# Nassif Ghoussoub

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## Graduate Student Supervision

##### Doctoral Student Supervision (Jan 2008 - May 2021)

In this work, we introduce and study a class of convex functionals on pairs ofprobability measures, the linear transfers, which have a structure that com-monly arises in the dual formulations of many well-studied variational prob-lems. We show that examples of linear transfers include a large number ofwell-known transport problems, including the weak, stochastic, martingale,and cost-minimising transports. Further examples include the balayage ofmeasures, and ergodic optimisation of expanding dynamical systems, amongothers. We also introduce an extension of the linear transfers, the convextransfers, and show that they include the relative entropy functional andp-powers (p ≥ 1) of linear transfers.We study the properties of linear and convex transfers and show that theinf-convolution operation preserves their structure. This allows dual formu-lations of transport-entropy and other related inequalities, to be computedin a systematic fashion.Motivated by connections of optimal transport to the theory of Aubry-Mather and weak KAM for Hamiltonian systems, we develop an analog inthe setting of linear transfers. We prove the existence of an idempotentoperator which maps into the set of weak KAM solutions, an idempotentlinear transfer that plays the role of the Peierls barrier, and we identifyanalogous objects in this setting such as the Mather measures and the Aubryset. We apply this to the framework of ergodic optimisation in the holonomiccase.

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In the first part of this thesis, we use the theory of self-duality to provide a variational approach for the resolution of a number of stochastic partial differential equations. We will be able to address the problem of existence of solutions to a class of semilinear stochastic partial differential equations in the form du+A(t,u(t))dt=B(t,u(t)) dW with u(0)=u₀, where for every t∊[0,T], A(t, ‧) is a maximal monotone operator on a reflexive Banach space V, and B is a linear or non-linear operator with values in a Hilbert space H. We use the fact that any maximal monotone operator A can be expressed as a potential of a self-dual Lagrangian L to associate to the equation a (completely) self-dual functional whose minimizer on a suitable path space yields a solution.One particular case of the above equation which already contains a large number of stochastic PDEs is when A is the subdifferential of a convex function φ. More generally, we can deal with equations of the form du(t)= -∂φ(t,u(t))dt + B(t,u(t))dW(t) with u(0)=u₀.We also prove the existence of solutions to SPDEs in divergence form involving a maximal monotone operator β on the n-dimensional Euclidean space which is not necessarily the gradient of a convex function:du= div(β(∇u(t,x)))dt +B(t)dW(t) if u∊[0,T]×D and u(0)=u₀ when u∊∂Dwhere D is a bounded domain in the n-dimensional Euclidean space.In the second part of the thesis, we use methods from optimal transport to address functional inequalities on the n-dimensional sphere. We prove Energy-Entropy duality formulas that yield and improve the celebrated Moser-Onofri inequalities on the 2-dimensional sphere.

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We obtain a few existence results for elliptic equations.We develop in Chapter 2 a new infinite dimensional gluing scheme for fractionalelliptic equations in the mildly non-local setting. Here it is applied to the catenoid. As a consequence of this method, a counter-example to a fractional analogue of De Giorgi conjecture can be obtained [51].Then, in Chapter 3, we construct singular solutions to the fractional Yamabe problem using conformal geometry. Fractional order ordinary differential equations are studied.Finally, in Chapter 4, we obtain the existence to a suitably perturbed doublycritical Hardy–Schr¨odinger equation in a bounded domain in the hyperbolic space.

