01970nas a2200169 4500008004100000022001400041245006200055210006100117260000800178300000700186490000800193520144100201100002401642700002101666700001901687856009401706 2023 eng d a0003-952700aBenjamin-Feir Instability of Stokes Waves in Finite Depth0 aBenjaminFeir Instability of Stokes Waves in Finite Depth cOCT a910 v2473 a

Whitham and Benjamin predicted in 1967 that small-amplitude periodic traveling Stokes waves of the 2d-gravity water waves equations are linearly unstable with respect to long-wave perturbations, if the depth h is larger than a critical threshold h(WB) approximate to 1.363. In this paper, we completely describe, for any finite value of h > 0, the four eigenvalues close to zero of the linearized equations at the Stokes wave, as the Floquet exponent mu is turned on. We prove, in particular, the existence of a unique depth h(WB), which coincides with the one predicted by Whitham and Benjamin, such that, for any 0 < h < h(WB), the eigenvalues close to zero are purely imaginary and, for any h > h(WB), a pair of non-purely imaginary eigenvalues depicts a closed figure ``8'', parameterized by the Floquet exponent. As h -> h(WB)(+) the ``8'' collapses to the origin of the complex plane. The complete bifurcation diagram of the spectrum is not deduced as in deep water, since the limits h -> +infinity (deep water) and mu -> 0 (long waves) do not commute. In finite depth, the four eigenvalues have all the same size O(mu), unlike in deep water, and the analysis of their splitting is much more delicate, requiring, as a new ingredient, a non-perturbative step of block-diagonalization. Along the whole proof, the explicit dependence of the matrix entries with respect to the depth h is carefully tracked.

1 aBerti, Massimiliano1 aMaspero, Alberto1 aVentura, Paolo uhttps://www.math.sissa.it/publication/benjamin-feir-instability-stokes-waves-finite-depth00441nas a2200109 4500008004100000245006500041210006500106100002400171700002100195700001900216856009600235 2023 eng d00aStokes waves at the critical depth are modulational unstable0 aStokes waves at the critical depth are modulational unstable1 aBerti, Massimiliano1 aMaspero, Alberto1 aVentura, Paolo uhttps://www.math.sissa.it/publication/stokes-waves-critical-depth-are-modulational-unstable01002nas a2200205 4500008004100000022001400041245007400055210006600129300001200195490000700207520033000214653002300544653003100567653002000598653001600618100002400634700002100658700001900679856009800698 2022 eng d a1120-633000aOn the analyticity of the Dirichlet-Neumann operator and Stokes waves0 aanalyticity of the DirichletNeumann operator and Stokes waves a611-6500 v333 a

We prove an analyticity result for the Dirichlet-Neumann operator under space periodic boundary conditions in any dimension in an unbounded domain with infinite depth. We derive an analytic bifurcation result of analytic Stokes waves-i.e., space periodic traveling solutions-of the water waves equations in deep water.

10aBifurcation theory10aDirichlet-Neumann operator10atraveling waves10awater waves1 aBerti, Massimiliano1 aMaspero, Alberto1 aVentura, Paolo uhttps://www.math.sissa.it/publication/analyticity-dirichlet-neumann-operator-and-stokes-waves01112nas a2200205 4500008004100000022001400041245004600055210004500101300001200146490000700158520050200165653002900667653002900696653002000725653001600745100002400761700002100785700001900806856008100825 2022 eng d a1120-633000aBenjamin-Feir instability of Stokes waves0 aBenjaminFeir instability of Stokes waves a399-4120 v333 a

We present the recent results in Berti et al. [Invent. Math. (2022), to appear] regarding the Benjamin-Feir instability of small amplitude Stokes waves in deep water. We completely describe the behavior of the four eigenvalues close to zero of the linearized water waves equations at the Stokes solution, as the Floquet exponent is turned on, proving the conjecture that a pair of non-purely imaginary eigenvalues depicts a closed figure ``8'', in full agreement with numerical simulations.

10aKato perturbation theory10amodulational instability10atraveling waves10awater waves1 aBerti, Massimiliano1 aMaspero, Alberto1 aVentura, Paolo uhttps://www.math.sissa.it/publication/benjamin-feir-instability-stokes-waves01605nas a2200169 4500008004100000020001400041245008000055210006900135260001500204300001400219490000800233520108300241100002401324700002101348700001901369856004701388 2022 eng d a1432-129700aFull description of Benjamin-Feir instability of stokes waves in deep water0 aFull description of BenjaminFeir instability of stokes waves in  c2022/11/01 a651 - 7110 v2303 a

Small-amplitude, traveling, space periodic solutions –called Stokes waves– of the 2 dimensional gravity water waves equations in deep water are linearly unstable with respect to long-wave perturbations, as predicted by Benjamin and Feir in 1967. We completely describe the behavior of the four eigenvalues close to zero of the linearized equations at the Stokes wave, as the Floquet exponent is turned on. We prove in particular the conjecture that a pair of non-purely imaginary eigenvalues depicts a closed figure “8”, parameterized by the Floquet exponent, in full agreement with numerical simulations. Our new spectral approach to the Benjamin-Feir instability phenomenon uses a symplectic version of Kato’s theory of similarity transformation to reduce the problem to determine the eigenvalues of a $ 4 \times 4 $ complex Hamiltonian and reversible matrix. Applying a procedure inspired by KAM theory, we block-diagonalize such matrix into a pair of $2 \times 2 $ Hamiltonian and reversible matrices, thus obtaining the full description of its eigenvalues.

