fractional Sobolev exponent, and $γ< γ_H:=2^α\frac{Γ^2(\frac{n+α}{4})}{Γ^2(\frac{n-α}{4})}$, the latter being the best fractional Hardy constant on $\R^n$.

math.AP↗

arXiv2015

A dual Moser-Onofri inequality and its extensions to higher dimensional spheres

We use optimal mass transport to provide a new proof and a dual formula to the Moser-Onofri inequality on $\s^2$ in the same spirit as the approach of Cordero-Erausquin, Nazaret and Villani to the Sobolev inequality and of Agueh-Ghoussoub-Kang to more general settings. There are however many hurdles to overcome once a stereographic projection on $\R^2$ is performed: Functions are not necessarily of compact support, hence boundary terms need to be evaluated. Moreover, the corresponding dual free energy of the reference probability density $μ_2(x)=\frac{1}{π(1+|x|^2)^2}$ is not finite on the whole space, which requires the introduction of a renormalized free energy into the dual formula. We also extend this duality to higher dimensions and establish an extension of the Onofri inequality to spheres $\s^n$ with $n\geq 2$. What is remarkable is that the corresponding free energy is again given by $F(ρ)=-nρ^{1-\frac{1}{n}}$, which means that both the {\it prescribed scalar curvature problem} and the {\it prescribed Gaussian curvature problem} lead essentially to the same dual problem whose extremals are stationary solutions of the fast diffusion equations.

math.AP↗

arXiv2014

On the Hardy-Schrödinger operator with a boundary singularity

We investigate the Hardy-Schrödinger operator $L_γ=-Δ-\fracγ{|x|^2}$ on domains $Ω\subset\rn$, whose boundary contain the singularity $0$. The situation is quite different from the well-studied case when $0$ is in the interior of $Ω$. For one, if $0\inΩ$, then $L_γ$ is positive if and only if $γ<\frac{(n-2)^2}{4}$, while if $0\in\partialΩ$ the operator $L_γ$ could be positive for larger value of $γ$, potentially reaching the maximal constant $\frac{n^2}{4}$ on convex domains. We prove optimal regularity and a Hopf-type Lemma for variational solutions of corresponding linear Dirichlet boundary value problems of the form $L_γ u=a(x)u$, but also for non-linear equations including $L_{_γ} u=\frac{|u|^{\crits-2}u}{|x|^s}$, where $γ<\frac{n^2}{4}$, $s\in [0,2)$ and $\crits:=\frac{2(n-s)}{n-2}$ is the critical Hardy-Sobolev exponent. We also provide a Harnack inequality and a complete description of the profile of all positive solutions --variational or not-- of the corresponding linear equation on the punctured domain. The value $γ=\frac{n^2-1}{4}$ turned out to be another critical threshold for the operator $L_γ$, and our analysis yields a corresponding notion of "Hardy singular boundary-mass" $m_γ(Ω)$ of a domain $Ω$ having $0\in \partial Ω$, which could be defined whenever $\frac{n^2-1}{4}<γ<\frac{n^2}{4}$.

math.AP↗

arXiv2013

Variational representations for N-cyclically monotone vector fields

Given a convex bounded domain $Ω$ in ${\mathbb{R}}^{d}$ and an integer $N\geq 2$, we associate to any jointly $N$-monotone $(N-1)$-tuplet $(u_1, u_2,..., u_{N-1})$ of vector fields from $% Ω$ into $\mathbb{R}^{d}$, a Hamiltonian $H$ on ${\mathbb{R}}^{d} \times {\mathbb{R}}^{d} ... \times {\mathbb{R}}^{d}$, that is concave in the first variable, jointly convex in the last $(N-1)$ variables such that for almost all $% x\in Ω$, \hbox{$(u_1(x), u_2(x),..., u_{N-1}(x))= \nabla_{2,...,N} H(x,x,...,x)$. Moreover, $H$ is $N$-sub-antisymmetric, meaning that $\sum% \limits_{i=0}^{N-1}H(σ^{i}(\mathbf{x}))\leq 0$ for all $\mathbf{x}% =(x_{1},...,x_{N})\in Ω^{N}$, $σ$ being the cyclic permutation on ${\mathbb{R}}^{d}$ defined by $σ(x_{1},x_2,...,x_{N})=(x_{2},x_{3},...,x_{N},x_{1})$. Furthermore, $H$ is $N$% -antisymmetric in a sense to be defined below. This can be seen as an extension of a theorem of E. Krauss, which associates to any monotone operator, a concave-convex antisymmetric saddle function. We also give various variational characterizations of vector fields that are almost everywhere $N$-monotone, showing that they are dual to the class of measure preserving $N$-involutions on $Ω$.