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In this thesis, we study properties of the fractional Hardy-Schrödinger operator L_(γ,α)≔(-∆)^(α/(2 ))- γ/〖|x|〗^α both on R^n and on its bounded domains. The following functional inequality is key to our variational approach. C〖(∫_(R^n)〖 〖|u|〗^(2_α^* (s))/〖|x|〗^s dx〗)〗^(2/(2_α^* (s)))≤ ∫_(R^n)〖 〖|(-∆)^(α/(4 )) u|〗^2 dx〗- γ∫_(R^n)〖 〖|u|〗^2/〖|x|〗^α dx,〗 where 0 ≤ s α, 2_α^* (s)=(2(n-s))/(n-α) and γ 0 attached to this inequality. This allows us to establish the existence of non-trivial weak solutions for the following doubly critical problem on R^n, L_(γ,α) u=〖|u|〗^(2_α^*-2)u + (〖|u|〗^(2_α^* (s)-2) u)/〖|x|〗^s in R^n, where 〖2_α^*≔2〗_α^* (0).We then look for least-energy solutions of the following linearly perturbed non-linear boundary value problem on bounded subdomains of R^n containing the singularity 0: (L_(γ,α)-λΙ)u= 〖|u|〗^(2_α^* (s)-1)/〖|x|〗^s on Ω, We show that if γ is below a certain threshold γ_crit(α), then such solutions exist for all 〖0

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In the first part of this thesis, we study the structure of solutions to the optimal martingale transport problem, when the marginals lie in higher dimensional Euclidean spaces (ℝ^d, d ≥ 2). The problem has been extensively studied in one-dimensional space (ℝ), but few results have been shown in higher dimensions. In this thesis, we propose two conjectures and provide key ideas that lead to solutions in important cases.In the second part, we study the structure of solutions to the optimal subharmonic martingale transport problem, again when the marginals lie in higher dimensional Euclidean spaces. First, we show that this problem has an equivalent formulation in terms of the celebrated Skorokhod embedding problem in probability theory. We then describe the fine structure of the solution provided the marginals are radially symmetric. The general case remains unsolved, and its potential solution calls for a deeper understanding of harmonic analysis and Brownian motion in higher dimensional spaces.

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This thesis which is a compendium of seven papers, focuses on the study of the semilinear elliptic equations and systems, on both bounded and unbounded domains of dimension n, most importantly the Allen-Cahn equation and the De Giorgi’s conjecture (1978). This conjecture brings together two groups of mathematicians: one specializing in nonlinear partial differential equations and another in differential geometry, more specifically on minimal surfaces and constant mean curvature surfaces. De Giorgi conjectured that the monotone and bounded solutions of the Allen-Cahn equation on the whole space of dimension n ≤ 8 must be 1-dimensional solutions. This is known to be true for n ≤ 3 and with extra (natural) assumptions for 4 ≤ n ≤ 8.Motivated by this conjecture, I have introduced two main concepts:The first concept is the “H-monotone solutions” that allows us to formulate a counterpart of the De Giorgi’s conjecture for system of equations stating that the H-monotone and bounded solutions of the gradient systems on the whole space of dimension n ≤ 8 must be 1-dimensional solutions. This seems to be in the right track to extend the De Giorgi's conjecture to systems.The second concept is the “m-Liouville theorem” for m = 0, · · · , n − 1 that allows us to formulate a counterpart of the De Giorgi’s conjecture for equations but this time for higher-dimensional solutions as opposed to 1-dimensional solutions. We use the induction idea that is to use 0-Liouville theorem (0- dimensional solutions) to prove 1-Liouville theorem (1-dimensional solutions) and then to prove (n − 1)- Liouville theorem ((n − 1)-dimensional solutions). The reason that we call this “m-Liouville theorem” is because of the great mathematician Joseph Liouville (1809-1882) who proved a classical theorem in complex analysis stating that "bounded harmonic functions on the whole space must be constant" and constants are 0-dimensional objects. 0-Liouville theorem is at the heart of this thesis and it includes various 0-Liouville theorems for various equations and system. In particular, we give a positive answer to the Henon-Lane-Emden conjecture in dimension three under an extra boundedness assumption. On the other hand, it is well known that there is a close relationship between the regularity of solutions on bounded domains and 0-Liouville theorem for related “limiting equations” on the whole space, via rescaling and blow up procedures. In this direction, we present regularity of solutions for gradient and twisted-gradient systems as well as the uniqueness results for nonlocal eigenvalue problems. The novelty here is a stability inequality for both gradient and twisted-gradient systems that gives us the chance to adjust the known techniques and ideas (for equations) to systems.