1 aBerti, Massimiliano1 aMaspero, Alberto1 aVentura, Paolo uhttps://doi.org/10.1007/s00222-022-01130-z01780nas a2200169 4500008004100000020002200041245009500063210006900158260005200227520110800279100001601387700002101403700002101424700002301445700001901468856012301487 2021 eng d a978-3-030-55874-100aDiscontinuous Galerkin Model Order Reduction of Geometrically Parametrized Stokes Equation0 aDiscontinuous Galerkin Model Order Reduction of Geometrically Pa aChambSpringer International Publishingc2021//3 a

The present work focuses on the geometric parametrization and the reduced order modeling of the Stokes equation. We discuss the concept of a parametrized geometry and its application within a reduced order modeling technique. The full order model is based on the discontinuous Galerkin method with an interior penalty formulation. We introduce the broken Sobolev spaces as well as the weak formulation required for an affine parameter dependency. The operators are transformed from a fixed domain to a parameter dependent domain using the affine parameter dependency. The proper orthogonal decomposition is used to obtain the basis of functions of the reduced order model. By using the Galerkin projection the linear system is projected onto the reduced space. During this process, the offline-online decomposition is used to separate parameter dependent operations from parameter independent operations. Finally this technique is applied to an obstacle test problem.The numerical outcomes presented include experimental error analysis, eigenvalue decay and measurement of online simulation time.

1 aShah, Nirav1 aHess, Martin, W.1 aRozza, Gianluigi1 aVermolen, Fred, J.1 aVuik, Cornelis uhttps://www.math.sissa.it/publication/discontinuous-galerkin-model-order-reduction-geometrically-parametrized-stokes-010834nas a2200109 45000080041000002450068000412100065001095201041400174100001810588700002210606856009610628 2021 eng d00aMonotonicity formulas for harmonic functions in RCD(0,N) spaces0 aMonotonicity formulas for harmonic functions in RCD0N spaces3 a

We generalize to the RCD(0,N) setting a family of monotonicity formulas by Colding and Minicozzi for positive harmonic functions in Riemannian manifolds with non-negative Ricci curvature. Rigidity and almost rigidity statements are also proven, the second appearing to be new even in the smooth setting. Motivated by the recent work in [AFM] we also introduce the notion of electrostatic potential in RCD spaces, which also satisfies our monotonicity formulas. Our arguments are mainly based on new estimates for harmonic functions in RCD(K,N) spaces and on a new functional version of the `(almost) outer volume cone implies (almost) outer metric cone' theorem.

1 aGigli, Nicola1 aViolo, Ivan, Yuri uhttps://www.math.sissa.it/publication/monotonicity-formulas-harmonic-functions-rcd0n-spaces01664nas a2200181 4500008004100000020002200041245016600063210006900229260005200298520089600350100002201246700001801268700001701286700002101303700002201324700001901346856011701365 2021 eng d a978-3-030-55874-100aReduced Order Methods for Parametrized Non-linear and Time Dependent Optimal Flow Control Problems, Towards Applications in Biomedical and Environmental Sciences0 aReduced Order Methods for Parametrized Nonlinear and Time Depend aChambSpringer International Publishingc2021//3 a

We introduce reduced order methods as an efficient strategy to solve parametrized non-linear and time dependent optimal flow control problems governed by partial differential equations. Indeed, the optimal control problems require a huge computational effort in order to be solved, most of all in physical and/or geometrical parametrized settings. Reduced order methods are a reliable and suitable approach, increasingly gaining popularity, to achieve rapid and accurate optimal solutions in several fields, such as in biomedical and environmental sciences. In this work, we employ a POD-Galerkin reduction approach over a parametrized optimality system, derived from the Karush-Kuhn-Tucker conditions. The methodology presented is tested on two boundary control problems, governed respectively by (1) time dependent Stokes equations and (2) steady non-linear Navier-Stokes equations.

1 aStrazzullo, Maria1 aZainib, Zakia1 aBallarin, F.1 aRozza, Gianluigi1 aVermolen, Fred, J1 aVuik, Cornelis uhttps://www.springerprofessional.de/en/reduced-order-methods-for-parametrized-non-linear-and-time-depen/1912267647999nas a2200097 45000080041000002450072000412100069001135204760600182100002247788856009147810 2021 eng d00aA remark on two notions of flatness for sets in the Euclidean space0 aremark on two notions of flatness for sets in the Euclidean spac3 a

In this note we compare two ways of measuring the n-dimensional "flatness" of a set SRd, where nN and d>n. The first one is to consider the classical Reifenberg-flat numbers α(x,r) (xSr>0), which measure the minimal scaling-invariant Hausdorff distances in Br(x) between S and n-dimensional affine subspaces of Rd. The second is an `intrinsic' approach in which we view the same set S as a metric space (endowed with the induced Euclidean distance). Then we consider numbers a(x,r)'s, that are the scaling-invariant Gromov-Hausdorff distances between balls centered at x of radius r in S and the n-dimensional Euclidean ball of the same radius. As main result of our analysis we make rigorous a phenomenon, first noted by David and Toro, for which the numbers a(x,r)'s behaves as the square of the numbers α(x,r)'s. Moreover we show how this result finds application in extending the Cheeger-Colding intrinsic-Reifenberg theorem to the biLipschitz case. As a by-product of our arguments, we deduce analogous results also for the Jones' numbers β's (i.e. the one-sided version of the numbers α's).

