Results for "Alessio Figalli"

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Sharp Extinction Rates for Fast Diffusion Equations on Generic Bounded DomainsFeb 08 2019We investigate the homogeneous Dirichlet problem for the Fast Diffusion Equation $u_t=\Delta u^m$, posed in a smooth bounded domain $\Omega\subset \mathbb{R}^N$, in the exponent range $m_s=(N-2)_+/(N+2)<m<1$. It is known that bounded positive solutions ... More
Invariant measures of Hamiltonian systems with prescribed asymptotic Maslov indexSep 29 2007Jan 19 2008We study the properties of the asymptotic Maslov index of invariant measures for time-periodic Hamiltonian systems on the cotangent bundle of a compact manifold M. We show that if M has finite fundamental group and the Hamiltonian satisfies some general ... More
Semiclassical limit of quantum dynamics with rough potentials and well posedness of transport equations with measure initial dataJun 28 2010In this paper we study the semiclassical limit of the Schr\"odinger equation. Under mild regularity assumptions on the potential $U$ which include Born-Oppenheimer potential energy surfaces in molecular dynamics, we establish asymptotic validity of classical ... More
On the Continuity of Center-Outward Distribution and Quantile FunctionsMay 13 2018To generalize the notion of distribution function to dimension $d\geq 2$, in the recent papers it was proposed a concept of center-outward distribution function based on optimal transportation ideas, and the inferential properties of the corresponding ... More
When is multidimensional screening a convex program?Dec 15 2009A principal wishes to transact business with a multidimensional distribution of agents whose preferences are known only in the aggregate. Assuming a twist (= generalized Spence-Mirrlees single-crossing) hypothesis and that agents can choose only pure ... More
Characterization of isoperimetric sets inside almost-convex conesMay 02 2016In this note we characterize isoperimetric regions inside almost-convex cones. More precisely, as in the case of convex cones, we show that isoperimetric sets are given by intersecting the cone with a ball centered at the origin.
Universality in several-matrix models via approximate transport mapsJul 10 2014Mar 18 2016We construct approximate transport maps for perturbative several-matrix models. As a consequence, we deduce that local statistics have the same asymptotic as in the case of independent GUE or GOE matrices, i.e., they are given by the Sine-kernel in the ... More
An overview of unconstrained free boundary problemsMay 24 2018In this paper we present a survey concerning unconstrained free boundary problems of type $$ \left\{ \begin{array}{ll} F_1(D^2u,\nabla u,u,x)=0 & \text{in }B_1 \cap \Omega ,\\ F_2 (D^2 u,\nabla u,u,x)=0 & \text{in }B_1\setminus\Omega ,\\ u \in \mathbb{S}(B_1), ... More
Total Variation Flow and Sign Fast Diffusion in one dimensionJul 11 2011Aug 17 2011We consider the dynamics of the Total Variation Flow (TVF) $u_t=\div(Du/|Du|)$ and of the Sign Fast Diffusion Equation (SFDE) $u_t=\Delta\sign(u)$ in one spatial dimension. We find the explicit dynamic and sharp asymptotic behaviour for the TVF, and we ... More
A sharp stability result for the relative isoperimetric inequality inside convex conesOct 11 2012The relative isoperimetric inequality inside an open, convex cone $\mathcal C$ states that, at fixed volume, $B_r \cap \mathcal C$ minimizes the perimeter inside $\mathcal C$. Starting from the observation that this result can be recovered as a corollary ... More
On the density function on moduli spaces of toric 4-manifoldsAug 07 2014Feb 14 2015The optimal density function assigns to each symplectic toric manifold $M$ a number $0 < d \leq 1$ obtained by considering the ratio between the maximum volume of $M$ which can be filled by symplectically embedded disjoint balls and the total symplectic ... More
A general class of free boundary problems for fully nonlinear parabolic equationsSep 03 2013In this paper we consider the fully nonlinear parabolic free boundary problem $$ \left\{\begin{array}{ll} F(D^2u) -\partial_t u=1 & \text{a.e. in}Q_1 \cap \Omega\\ |D^2 u| + |\partial_t u| \leq K & \text{a.e. in}Q_1\setminus\Omega, \end{array} \right. ... More