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This thesis consists of three parts and manuscripts of seven research papers studying improved Hardy and Hardy-Rellich inequalities, nonlinear eigenvalue problems, and simultaneouspreconditioning and symmetrization of linear systems.In the first part that consists of three research papers we study improved Hardy and Hardy-Rellich inequalities. In sections 2 and 3, we give necessary and sufficient conditions on a pair of positive radial functions V and W on a ball B of radius R in Rn, n ≥ 1, so that the following inequalities hold for all u \in C_{0}^{\infty}(B):\begin{equation*} \label{one}\hbox{$\int_{B}V(x)|\nabla u²dx \geq \int_{B} W(x)u²dx,}\end{equation*}\begin{equation*} \label{two}\hbox{$\int_{B}V(x)|\Delta u|²dx \geq \int_{B} W(x)|\nabla %@u|^²dx+(n-1)\int_{B}(\frac{V(x)}{|x|²}-\frac{V_r(|x|)}{|x|})|\nablau|²dx.\end{equation*}This characterization makes a very useful connection between Hardy-type inequalities and the oscillatory behavior of certain ordinary differential equations. This allows us to improve, extend, and unify many results about Hardy and Hardy-Rellich type inequalities. In section 4, with a similar approach, we presentvarious classes of Hardy-Rellich inequalities on H²\cap H¹₀The second part of the thesis studies the regularity of the extremal solution of fourth order semilinear equations. In sections 5 and 6we study the extremal solution u_{\lambda^*}$ of the semilinear biharmonic equation $\Delta² u=\frac{\lambda}{(1-u)², which models a simple Micro-Electromechanical System (MEMS) device on aball B\subset R^N, under Dirichlet or Navier boundary conditions. We show that u* is regular provided N ≤ 8 while u_{\lambda^*} is singular for N ≥ 9. In section 7, by a rigorous mathematical proof, we show that the extremal solutions of the bilaplacian with exponential nonlinearity is singular in dimensions N ≥ 13.In the third part, motivated by the theory of self-duality we propose new templates for solving non-symmetric linear systems. Our approach is efficient when dealing with certain ill-conditioned and highly non-symmetric systems.

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We apply self-dual variational calculus to inverse problems, optimal control problems and homogenization problems in partial differential equations.Self-dual variational calculus allows for the variational formulation of equations which do not have to be of Euler-Lagrange type. Instead, a monotonicity condition permits the construction of a self-dual Lagrangian. This Lagrangian then permits the construction of a non-negative functional whose minimum value is zero, and its minimizer is a solution to the corresponding equation.In the case of inverse and optimal control problems, we use the variational functional given by the self-dual Lagrangian as a penalization functional, which naturally possesses the ideal qualities for such a role. This allows for the application of standard variational techniques in a convex setting, as opposed to working with more complex constrained optimization problems. This extends work pioneered by Barbu and Kunisch.In the case of homogenization problems, we recover existing results by dal Maso, Piat, Murat and Tartar with the use of simpler machinery. In this context self-dual variational calculus permits one to study the asymptotic properties of the potential functional using classical Gamma-convergence techniques which are simpler to handle than the direct techniques required to study the asymptotic properties of the equation itself. The approach also allows for the seamless handling of multivalued equations. The study of such problems introduces naturally the study of the topological structures of the spaces of maximal monotone operators and their corresponding self-dual potentials. We use classical tools such as Gamma-convergence, Mosco convergence and Kuratowski-Painlevé convergence and show that these tools are well suited for the task. Results from convex analysis regarding these topologies are extended to the more general case of maximal monotone operators in a natural way. Of particular interest is that the Gamma-convergence of self-dual Lagrangians is equivalent to the Mosco convergence, and this in turn implies the Kuratowski-Painlevé convergence of their corresponding maximal monotone operators; this partially extends a classical result by Attouch relating the convergence of convex functions to the convergence of their corresponding subdifferentials.