1 aViolo, Ivan, Yuri uhttps://www.math.sissa.it/publication/remark-two-notions-flatness-sets-euclidean-space36575nas a2200109 45000080041000002450109000412100069001505203607300219100002236292700002236314856012936336 2021 eng d00aRigidity and almost rigidity of Sobolev inequalities on compact spaces with lower Ricci curvature bounds0 aRigidity and almost rigidity of Sobolev inequalities on compact 3 a

 

We prove that if M is a closed n-dimensional Riemannian manifold, n3, with Ricn1 and for which the optimal constant in the critical Sobolev inequality equals the one of the n-dimensional sphere Sn, then M is isometric to Sn. An almost-rigidity result is also established, saying that if equality is almost achieved, then M is close in the measure Gromov-Hausdorff sense to a spherical suspension. These statements are obtained in the RCD-setting of (possibly non-smooth) metric measure spaces satisfying synthetic lower Ricci curvature bounds.
An independent result of our analysis is the characterization of the best constant in the Sobolev inequality on any compact CD space, extending to the non-smooth setting a classical result by Aubin. Our arguments are based on a new concentration compactness result for mGH-converging sequences of RCD spaces and on a Polya-Szego inequality of Euclidean-type in CD spaces.
As an application of the technical tools developed we prove both an existence result for the Yamabe equation and the continuity of the generalized Yamabe constant under measure Gromov-Hausdorff convergence, in the RCD-setting.
1 aNobili, Francesco1 aViolo, Ivan, Yuri uhttps://www.math.sissa.it/publication/rigidity-and-almost-rigidity-sobolev-inequalities-compact-spaces-lower-ricci-curvature00406nas a2200097 4500008004100000245006600041210006600107100002500173700002000198856009000218 2020 eng d00aMinimality of the ball for a model of charged liquid droplets0 aMinimality of the ball for a model of charged liquid droplets1 aMukoseeva, Ekaterina1 aVescovo, Giulia uhttps://www.math.sissa.it/publication/minimality-ball-model-charged-liquid-droplets-001696nas a2200157 4500008004100000245012100041210007300162300001200235490000700247520104800254100001401302700002201316700002101338700002701359856015201386 2020 eng d00aPOD–Galerkin reduced order methods for combined Navier–Stokes transport equations based on a hybrid FV-FE solver0 aPOD–Galerkin reduced order methods for combined Navier–Stokes tr a256-2730 v793 a

The purpose of this work is to introduce a novel POD–Galerkin strategy for the semi-implicit hybrid high order finite volume/finite element solver introduced in Bermúdez et al. (2014) and Busto et al. (2018). The interest is into the incompressible Navier–Stokes equations coupled with an additional transport equation. The full order model employed in this article makes use of staggered meshes. This feature will be conveyed to the reduced order model leading to the definition of reduced basis spaces in both meshes. The reduced order model presented herein accounts for velocity, pressure, and a transport-related variable. The pressure term at both the full order and the reduced order level is reconstructed making use of a projection method. More precisely, a Poisson equation for pressure is considered within the reduced order model. Results are verified against three-dimensional manufactured test cases. Moreover a modified version of the classical cavity test benchmark including the transport of a species is analysed.

1 aBusto, S.1 aStabile, Giovanni1 aRozza, Gianluigi1 aVázquez-Cendón, M.E. uhttps://www.scopus.com/inward/record.uri?eid=2-s2.0-85068068567&doi=10.1016%2fj.camwa.2019.06.026&partnerID=40&md5=a8dcce1b53b8ee872d174bbc4c20caa302563nas a2200169 4500008004100000245009100041210006900132520193100201100001702132700001902149700002102168700002502189700001902214700002102233700002102254856011802275 2019 eng d00aEfficient reduction in shape parameter space dimension for ship propeller blade design0 aEfficient reduction in shape parameter space dimension for ship 3 a

In this work, we present the results of a ship propeller design optimization campaign carried out in the framework of the research project PRELICA, funded by the Friuli Venezia Giulia regional government. The main idea of this work is to operate on a multidisciplinary level to identify propeller shapes that lead to reduced tip vortex-induced pressure and increased efficiency without altering the thrust. First, a specific tool for the bottom-up construction of parameterized propeller blade geometries has been developed. The algorithm proposed operates with a user defined number of arbitrary shaped or NACA airfoil sections, and employs arbitrary degree NURBS to represent the chord, pitch, skew and rake distribution as a function of the blade radial coordinate. The control points of such curves have been modified to generate, in a fully automated way, a family of blade geometries depending on as many as 20 shape parameters. Such geometries have then been used to carry out potential flow simulations with the Boundary Element Method based software PROCAL. Given the high number of parameters considered, such a preliminary stage allowed for a fast evaluation of the performance of several hundreds of shapes. In addition, the data obtained from the potential flow simulation allowed for the application of a parameter space reduction methodology based on active subspaces (AS) property, which suggested that the main propeller performance indices are, at a first but rather accurate approximation, only depending on a single parameter which is a linear combination of all the original geometric ones. AS analysis has also been used to carry out a constrained optimization exploiting response surface method in the reduced parameter space, and a sensitivity analysis based on such surrogate model. The few selected shapes were finally used to set up high fidelity RANS simulations and select an optimal shape.