On stable solutions for boundary reactions: a De Giorgi-type result in dimension 4+1May 08 2017We prove that every bounded stable solution of \[ (-\Delta)^{1/2} u + f(u) =0 \qquad \mbox{in }\mathbb R^3\] is a 1D profile, i.e., $u(x)= \phi(e\cdot x)$ for some $e\in \mathbb S^2$, where $\phi:\mathbb R\to \mathbb R$ is a nondecreasing bounded stable ... More
Regularity of solutions to the parabolic fractional obstacle problemJan 26 2011In this paper we study a parabolic version of the fractional obstacle problem, proving almost optimal regularity for the solution. This problem is motivated by an American option model proposed by Menton which introduces, into the theory of option evaluation, ... More
Gradient stability for the Sobolev inequality: the case $p\geq 2$Oct 07 2015Oct 02 2016We prove a strong form of the quantitative Sobolev inequality in $\mathbb{R}^n$ for $p\geq 2$, where the deficit of a function $u\in \dot W^{1,p} $ controls $\| \nabla u -\nabla v\|_{L^p}$ for an extremal function $v$ in the Sobolev inequality.
Lecture notes on variational models for incompressible Euler equationsSep 17 2010These notes briefly summarize the lectures for the Summer School "Optimal transportation: Theory and applications" held by the second author in Grenoble during the week of June 22-26, 2009. Their goal is to describe some recent results on Brenier's variational ... More
On the fine structure of the free boundary for the classical obstacle problemSep 12 2017Nov 25 2017In the classical obstacle problem, the free boundary can be decomposed into "regular" and "singular" points. As shown by Caffarelli in his seminal papers \cite{C77,C98}, regular points consist of smooth hypersurfaces, while singular points are contained ... More
Surface measures and convergence of the Ornstein-Uhlenbeck semigroup in Wiener spacesSep 01 2010We study points of density 1/2 of sets of finite perimeter in infinite-dimensional Gaussian spaces and prove that, as in the finite-dimensional theory, the surface measure is concentrated on this class of points. Here density 1/2 is formulated in terms ... More
Almost everywhere well-posedness of continuity equations with measure initial dataOct 19 2009The aim of this note is to present some new results concerning "almost everywhere" well-posedness and stability of continuity equations with measure initial data. The proofs of all such results can be found in \cite{amfifrgi}, together with some application ... More
On flows associated to Sobolev vector fields in Wiener spaces: an approach à la DiPerna-LionsMar 10 2008In this paper we extend the DiPerna-Lions theory on ODEs with Sobolev vector fields to the setting of abstract Wiener spaces.
High action orbits for Tonelli Lagrangians and superlinear Hamiltonians on compact configuration spacesSep 22 2006Multiplicity results for solutions of various boundary value problems are known for dynamical systems on compact configuration manifolds, given by Lagrangians or Hamiltonians which have quadratic growth in the velocities or in the momenta. Such results ... More
Regularity and Bernstein-type results for nonlocal minimal surfacesJun 30 2013We prove that, in every dimension, Lipschitz nonlocal minimal surfaces are smooth. Also, we extend to the nonlocal setting a famous theorem of De Giorgi stating that the validity of Bernstein's theorem in dimension $n+1$ is a consequence of the nonexistence ... More
Mass Transportation on Sub-Riemannian ManifoldsMar 20 2008Oct 15 2009We study the optimal transport problem in sub-Riemannian manifolds where the cost function is given by the square of the sub-Riemannian distance. Under appropriate assumptions, we generalize Brenier-McCann's Theorem proving existence and uniqueness of ... More
Rigidity and stability of Caffarelli's log-concave perturbation theoremMay 31 2016In this note we establish some rigidity and stability results for Caffarelli's log-concave perturbation theorem. As an application we show that if a 1-log-concave measure has almost the same Poincar\'e constant as the Gaussian measure, then it almost ... More