math.OC↗

arXiv2013

Symmetric Monge-Kantorovich problems and polar decompositions of vector fields

For any given integer $N\geq 2$, we show that every bounded measurable vector field from a bounded domain $Ω$ into $\R^d$ is $N$-cyclically monotone up to a measure preserving $N$-involution. The proof involves the solution of a multidimensional symmetric Monge-Kantorovich problem, which we first study in the case of a general cost function on a product domain $Ω^N$. The polar decomposition described above corresponds to a special cost function derived from the vector field in question (actually $N-1$ of them). In this case, we show that the supremum over all probability measures on $Ω^N$ which are invariant under cyclic permutations and with a given first marginal $μ$, is attained on a probability measure that is supported on the graph of a function of the form $x\to (x, Sx, S^2x,..., S^{N-1}x)$, where $S$ is a $μ$-measure preserving transformation on $Ω$ such that $S^N=I$ a.e. The proof exploits a remarkable duality between such involutions and those Hamiltonians that are $N$-cyclically antisymmetric.

math.AP↗

arXiv2013

Decoupling of DeGiorgi-type systems via multi-marginal optimal transport

We exhibit a surprising relationship between elliptic gradient systems of PDEs, multi-marginal Monge-Kantorovich optimal transport problem, and multivariable Hardy-Littlewood inequalities. We show that the notion of an orientable elliptic system, conjectured in [6] to imply that (in low dimensions) solutions with certain monotonicity properties are essentially 1-dimensional, is equivalent to the definition of a compatible cost function, known to imply uniqueness and structural results for optimal measures to certain Monge-Kantorovich problems [11]. Orientable nonlinearities and compatible cost functions turned out to be also related to submodular functions, which appear in rearrangement inequalities of Hardy-Littlewood type. We use this equivalence to establish a decoupling result for certain solutions to elliptic PDEs and show that under the orientability condition, the decoupling has additional properties, due to the connection to optimal transport.

math.AP↗

arXiv2012

Remarks on multi-marginal symmetric Monge-Kantorovich problems

Symmetric Monge-Kantorovich transport problems involving a cost function given by a family of vector fields were used by Ghoussoub-Moameni to establish polar decompositions of such vector fields into $m$-cyclically monotone maps composed with measure preserving $m$-involutions ($m\geq 2$). In this note, we relate these symmetric transport problems to the Brenier solutions of the Monge and Monge-Kantorovich problem, as well as to the Gangbo-Świȩch solutions of their multi-marginal counterparts, both of which involving quadratic cost functions.

math.AP↗

arXiv2012

On the Hénon-Lane-Emden conjecture

We consider Liouville-type theorems for the following Hénon-Lane-Emden system \hfill -Δu&=& |x|^{a}v^p \text{in} \mathbb{R}^N, \hfill -Δv&=& |x|^{b}u^q \text{in} \mathbb{R}^N, when $pq>1$, $p,q,a,b\ge0$. The main conjecture states that there is no non-trivial non-negative solution whenever $(p,q)$ is under the critical Sobolev hyperbola, i.e. $ \frac{N+a}{p+1}+\frac{N+b}{q+1}>{N-2}$. We show that this is indeed the case in dimension N=3 provided the solution is also assumed to be bounded, extending a result established recently by Phan-Souplet in the scalar case. Assuming stability of the solutions, we could then prove Liouville-type theorems in higher dimensions. For the scalar cases, albeit of second order ($a=b$ and $p=q$) or of fourth order ($a\ge 0=b$ and $p>1=q$), we show that for all dimensions $N\ge 3$ in the first case (resp., $N\ge 5$ in the second case), there is no positive solution with a finite Morse index, whenever $p$ is below the corresponding critical exponent, i.e $ 1<p<\frac{N+2+2a}{N-2}$ (resp., $ 1<p<\frac{N+4+2a}{N-4}$). Finally, we show that non-negative stable solutions of the full Hénon-Lane-Emden system are trivial provided \label{sysdim00} N<2+2(\frac{p(b+2)+a+2}{pq-1}) (\sqrt{\frac{pq(q+1)}{p+1}}+ \sqrt{\frac{pq(q+1)}{p+1}-\sqrt\frac{pq(q+1)}{p+1}}).

math.AP↗

arXiv2012

Regularity of semi-stable solutions to fourth order nonlinear eigenvalue problems on general domains

We examine the fourth order problem $Δ^2 u = λf(u) $ in $ Ω$ with $ Δu = u =0 $ on $ \partial Ω$, where $ λ> 0$ is a parameter, $ Ω$ is a bounded domain in $ R^N$ and where $f$ is one of the following nonlinearities: $ f(u)=e^u$, $ f(u)=(1+u)^p $ or $ f(u)= \frac{1}{(1-u)^p}$ where $ p>1$. We show the regularity of all semi-stable solutions and hence of the extremal solutions, provided [N < 2 + 4 \sqrt{2} + 4 \sqrt{2 - \sqrt{2}} \approx 10.718 when $ f(u)=e^u$,] and [\frac{N}{4} < \frac{p}{p-1} + \frac{p+1}{p-1} (\sqrt{\frac{2p}{p+1}} + \sqrt{\frac{2p}{p+1} - \sqrt{\frac{2p}{p+1}}} - 1/2)] when $ f(u)=(u+1)^p$. New results are also obtained in the case where $ f(u)=(1-u)^{-p}$. These are substantial improvements to various results on critical dimensions obtained recently by various authors. We view the equation as a system and then derive a new stability inequality, valid for minimal solutions, which allows a method of proof which is reminiscent of the second order case.

math.AP↗