1 aMola, Andrea1 aTezzele, Marco1 aGadalla, Mahmoud1 aValdenazzi, Federica1 aGrassi, Davide1 aPadovan, Roberta1 aRozza, Gianluigi uhttps://www.scopus.com/inward/record.uri?eid=2-s2.0-85075395143&partnerID=40&md5=b6aa0fcedc2f88e78c295d0f437824d000495nas a2200157 4500008004100000245007500041210006900116260001500185300001300200490000800213100002000221700002200241700001600263700001500279856004300294 2019 eng d00aA neutrally stable shell in a Stokes flow: a rotational Taylor's sheet0 aneutrally stable shell in a Stokes flow a rotational Taylors she c2019/07/26 a201901780 v4751 aCorsi, Giovanni1 aDeSimone, Antonio1 aMaurini, C.1 aVidoli, S. uhttps://doi.org/10.1098/rspa.2019.017800422nas a2200109 4500008004100000245006100041210006100102100002400163700001800187700002000205856008700225 2019 eng d00aRegularity of minimizers for a model of charged droplets0 aRegularity of minimizers for a model of charged droplets1 aDe Philippis, Guido1 aHirsch, Jonas1 aVescovo, Giulia uhttps://www.math.sissa.it/publication/regularity-minimizers-model-charged-droplets01267nas a2200145 4500008004100000245010200041210006900143300001000212520067900222100001600901700001400917700001700931700002100948856015200969 2019 eng d00aWeighted Reduced Order Methods for Parametrized Partial Differential Equations with Random Inputs0 aWeighted Reduced Order Methods for Parametrized Partial Differen a27-403 a

In this manuscript we discuss weighted reduced order methods for stochastic partial differential equations. Random inputs (such as forcing terms, equation coefficients, boundary conditions) are considered as parameters of the equations. We take advantage of the resulting parametrized formulation to propose an efficient reduced order model; we also profit by the underlying stochastic assumption in the definition of suitable weights to drive to reduction process. Two viable strategies are discussed, namely the weighted reduced basis method and the weighted proper orthogonal decomposition method. A numerical example on a parametrized elasticity problem is shown.

1 aVenturi, L.1 aTorlo, D.1 aBallarin, F.1 aRozza, Gianluigi uhttps://www.scopus.com/inward/record.uri?eid=2-s2.0-85084009379&doi=10.1007%2f978-3-030-04870-9_2&partnerID=40&md5=446bcc1f331167bbba67bc00fb17015001813nas a2200205 4500008004100000245005400041210005400095260001400149300000700163520117300170100002601343700001901369700002001388700002101408700002201429700002101451700002601472700002501498856008401523 2018 eng d00aComputational methods in cardiovascular mechanics0 aComputational methods in cardiovascular mechanics bCRC Press a543 a

The introduction of computational models in cardiovascular sciences has been progressively bringing new and unique tools for the investigation of the physiopathology. Together with the dramatic improvement of imaging and measuring devices on one side, and of computational architectures on the other one, mathematical and numerical models have provided a new, clearly noninvasive, approach for understanding not only basic mechanisms but also patient-specific conditions, and for supporting the design and the development of new therapeutic options. The terminology in silico is, nowadays, commonly accepted for indicating this new source of knowledge added to traditional in vitro and in vivo investigations. The advantages of in silico methodologies are basically the low cost in terms of infrastructures and facilities, the reduced invasiveness and, in general, the intrinsic predictive capabilities based on the use of mathematical models. The disadvantages are generally identified in the distance between the real cases and their virtual counterpart required by the conceptual modeling that can be detrimental for the reliability of numerical simulations.

1 aAuricchio, Ferdinando1 aConti, Michele1 aLefieux, Adrian1 aMorganti, Simone1 aReali, Alessandro1 aRozza, Gianluigi1 aVeneziani, Alessandro1 aLabrosse, Michel, F. uhttps://www.taylorfrancis.com/books/e/9781315280288/chapters/10.1201%2Fb21917-501884nas a2200145 4500008004100000245005300041210005300094260003400147300001100181490000700192520140800199100002001607700002501627856008601652 2016 eng d00aCurrents and dislocations at the continuum scale0 aCurrents and dislocations at the continuum scale bInternational Press of Boston a1–340 v233 a

A striking geometric property of elastic bodies with dislocations is that the deformation tensor cannot be written as the gradient of a one-to-one immersion, its curl being nonzero and equal to the density of the dislocations, a measure concentrated in the dislocation lines. In this work, we discuss the mathematical properties of such constrained deformations and study a variational problem in finite-strain elasticity, where Cartesian maps allow us to consider deformations in $L^p$ with $1\leq p<2$, as required for dislocation-induced strain singularities. Firstly, we address the problem of mathematical modeling of dislocations. It is a key purpose of the paper to build a framework where dislocations are described in terms of integral 1-currents and to extract from this theoretical setting a series of notions having a mechanical meaning in the theory of dislocations. In particular, the paper aims at classifying integral 1-currents, with modeling purposes. In the second part of the paper, two variational problems are solved for two classes of dislocations, at the mesoscopic and at the continuum scale. By continuum it is here meant that a countable family of dislocations is considered, allowing for branching and cluster formation, with possible complex geometric patterns. Therefore, modeling assumptions of the defect part of the energy must also be provided, and discussed.

1 aScala, Riccardo1 aVan Goethem, Nicolas uhttps://www.math.sissa.it/publication/currents-and-dislocations-continuum-scale-001802nas a2200229 4500008004100000245010400041210006900145300001400214490000700228520107100235653001001306653001001316653002901326653001501355653002001370653002501390653001801415100003301433700002001466700002501486856006101511 2015 eng d00aA compatible-incompatible decomposition of symmetric tensors in Lp with application to elasticity0 acompatibleincompatible decomposition of symmetric tensors in Lp  a5217-52300 v383 a

In this paper, we prove the Saint-Venant compatibility conditions in $L^p$ for $p\in(1,∞)$, in a simply connected domain of any space dimension. As a consequence, alternative, simple, and direct proofs of some classical Korn inequalities in Lp are provided. We also use the Helmholtz decomposition in $L^p$ to show that every symmetric tensor in a smooth domain can be decomposed in a compatible part, which is the symmetric part of a displacement gradient, and in an incompatible part, which is the incompatibility of a certain divergence-free tensor. Moreover, under a suitable Dirichlet boundary condition, this Beltrami-type decomposition is proved to be unique. This decomposition result has several applications, one of which being in dislocation models, where the incompatibility part is related to the dislocation density and where $1 < p < 2$. This justifies the need to generalize and prove these rather classical results in the Hilbertian case ($p = 2$), to the full range $p\in(1,∞)$. Copyright © 2015 John Wiley & Sons, Ltd.