Second order stability for the Monge-Ampere equation and strong Sobolev convergence of optimal transport mapsFeb 24 2012Nov 16 2012The aim of this note is to show that Alexandrov solutions of the Monge-Ampere equation, with right hand side bounded away from zero and infinity, converge strongly in $W^{2,1}_{loc}$ if their right hand side converge strongly in $L^1_{loc}$. As a corollary ... More
$W^{2,1}$ regularity for solutions of the Monge-Ampère equationNov 30 2011Apr 14 2012In this paper we prove that a strictly convex Alexandrov solution u of the Monge-Amp\`ere equation, with right hand side bounded away from zero and infinity, is $W_{\rm loc}^{2,1}$. This is obtained by showing higher integrability a-priori estimates for ... More
Stability for a GNS inequality and the Log-HLS inequality, with application to the critical mass Keller-Segel equationJul 29 2011Starting from the quantitative stability result of Bianchi and Egnell for the 2-Sobolev inequality, we deduce several different stability results for a Gagliardo-Nirenberg-Sobolev inequality in the plane. Then, exploiting the connection between this inequality ... More
Sobolev regularity for Monge-Ampère type equationsNov 10 2012In this note we prove that, if the cost function satisfies some necessary structural conditions and the densities are bounded away from zero and infinity, then strictly $c$-convex potentials arising in optimal transportation belong to $W^{2,1+\kappa}_{\rm ... More
Partial Regularity for optimal transport mapsSep 25 2012We prove that, for general cost functions on $\mathbb{R}^n$, or for the cost $d^2/2$ on a Riemannian manifold, optimal transport maps between smooth densities are always smooth outside a closed singular set of measure zero.
Regularity of monotone transport maps between unbounded domainsFeb 20 2019The regularity of monotone transport maps plays an important role in several applications to PDE and geometry. Unfortunately, the classical statements on this subject are restricted to the case when the measures are compactly supported. In this note we ... More
A note on the dimension of the singular set in free interface problemsMay 30 2014The aim of this note is to investigate the size of the singular set of a general class of free interface problems. We show porosity of the singular set, obtaining as a corollary that both its Hausdorff and Minkowski dimensions are strictly smaller than ... More
Higher integrability for minimizers of the Mumford-Shah functionalMar 05 2013We prove higher integrability for the gradient of local minimizers of the Mumford-Shah energy functional, providing a positive answer to a conjecture of De Giorgi.
The Monge-Ampère equation and its link to optimal transportationOct 23 2013We survey the (old and new) regularity theory for the Monge-Amp\`ere equation, show its connection to optimal transportation, and describe the regularity properties of a general class of Monge-Amp\`ere type equations arising in that context.
A rigorous derivation from the kinetic Cucker-Smale model to the pressureless Euler system with nonlocal alignmentFeb 26 2017Sep 12 2018We consider the kinetic Cucker-Smale model with local alignment as a mesoscopic description for the flocking dynamics. The local alignment was first proposed by Karper, Mellet and Trivisa \cite{K-M-T-3}, as a singular limit of a normalized non-symmetric ... More
Free boundary regularity in the parabolic fractional obstacle problemMay 02 2016The parabolic obstacle problem for the fractional Laplacian naturally arises in American option models when the assets prices are driven by pure jump L\'evy processes. In this paper we study the regularity of the free boundary. Our main result establishes ... More
On the convexity of injectivity domains on nonfocal manifoldsApr 01 2014Given a smooth nonfocal compact Riemannian manifold, we show that the so-called Ma--Trudinger--Wang condition implies the convexity of injectivity domains. This improves a previous result by Loeper and Villani.