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This thesis consists of six research papers.In ``Regularity of the extremal solution in a MEMS model with advection,'' we examine the equation given by $ -\Delta u + c(x) \cdot \nabla u = \lambda f(u) $ in $ \Omega$ with Dirichlet boundary conditions and where $ f(u) = (1-u)^{-2}$ or $ f(u) =e^u$. Our main result is that the associated extremal solution is smooth provided this is the case for the advection free case; $c(x)=0$.In ``Estimates on pull-in distances in MEMS models and other nonlinear eigenvalue problems'' we prove some results, which were observed numerically, regarding equations of the form $ -\Delta u = \lambda |x|^\alpha F(u) $ in $B$ where $B$ is the unit ball in $ \IR^N$. In addition we obtain upper and lower estimates on the extremal solutions associated with various nonlinear eigenvalue problems. In ``The critical dimension for a fourth order elliptic problem with singular nonlinearity,'' we examine the equation given by $ \Delta^2 u = \lambda (1-u)^{-2}$ in $ B$ with Dirichlet boundary conditions where $B$ is the unit ball in $ \IR^N$. Our main result is that the extremal solution $u^*$ is smooth if and only if $ N \le 8$. In ``Regularity of extremal solutions in fourth order nonlinear eigenvalue problems on general domains'' we examine the equation $ \Delta^2 u = \lambda f(u)$ in $ \Omega$ with Navier boundary conditions where $ \Omega$ is a general bounded domain in $ \IR^N$. We obtain various results concerning the regularity of the associated extremal solution. In ``Regularity of the extremal solutions in elliptic systems'' we examine the elliptic system given by $ -\Delta u = \lambda e^v$, \; $ -\Delta v = \gamma e^u$ in $ \Omega$ where $\lambda$ and $\gamma$ are positive constants and we obtain results concering the regularity of the extremal solutions. In ``Optimal Hardy inequalities for general elliptic operators with improvements'' we examine some very general Hardy inequalities. Optimal constants are obtained and we characterize the improvements of these general Hardy inequalities. In addition we prove various weighted versions of these inequalities with improvements and many other results.

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##### Master's Student Supervision (2010 - 2020)

I present analysis of how the mass transports that optimize the inner product cost---considered by Y. Brenier---propagate in time along a given Lagrangian in both deterministic and stochastic settings. While for the minimizing transports one may easily obtain Hopf-Lax formulas on Wasserstein space by inf-convolution, this is not the case for the maximizing transports, which are sup-inf problems. In this case, we assume that the Lagrangian is jointly convex on phase space, which allow us to use Bolza-type duality, a well known phenomenon in the deterministic case but, as far as I know, novel in the stochastic case. Hopf-Lax formulas help relate optimal ballistic transports to those associated with the dynamic fixed-end transports studied by Bernard-Buffoni and Fathi-Figalli in the deterministic case, and by Mikami-Thieullen in the stochastic setting.

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This essay focuses on exploring dilatons as an alternate model to the Higgs mechanism. An introductory analysis to the Higgs mechanism, effective potential method, and dilatons is provided. Then, three different models are explored on how to obtain a light dilaton that emerges as a pseudo Goldstone boson because of the spontaneously broken approximate scale invariance. This light dilaton is shown to have properties that are, in general, similar to the Higgs boson with minor differences that can differentiate between the two models in collider experiments.

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We study the link between two different factorization theorems and their proofs : Brenier's Theorem which states that for any u ∈L^p(Ω), where Ω is a bounded domain in R ^d and 1 ≤ p ≤ ∞, u can be written as u=∇ ɸ ο s where ɸ is a convex function, and s a measure preserving transformation, and on the other hand Ghoussoub and Moameni's theorem which states that for any u ∈ L∞ (Ω), u(x) = ∇₁H(S(x),x), where H is a convex concave anti-symmetric function, and S is a measure preserving involution.In a second time we prove that Ghoussoub and Moameni's theorem is true in L², and find the decomposition for particular example : u(x) = |x-1/2|.

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In this thesis we will review some recent results of Optimal Mass Transportation emphasizing on the role of displacement interpolation and displacementconvexity. We will show some of its recent applications, specially the ones by Bernard, and Agueh-Ghoussoub-Kang.

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