10a35J5810a35Q7410acompatibility conditions10aelasticity10aKorn inequality10astrain decomposition10asubclass74B051 aMaggiani, Giovanni, Battista1 aScala, Riccardo1 aVan Goethem, Nicolas uhttps://onlinelibrary.wiley.com/doi/abs/10.1002/mma.345001384nas a2200133 4500008004100000245008700041210006900128260001000197520092000207100002701127700002301154700002201177856005101199 2015 en d00aDynamics of screw dislocations: a generalised minimising-movements scheme approach0 aDynamics of screw dislocations a generalised minimisingmovements bSISSA3 aThe gradient flow structure of the model introduced in [CG99] for the dynamics of screw dislocations is investigated by means of a generalised minimising-movements scheme approach. The assumption of a finite number of available glide directions, together with the "maximal dissipation criterion" that governs the equations of motion, results into solving a differential inclusion rather than an ODE. This paper addresses how the model in [CG99] is connected to a time-discrete evolution scheme which explicitly confines dislocations to move each time step along a single glide direction. It is proved that the time-continuous model in [CG99] is the limit of these time-discrete minimising-movement schemes when the time step converges to 0. The study presented here is a first step towards a generalization of the setting in [AGS08, Chap. 2 and 3] that allows for dissipations which cannot be described by a metric.1 aBonaschi, Giovanni, A.1 aVan Meurs, Patrick1 aMorandotti, Marco uhttp://urania.sissa.it/xmlui/handle/1963/3449501209nas a2200133 4500008004100000245010300041210006900144260001000213520075500223100002100978700002000999700002001019856003601039 2014 eng d00aAdler-Gelfand-Dickey approach to classical W-algebras within the theory of Poisson vertex algebras0 aAdlerGelfandDickey approach to classical Walgebras within the th bSISSA3 aWe put the Adler-Gelfand-Dickey approach to classical W-algebras in the framework of Poisson vertex algebras. We show how to recover the bi-Poisson structure of the KP hierarchy, together with its generalizations and reduction to the N-th KdV hierarchy, using the formal distribution calculus and the lambda-bracket formalism. We apply the Lenard-Magri scheme to prove integrability of the corresponding hierarchies. We also give a simple proof of a theorem of Kupershmidt and Wilson in this framework. Based on this approach, we generalize all these results to the matrix case. In particular, we find (non-local) bi-Poisson structures of the matrix KP and the matrix N-th KdV hierarchies, and we prove integrability of the N-th matrix KdV hierarchy.1 aDe Sole, Alberto1 aKac, Victor, G.1 aValeri, Daniele uhttp://hdl.handle.net/1963/724201294nas a2200133 4500008004100000245010400041210006900145260001000214520083900224100002101063700002001084700002001104856003601124 2014 en d00aClassical W-algebras and generalized Drinfeld-Sokolov hierarchies for minimal and short nilpotents0 aClassical Walgebras and generalized DrinfeldSokolov hierarchies  bSISSA3 aWe derive explicit formulas for lambda-brackets of the affine classical W-algebras attached to the minimal and short nilpotent elements of any simple Lie algebra g. This is used to compute explicitly the first non-trivial PDE of the corresponding intgerable generalized Drinfeld-Sokolov hierarchies. It turns out that a reduction of the equation corresponding to a short nilpotent is Svinolupov's equation attached to a simple Jordan algebra, while a reduction of the equation corresponding to a minimal nilpotent is an integrable Hamiltonian equation on 2h-3 functions, where h is the dual Coxeter number of g. In the case when g is sl_2 both these equations coincide with the KdV equation. In the case when g is not of type C_n, we associate to the minimal nilpotent element of g yet another generalized Drinfeld-Sokolov hierarchy.1 aDe Sole, Alberto1 aKac, Victor, G.1 aValeri, Daniele uhttp://hdl.handle.net/1963/697900720nas a2200109 4500008004100000245007700041210006900118260003100187520031700218100002400535856005100559 2014 en d00aA correction and an extension of Stampacchia's work on the geometric BVP0 acorrection and an extension of Stampacchias work on the geometri bAdvanced Nonlinear Studies3 aG. Stampacchia introduced the geometric boundary value problem for ODEs in his doctoral thesis and published four papers related to it. Here we point out that the proof of his last theorem on the subject is incorrect and we provide a substitute for it as well as a generalizations of some of his earlier results.1 aVidossich, Giovanni uhttp://urania.sissa.it/xmlui/handle/1963/3502300706nas a2200133 4500008004100000245004800041210004800089260001000137520032800147100002100475700002000496700002000516856003600536 2014 en d00aDirac reduction for Poisson vertex algebras0 aDirac reduction for Poisson vertex algebras bSISSA3 aWe construct an analogue of Dirac's reduction for an arbitrary local or non-local Poisson bracket in the general setup of non-local Poisson vertex algebras. This leads to Dirac's reduction of an arbitrary non-local Poisson structure. We apply this construction to an example of a generalized Drinfeld-Sokolov hierarchy.1 aDe Sole, Alberto1 aKac, Victor, G.1 aValeri, Daniele uhttp://hdl.handle.net/1963/698000376nas a2200097 4500008004100000245008700041210006900128100002000197700002500217856003600242 2014 eng d00aDislocations at the continuum scale: functional setting and variational properties0 aDislocations at the continuum scale functional setting and varia1 aScala, Riccardo1 aVan Goethem, Nicolas uhttp://cvgmt.sns.it/paper/2294/00316nas a2200121 4500008004100000245001400041210001400055260001300069100002000082700002100102700002000123856005100143 2014 en d00aEditorial0 aEditorial bSpringer1 aCiliberto, Ciro1 aDal Maso, Gianni1 aVetro, Pasquale uhttp://urania.sissa.it/xmlui/handle/1963/3471200754nas a2200133 4500008004100000245006000041210005900101260001000160520035300170100002100523700002000544700002000564856003600584 2014 en d00aIntegrability of Dirac reduced bi-Hamiltonian equations0 aIntegrability of Dirac reduced biHamiltonian equations bSISSA3 aFirst, we give a brief review of the theory of the Lenard-Magri scheme for a non-local bi-Poisson structure and of the theory of Dirac reduction. These theories are used in the remainder of the paper to prove integrability of three hierarchies of bi-Hamiltonian PDE's, obtained by Dirac reduction from some generalized Drinfeld-Sokolov hierarchies.1 aDe Sole, Alberto1 aKac, Victor, G.1 aValeri, Daniele uhttp://hdl.handle.net/1963/724700566nas a2200133 4500008004100000245010000041210006900141300001000210100002100220700002200241700002100263700002100284856012700305 2014 eng d00aReduced basis method for the Stokes equations in decomposable domains using greedy optimization0 aReduced basis method for the Stokes equations in decomposable do a1–71 aIapichino, Laura1 aQuarteroni, Alfio1 aRozza, Gianluigi1 aVolkwein, Stefan uhttps://www.math.sissa.it/publication/reduced-basis-method-stokes-equations-decomposable-domains-using-greedy-optimization01519nas a2200145 4500008004100000245015300041210006900194260001300263520096300276100002001239700002301259700002401282700001601306856005101322 2014 en d00aSix-dimensional supersymmetric gauge theories, quantum cohomology of instanton moduli spaces and gl(N) Quantum Intermediate Long Wave Hydrodynamics0 aSixdimensional supersymmetric gauge theories quantum cohomology  bSpringer3 aWe show that the exact partition function of U(N) six-dimensional gauge theory with eight supercharges on C^2 x S^2 provides the quantization of the integrable system of hydrodynamic type known as gl(N) periodic Intermediate Long Wave (ILW). We characterize this system as the hydrodynamic limit of elliptic Calogero-Moser integrable system. We compute the Bethe equations from the effective gauged linear sigma model on S^2 with target space the ADHM instanton moduli space, whose mirror computes the Yang-Yang function of gl(N) ILW. The quantum Hamiltonians are given by the local chiral ring observables of the six-dimensional gauge theory. As particular cases, these provide the