Infinite speed of propagation and regularity of solutions to the fractional porous medium equation in general domainsOct 13 2015Jun 21 2016We study the positivity and regularity of solutions to the fractional porous medium equations $u_t+(-\Delta)^su^m=0$ in $(0,\infty)\times\Omega$, for $m>1$ and $s\in (0,1)$ and with Dirichlet boundary data $u=0$ in $(0,\infty)\times({\mathbb R}^N\setminus\Omega)$, ... More
Nonlinear Bounds in Hölder Spaces for the Monge-Ampère EquationAug 11 2015Mar 29 2016We demonstrate that $C^{2,\alpha}$ estimates for the Monge-Amp\`{e}re equation depend in a highly nonlinear way both on the $C^{\alpha}$ norm of the right-hand side and $1/\alpha$. First, we show that if a solution is strictly convex, then the $C^{2,\alpha}$ ... More
On sets of finite perimeter in Wiener spaces: reduced boundary and convergence to halfspacesJun 27 2012We study sets of finite perimeter in Wiener space, and prove that at almost every point (with respect to the perimeter measure) a set of finite perimeter blows-up to a halfspace.
Non-Local Tug-of-War and the Infinity Fractional LaplacianNov 09 2010May 03 2011Motivated by the "tug-of-war" game studied in [12], we consider a "non-local" version of the game which goes as follows: at every step two players pick respectively a direction and then, instead of flipping a coin in order to decide which direction to ... More
Transport maps for Beta-matrix models and UniversalityNov 10 2013Jan 26 2015We construct approximate transport maps for non-critical Beta-matrix models, that is, maps so that the push forward of a non-critical Beta-matrix model with a given potential is a non-critical Beta-matrix model with another potential, up to a small error ... More
On the Hausdorff Dimension of the Mather QuotientNov 08 2007Under appropriate assumptions on the dimension of the ambient manifold and the regularity of the Hamiltonian, we show that the Mather quotient is small in term of Hausdorff dimension. Then, we present applications in dynamics.
Global well-posedness of the spatially homogeneous Kolmogorov-Vicsek model as a gradient flowSep 09 2015We consider the so-called spatially homogenous Kolmogorov-Vicsek model, a non-linear Fokker-Planck equation of self-driven stochastic particles with orientation interaction under the space-homogeneity. We prove the global existence and uniqueness of weak ... More
Sharp boundary behaviour of solutions to semilinear nonlocal elliptic equationsOct 07 2017Feb 12 2018We investigate quantitative properties of nonnegative solutions $u(x)\ge 0$ to the semilinear diffusion equation $\mathcal{L} u= f(u)$, posed in a bounded domain $\Omega\subset {\mathbb R}^N$ with appropriate homogeneous Dirichlet or outer boundary conditions. ... More
Existence and uniqueness of maximal regular flows for non-smooth vector fieldsJun 14 2014In this paper we provide a complete analogy between the Cauchy-Lipschitz and the DiPerna-Lions theories for ODE's, by developing a local version of the DiPerna-Lions theory. More precisely, we prove existence and uniqueness of a maximal regular flow for ... More
On the Lagrangian structure of transport equations: the Vlasov-Poisson systemDec 11 2014Jun 16 2015The Vlasov-Poisson system is a classical model in physics used to describe the evolution of particles under their self-consistent electric or gravitational field. The existence of classical solutions is limited to dimensions $d\leq 3$ under strong assumptions ... More
A note on interior $W^{2,1+\varepsilon}$ estimates for the Monge-Ampere equationFeb 24 2012Oct 30 2012By a variant of the techniques introduced by the first two authors in [DF] to prove that second derivatives of solutions to the Monge-Ampere equation are locally in $L\log L$, we obtain interior $W^{2,1+\varepsilon}$ estimates.