gl(N) Benjamin-Ono and Korteweg-de Vries quantum Hamiltonians. In the four dimensional limit, we identify the local chiral ring observables with the conserved charges of Heisenberg plus W_N algebrae, thus providing a gauge theoretical proof of AGT correspondence.1 aBonelli, Giulio1 aSciarappa, Antonio1 aTanzini, Alessandro1 aVasko, Petr uhttp://urania.sissa.it/xmlui/handle/1963/3454601769nas a2200133 4500008004100000245008300041210006900124260001900193520130400212100002001516700002101536700002701557856005101584 2014 en d00aSome remarks on the seismic behaviour of embedded cantilevered retaining walls0 aSome remarks on the seismic behaviour of embedded cantilevered r bThomas Telford3 aThis paper is a numerical investigation of the physical phenomena that control the dynamic behaviour of embedded cantilevered retaining walls. Recent experimental observations obtained from centrifuge tests have shown that embedded cantilevered retaining walls experience permanent displacements even before the acceleration reaches its critical value, corresponding to full mobilisation of the soil strength. The motivation for this work stems from the need to incorporate these observations in simplified design procedures. A parametric study was carried out on a pair of embedded cantilevered walls in dry sand, subjected to real earthquakes scaled at different values of the maximum acceleration. The results of these analyses indicate that, for the geotechnical design of the wall, the equivalent acceleration to be used in pseudo-static calculations can be related to the maximum displacement that the structure can sustain, and can be larger than the maximum acceleration expected at the site. For the structural design of the wall, it is suggested that the maximum bending moments of the wall can be computed using a realistic distribution of contact stress and a conservative value of the pseudo-static acceleration, taking into account two-dimensional amplification effects near the walls.1 aConti, Riccardo1 aD'Arezzo, Burali1 aViggiani, Giulia, M.B. uhttp://urania.sissa.it/xmlui/handle/1963/3507301412nas a2200145 4500008004100000245004500041210004100086260001300127520099200140100002001132700002301152700002401175700001601199856005101215 2014 en d00aThe stringy instanton partition function0 astringy instanton partition function bSpringer3 aWe perform an exact computation of the gauged linear sigma model associated to a D1-D5 brane system on a resolved A_1 singularity. This is accomplished via supersymmetric localization on the blown-up two-sphere. We show that in the blow-down limit C^2/Z_2 the partition function reduces to the Nekrasov partition function evaluating the equivariant volume of the instanton moduli space. For finite radius we obtain a tower of world-sheet instanton corrections, that we identify with the equivariant Gromov-Witten invariants of the ADHM moduli space. We show that these corrections can be encoded in a deformation of the Seiberg-Witten prepotential. From the mathematical viewpoint, the D1-D5 system under study displays a twofold nature: the D1-branes viewpoint captures the equivariant quantum cohomology of the ADHM instanton moduli space in the Givental formalism, and the D5-branes viewpoint is related to higher rank equivariant Donaldson-Thomas invariants of P^1 x C^2.1 aBonelli, Giulio1 aSciarappa, Antonio1 aTanzini, Alessandro1 aVasko, Petr uhttp://urania.sissa.it/xmlui/handle/1963/3458901002nas a2200133 4500008004100000245005800041210005500099260001000154520060700164100002100771700002000792700002000812856003600832 2014 en d00aStructure of classical (finite and affine) W-algebras0 aStructure of classical finite and affine Walgebras bSISSA3 aFirst, we derive an explicit formula for the Poisson bracket of the classical finite W-algebra W^{fin}(g,f), the algebra of polynomial functions on the Slodowy slice associated to a simple Lie algebra g and its nilpotent element f. On the other hand, we produce an explicit set of generators and we derive an explicit formula for the Poisson vertex algebra structure of the classical affine W-algebra W(g,f). As an immediate consequence, we obtain a Poisson algebra isomorphism between W^{fin}(g,f) and the Zhu algebra of W(g,f). We also study the generalized Miura map for classical W-algebras.1 aDe Sole, Alberto1 aKac, Victor, G.1 aValeri, Daniele uhttp://hdl.handle.net/1963/731401601nas a2200145 4500008004100000245008900041210007100130260001300201520110700214100002001321700002301341700002401364700001601388856005101404 2014 en d00aVortex Partition Functions, Wall Crossing and Equivariant Gromov–Witten Invariants0 aVortex Partition Functions Wall Crossing and Equivariant Gromov– bSpringer3 aIn this paper we identify the problem of equivariant vortex counting in a (2,2) supersymmetric two dimensional quiver gauged linear sigma model with that of computing the equivariant Gromov–Witten invariants of the GIT quotient target space determined by the quiver. We provide new contour integral formulae for the I and J-functions encoding the equivariant quantum cohomology of the target space. Its chamber structure is shown to be encoded in the analytical properties of the integrand. This is explained both via general arguments and by checking several key cases. We show how several results in equivariant Gromov–Witten theory follow just by deforming the integration contour. In particular, we apply our formalism to compute Gromov–Witten invariants of the C3/Zn orbifold, of the Uhlembeck (partial) compactification of the moduli space of instantons on C2, and of An and Dn singularities both in the orbifold and resolved phases. Moreover, we analyse dualities of quantum cohomology rings of holomorphic vector bundles over Grassmannians, which are relevant to BPS Wilson loop algebrae.1 aBonelli, Giulio1 aSciarappa, Antonio1 aTanzini, Alessandro1 aVasko, Petr uhttp://urania.sissa.it/xmlui/handle/1963/3465201065nas a2200133 4500008004100000245012700041210006900168260001300237520058400250100002100834700002000855700002000875856003600895 2013 en d00aClassical W-algebras and generalized Drinfeld-Sokolov bi-Hamiltonian systems within the theory of Poisson vertex algebras0 aClassical Walgebras and generalized DrinfeldSokolov biHamiltonia bSpringer3 aWe provide a description of the Drinfeld-Sokolov Hamiltonian reduction for the construction of classical W-algebras within the framework of Poisson vertex algebras. In this context, the gauge group action on the phase space is translated in terms of (the exponential of) a Lie conformal algebra action on the space of functions. Following the ideas of Drinfeld and Sokolov, we then establish under certain sufficient conditions the applicability of the Lenard-Magri scheme of integrability and the existence of the corresponding integrable hierarchy of bi-Hamiltonian equations.1 aDe Sole, Alberto1 aKac, Victor, G.1 aValeri, Daniele uhttp://hdl.handle.net/1963/697800951nas a2200145 4500008004100000245009100041210006900132260001000201520045700211653003300668100002100701700002500722700002200747856003600769 2013 en d00aDislocation dynamics in crystals: a macroscopic theory in a fractional Laplace setting0 aDislocation dynamics in crystals a macroscopic theory in a fract bSISSA3 aWe consider an evolution equation arising in the Peierls-Nabarro model for crystal dislocation. We study the evolution of such dislocation function and show that, at a macroscopic scale, the dislocations have the tendency to concentrate at single points of the crystal, where the size of the slip coincides with the natural periodicity of the medium. These dislocation points evolve according to the external stress and an interior repulsive potential.10anonlocal Allen-Cahn equation1 aDipierro, Serena1 aPalatucci, Giampiero1 aValdinoci, Enrico uhttp://hdl.handle.net/1963/712400989nas a2200133 4500008004100000245010200041210006900143260002600212520042000238100002100658700002500679700002200704856012900726 2013 en d00aExistence and symmetry results for a Schrodinger type problem involving the fractional Laplacian0 aExistence and symmetry results for a Schrodinger type problem in bUniversity of Catania3 a

This paper deals with the following class of nonlocal Schr\"odinger equations $$ \displaystyle (-\Delta)^s u + u = |u|^{p-1}u \ \ \text{in} \ \mathbb{R}^N, \quad \text{for} \ s\in (0,1). $$ We prove existence and symmetry results for the solutions $u$ in the fractional Sobolev space $H^s(\mathbb{R}^N)$. Our results are in clear accordance with those for the classical local counterpart, that is when $s=1$.

1 aDipierro, Serena1 aPalatucci, Giampiero1 aValdinoci, Enrico uhttps://www.math.sissa.it/publication/existence-and-symmetry-results-schrodinger-type-problem-involving-fractional-laplacian01391nas a2200109 4500008004100000245006200041210006200103260001000165520104500175100002501220856003601245 2013 en d00aFields of bounded deformation for mesoscopic dislocations0 aFields of bounded deformation for mesoscopic dislocations bSISSA3 aIn this paper we discuss the consequences of the distributional approach to dislocations in terms of the mathematical properties\\r\\nof the auxiliary model fields such as displacement and displacement gradient which are obtained directly from \\r\\nthe main model field here considered as the linear strain. We show that these fields cannot be introduced rigourously without \\r\\nthe introduction of gauge fields, or equivalently, without cuts in the Riemann foliation associated to the dislocated crystal.\\r\\nIn a second step we show that the space of bounded deformations follows from the distributional approach in a natural way and \\r\\ndiscuss the reasons why it is adequate to model dislocations. The case of dislocation clusters is also addressed, as it represents an important issue in industrial crystal growth while from a mathematical point of view, peculiar phenomena might appear at the set of accumulation points. \\r\\nThe elastic-plastic decomposition of the strain within this approach is also given a precise meaning.1 aVan Goethem, Nicolas uhttp://hdl.handle.net/1963/637801676nas a2200145 4500008004100000245009400041210006900135260001000204520112400214653008201338100002001420700002501440700002901465856003601494 2013 en d00aMinimal partitions and image classification using a gradient-free perimeter approximation0 aMinimal partitions and image classification using a gradientfree bSISSA3 aIn this paper a new mathematically-founded method for the optimal partitioning of domains, with applications to the classification of greyscale and color images, is proposed. Since optimal partition problems are in general ill-posed, some regularization strategy is required. Here we regularize by a non-standard approximation of the total interface length, which does not involve the gradient of approximate characteristic functions, in contrast to the classical Modica-Mortola approximation. Instead, it involves a system of uncoupled linear partial differential equations and nevertheless shows $\Gamma$-convergence properties in appropriate function spaces. This approach leads to an alternating algorithm that ensures a decrease of the objective function at each iteration, and which always provides a partition, even during the iterations. The efficiency of this algorithm is illustrated by various numerical examples. Among them we consider binary and multilabel minimal partition problems including supervised or automatic image classification, inpainting, texture pattern identification and deblurring.10aImage classification, deblurring, optimal partitions, perimeter approximation1 aAmstutz, Samuel1 aVan Goethem, Nicolas1 aNovotny, Antonio, André uhttp://hdl.handle.net/1963/697600868nas a2200145 4500008004100000245004700041210004400088260004800132520037400180100002100554700002100575700002500596700002200621856007900643 2012 en d00aAsymptotics of the s-perimeter as s →0 0 aAsymptotics of the sperimeter as s →0 bAmerican Institute of Mathematical Sciences3 a