Strongly nonlocal dislocation dynamics in crystalsNov 14 2013We consider the equation $$v_t=L_s v-W'(v)+\sigma_\epsilon(t,x) \quad {\mbox{ in }} (0,+\infty)\times\R,$$ where $L_s$ is an integro-differential operator of order $2s$, with $s\in(0,1)$, $W$ is a periodic potential, and $\sigma_\epsilon$ is a small external ... More
A quantitative analysis of metrics on $\R^n$ with almost constant positive scalar curvature, with applications to Yamabe and fast diffusion flowsFeb 05 2016Feb 18 2016We prove a quantitative version of Struwe's theorem on the structure of metrics on $\R^n$ which are conformal to the flat metric and have almost constant positive scalar curvature. As two applications of our result, we prove the convergence of the volume-preserving ... More
The sharp quantitative Euclidean concentration inequalityJan 16 2016Aug 10 2016The Euclidean concentration inequality states that, among sets with fixed volume, balls have $r$-neighborhoods of minimal volume for every $r>0$. On an arbitrary set, the deviation of this volume growth from that of a ball is shown to control the square ... More
Lipschitz Changes of Variables between Perturbations of Log-concave MeasuresOct 13 2015Extending a result of Caffarelli, we provide global Lipschitz changes of variables between compactly supported perturbations of log-concave measures. The result is based on a combination of ideas from optimal transportation theory and a new Pogorelov-type ... More
Sharp global estimates for local and nonlocal porous medium-type equations in bounded domainsOct 31 2016Nov 25 2017This paper provides a quantitative study of nonnegative solutions to nonlinear diffusion equations of porous medium-type of the form $\partial_t u + {\mathcal L}u^m=0$, $m>1$, where the operator ${\mathcal L}$ belongs to a general class of linear operators, ... More
Sharp global estimates for local and nonlocal porous medium-type equations in bounded domainsOct 31 2016This paper provides a quantitative study of nonnegative solutions to nonlinear diffusion equations of porous medium-type of the form $\partial_t u + \mathcal{L} u^m=0$, $m>1$, where the operator $\mathcal{L}$ belongs to a general class of linear operators, ... More
Bootstrap regularity for integro-differential operators and its application to nonlocal minimal surfacesFeb 21 2012We prove that $C^{1,\alpha}$ $s$-minimal surfaces are automatically $C^\infty$. For this, we develop a new bootstrap regularity theory for solutions of integro-differential equations of very general type, which we believe is of independent interest.
Semiclassical limit for mixed states with singular and rough potentialsDec 11 2010We consider the semiclassical limit for the Heisenberg-von Neumann equation with a potential which consists of the sum of a repulsive Coulomb potential, plus a Lipschitz potential whose gradient belongs to $BV$; this assumption on the potential guarantees ... More
Optimal regularity and structure of the free boundary for minimizers in cohesive zone modelsDec 21 2017We study optimal regularity and free boundary for minimizers of an energy functional arising in cohesive zone models for fracture mechanics. Under smoothness assumptions on the boundary conditions and on the fracture energy density, we show that minimizers ... More
Global regularity for the free boundary in the obstacle problem for the fractional LaplacianJun 15 2015We study the regularity of the free boundary in the obstacle problem for the fractional Laplacian under the assumption that the obstacle $\varphi$ satisfies $\Delta \varphi\leq 0$ near the contact region. Our main result establishes that the free boundary ... More
Symplectic $G$-capacities and integrable systemsNov 14 2015For any Lie group $G$, we construct a $G$-equivariant analogue of symplectic capacities and give examples when $G = \mathbb{T}^k\times\mathbb{R}^{d-k}$, in which case the capacity is an invariant of integrable systems. Then we study the continuity of ... More
On supporting hyperplanes to convex bodiesJul 06 2011Given a convex set and an interior point close to the boundary, we prove the existence of a supporting hyperplane whose distance to the point is controlled, in a dimensionally quantified way, by the thickness of the convex set in the orthogonal direction. ... More
Isoperimetry and stability properties of balls with respect to nonlocal energiesMar 03 2014We obtain a sharp quantitative isoperimetric inequality for nonlocal $s$-perimeters, uniform with respect to $s$ bounded away from $0$. This allows us to address local and global minimality properties of balls with respect to the volume-constrained minimization ... More
Hölder continuity and injectivity of optimal mapsJul 06 2011Consider transportation of one distribution of mass onto another, chosen to optimize the total expected cost, where cost per unit mass transported from x to y is given by a smooth function c(x,y). If the source density f^+(x) is bounded away from zero ... More
Existence of Eulerian solutions to the semigeostrophic equations in physical space: the 2-dimensional periodic caseNov 30 2011Oct 13 2012In this paper we use the new regularity and stability estimates for Alexandrov solutions to Monge-Ampere equations estabilished by G.De Philippis and A.Figalli to provide a global in time existence of distributional solutions to a semigeostrophic equation ... More
Continuity and injectivity of optimal maps for non-negatively cross-curved costsNov 20 2009Consider transportation of one distribution of mass onto another, chosen to optimize the total expected cost, where cost per unit mass transported from x to y is given by a smooth function c(x,y). If the source density f^+(x) is bounded away from zero ... More
Some new well-posedness results for continuity and transport equations, and applications to the chromatography systemApr 02 2009Oct 01 2009We obtain various new well-posedness results for continuity and transport equations, among them an existence and uniqueness theorem (in the class of strongly continuous solutions) in the case of nearly incompressible vector fields, possibly having a blow-up ... More
A global existence result for the semigeostrophic equations in three dimensional convex domainsMay 24 2012Oct 13 2012Exploiting recent regularity estimates for the Monge-Amp\`ere equation, under some suitable assumptions on the initial data we prove global-in-time existence of Eulerian distributional solutions to the semigeostrophic equations in 3-dimensional convex ... More
Rigidity and sharp stability estimates for hypersurfaces with constant and almost-constant nonlocal mean curvatureMar 02 2015We prove that the boundary of a (not necessarily connected) bounded smooth set with constant nonlocal mean curvature is a sphere. More generally, and in contrast with what happens in the classical case, we show that the Lipschitz constant of the nonlocal ... More
Asymptotics of the $s$-perimeter as $s\searrow 0$Apr 03 2012Oct 20 2012We deal with the asymptotic behavior of the $s$-perimeter of a set $E$ inside a domain $\Omega$ as $s\searrow0$. We prove necessary and sufficient conditions for the existence of such limit, by also providing an explicit formulation in terms of the Lebesgue ... More
BMO-type norms related to the perimeter of setsJul 08 2014In this paper we consider an isotropic variant of the $BMO$-type norm recently introduced by Bourgain, Brezis and Mironescu. We prove that, when considering characteristic functions of sets, this norm is related to the perimeter. A byproduct of our analysis ... More
Regularity of optimal transport maps on multiple products of spheresJun 10 2010This article addresses regularity of optimal transport maps for cost="squared distance" on Riemannian manifolds that are products of arbitrarily many round spheres with arbitrary sizes and dimensions. Such manifolds are known to be non-negatively cross-curved ... More
On a conjecture of Wilf about the Frobenius numberAug 22 2014May 20 2015Given coprime positive integers $a_1 < ...< a_d$, the Frobenius number $F$ is the largest integer which is not representable as a non-negative integer combination of the $a_i$. Let $g$ denote the number of all non-representable positive integers: Wilf ... More
On a Coarse-Graining Concept in Colloidal Physics with Application to Fluid and Arrested Colloidal Suspensions in Shearing FieldsApr 13 2010We poorly understand the macroscopic properties of complex fluids and of amorphous bodies in general. This is mainly due to the interplay between phenomena at different levels and length-scales. In particular, it is not necessarily true that the microscopic ... More
Simple model for the static structure and the mean coordination of amorphous solidsMay 07 2009We propose a simple route to evaluate the static structure, in terms of average coordination, of completely disordered solids with spherical constituents, from ca. 55% volume fraction up to random close packing, in the absence of structural heterogeneities. ... More
Detecting multimode entanglement by symplectic uncertainty relationsAug 31 2005Aug 17 2006A hierarchy of multimode uncertainty relations on the second moments of n pairs of canonical operators is derived in terms of quantities invariant under linear canonical (i.e. symplectic) transformations. Conditions for the separability of multimode continuous ... More
Vanishing ideals of binary Hamming spheresFeb 08 2018We show how to efficiently obtain the Algebraic Normal Form of Boolean functions vanishing on Hamming spheres centred at zero. By exploiting the symmetry of the problem we obtain formulas for particular cases, and a computational method to address the ... More
Inference for Additive Models in the Presence of Infinite Dimensional Nuisance ParametersNov 07 2016A framework for hypothesis testing of functional restrictions against general alternatives is proposed. The parameter space is a reproducing kernel Hilbert space. The null the hypothesis does not necessarily define a parametric model. The tests allow ... More
Torsional instability and sensitivity analysis in a suspension bridge model related to the Melan equationJul 19 2018Inspired by the Melan equation we propose a model for suspension bridges with two cables linked to a deck, through inextensible hangers. We write the energy of the system and we derive from variational principles two nonlinear and nonlocal hyperbolic ... More
Coupled nonlinear Schrodinger systems with potentialsJun 01 2005Coupled nonlinear Schrodinger systems describe some physical phenomena such as the propagation in birefringent optical fibers, Kerr-like photorefractive media in optics and Bose-Einstein condensates. In this paper, we study the existence of concentrating ... More
Ground states for a system of nonlinear Schrodinger equations with three waves interactionOct 19 2009We consider a system of nonlinear Schrodinger equations with three waves interaction studying the existence of ground state solutions. In particular, we find a vector ground state, namely a ground state with the three components all different from zero. ... More
Semistable 3-fold flipsMay 31 1995We piece together ingredients, which are well known and documented in the literature, into a new proof of the existence of semistable 3-fold flips
Weak Convergence of Laws on R^{K} with Common MarginalsJun 19 2006We present a result on topologically equivalent integral metrics (Rachev, 1991, Muller, 1997) that metrize weak convergence of laws with common marginals. This result is relevant for applications, as shown in a few simple examples.
Stationary layered solutions for a system of Allen-Cahn type equationsNov 25 2012Dec 03 2012We consider a class of semilinear elliptic system of the form $-\Delta u(x,y)+\nabla W(u(x,y))=0,\quad (x,y)\in\R^{2}$ where $W:\R^{2}\to\R$ is a double well non negative symmetric potential. We show, via variational methods, that if the set of solutions ... More
The $ω$-Borel invariant for representations into $SL(n,\mathbb{C}_ω)$Sep 22 2017Let $\Gamma$ be the fundamental group of a complete hyperbolic $3$-manifold $M$ with toric cusps. We define the $\omega$-Borel invariant $\beta_n^\omega(\rho_\omega)$ associated to a representation $\rho_\omega: \Gamma \rightarrow SL(n,\mathbb{C}_\omega)$, ... More
Greedy algorithms for predictionFeb 05 2016In many prediction problems, it is not uncommon that the number of variables used to construct a forecast is of the same order of magnitude as the sample size, if not larger. We then face the problem of constructing a prediction in the presence of potentially ... More
Minimal relations and the Diophantine Frobenius Problem in embedding dimension threeAug 22 2016Aug 24 2016This paper provides a formula for the minimal relations and the Frobenius number of a numerical semigroup minimally generated by three pairwise coprime positive integers.
Singularly perturbed Neumann problems with potentialsOct 08 2003We study the existence of concentrating solutions of a singularly perturbed Neumann problems with two potentials.
Algebras of differential operators on Lie groups and spectral multipliersJul 07 2010This thesis is devoted to the study of joint spectral multipliers for a system of pairwise commuting, self-adjoint left-invariant differential operators L_1,...,L_n on a connected Lie group G. Under the assumption that the algebra generated by L_1,...,L_n ... More
Generators of a fraction of a numerical semigroupFeb 20 2014May 30 2015Given a numerical semigroup $S$ and a positive integer $d$, the fraction $\frac{S}{d}=\{ x \in \mathbb{N} \ | \ dx \in S\}$ is again a numerical semigroup. In this paper we determine a generating set for $\frac{S}{d}$ in terms of the minimal generators ... More
On the least positive solution to a proportionally modular Diophantine inequalitDec 18 2013Feb 03 2014Given three positive integers $a,b,c$, a proportionally modular Diophantine inequality is an expression of the form $ax \mod{b} \le cx$. Our aim is to give a recursive formula for the least solution to such an inequality. We then use the formula to derive ... More
Spectral multipliers on Heisenberg-Reiter and related groupsDec 04 2012Let $L$ be a homogeneous sublaplacian on a 2-step stratified Lie group $G$ of topological dimension $d$ and homogeneous dimension $Q$. By a theorem due to Christ and to Mauceri and Meda, an operator of the form $F(L)$ is bounded on $L^p$ for $1 < p < ... More
The symmetric signatureJun 10 2016This is the author's Ph.D. thesis. We introduce two related invariants for local (and standard graded) rings called differential and syzygy symmetric signature. These are defined by looking at the maximal free splitting of the module of K\"ahler differentials ... More
The LHCb UpgradeOct 01 2013The LHCb experiment is designed to perform high-precision measurements of CP violation and search for New Physics using the enormous flux involving beauty and charm quarks produced at the LHC. The operation and the results obtained from the data collected ... More
Weber's formula for the bitangents of a smooth plane quarticDec 06 2016In a section of his 1876 treatise Theorie der Abel'schen Functionen vom Geschlecht 3 Weber proved a formula that expresses the bitangents of a non-singular plane quartic in terms of Riemann theta constants (Thetanullwerte). The present note is devoted ... More
Blowup algebras of rational normal scrollsOct 13 2016We investigate the algebraic relations among the minors of a $2 \times c$ matrix with $d$ catalecticant blocks, which define a $d$-fold rational normal scroll $\mathcal{S}\subseteq \mathbb{P}^{c+d- 1}$. We determine the equations of the blowup of $\mathbb{P}^{c+d-1}$ ... More
Analysis of joint spectral multipliers on Lie groups of polynomial growthOct 06 2010Dec 19 2010We study the problem of L^p-boundedness (1 < p < \infty) of operators of the form m(L_1,...,L_n) for a commuting system of self-adjoint left-invariant differential operators L_1,...,L_n on a Lie group G of polynomial growth, which generate an algebra ... More
Oscillating solutions for prescribed mean curvature equations: Euclidean and Lorentz-Minkowski casesSep 20 2017Feb 25 2018This paper deals with the prescribed mean curvature equations both in the Euclidean case and in the Lorentz-Minkowski case in presence of a nonlinearity $g$ such that $g'(0)>0$. We show the existence of oscillating solutions, namely with an unbounded ... More
On integers which are representable as sums of large squaresAug 06 2014Apr 11 2015We prove that the greatest positive integer that is not expressible as a linear combination with integer coefficients of elements of the set $\{n^2,(n+1)^2,\ldots \}$ is asymptotically $O(n^2)$, verifying thus a conjecture of Dutch and Rickett. Furthermore ... More
Consecutive cancellations in Tor modules over local ringsMay 17 2016Let $M, N$ be finite modules over a Noetherian local ring $R$, and let $G$ be the associated graded ring of $R$. We show that the bigraded Hilbert series of $gr(Tor^R(M,N))$ is obtained from that of $Tor^G(gr(M),gr(N))$ by negative consecutive cancellations, ... More
Finite generation of adjoint rings after Lazic: an introductionJun 26 2010An introduction to all the key ideas of Lazic's proof of the theorem on the finite generation of adjoint rings.
The shear modulus of metastable amorphous solids with strong central and bond-bending interactionsJul 23 2008Feb 17 2009We derive expressions for the shear modulus of deeply-quenched, glassy solids, in terms of a Cauchy-Born free energy expansion around a rigid (quenched) reference state, following the approach due to Alexander [Alexander, Phys. Rep. 296, 1998]. Continuum-limit ... More
A System of Interaction and StructureOct 28 1999Jan 27 2007This paper introduces a logical system, called BV, which extends multiplicative linear logic by a non-commutative self-dual logical operator. This extension is particularly challenging for the sequent calculus, and so far it is not achieved therein. It ... More