We 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 measure of $E$ and $\Omega$. Moreover, we construct examples of sets for which the limit does not exist.

1 aDipierro, Serena1 aFigalli, Alessio1 aPalatucci, Giampiero1 aValdinoci, Enrico uhttps://www.math.sissa.it/publication/asymptotics-s-perimeter-s-%E2%86%92001737nas a2200145 4500008004100000245007100041210006400112260001900176520126500195653002501460100002001485700002301505700002701528856003601555 2012 en d00aOn the behaviour of flexible retaining walls under seismic actions0 abehaviour of flexible retaining walls under seismic actions bICE Publishing3 aThis paper describes an experimental investigation of the behaviour of embedded retaining walls under seismic actions. Nine centrifuge tests were carried out on reduced-scale models of pairs of retaining walls in dry sand, either cantilevered or with one level of props near the top. The experimental data indicate that, for maximum accelerations that are smaller than the critical limit equilibrium value, the retaining walls experience significant permanent displacements under increasing structural loads, whereas for larger accelerations the walls rotate under constant internal forces. The critical acceleration at which the walls start to rotate increases with increasing maximum acceleration. No significant displacements are measured if the current earthquake is less severe than earthquakes previously experienced by the wall. The increase of critical acceleration is explained in terms of redistribution of earth pressures and progressive mobilisation of the passive strength in front of the wall. The experimental data for cantilevered retaining walls indicate that the permanent displacements of the wall can be reasonably predicted adopting a Newmark-type calculation with a critical acceleration that is a fraction of the limit equilibrium value.10aCentrifuge modelling1 aConti, Riccardo1 aMadabhushi, G.S.P.1 aViggiani, Giulia, M.B. uhttp://hdl.handle.net/1963/693301861nas a2200145 4500008004100000245007600041210006900117260001300186520138500199653002701584100002001611700002101631700002701652856003601679 2012 en d00aNumerical modelling of installation effects for diaphragm walls in sand0 aNumerical modelling of installation effects for diaphragm walls  bSpringer3 aThe scopes of this work are to study the mechanisms of load transfer and the deformations of the ground during slurry trenching and concreting in dry sand and to evaluate their effects on service structural loads, wall deflections and ground displacements behind the wall caused by subsequent excavation. A series of three-dimensional finite element analyses was carried out modelling the installation of diaphragm walls consisting of panels of different length. The soil was modelled as either linearly elastic-perfectly plastic or incrementally non-linear (hypoplastic) with elastic strain range. Plane strain analyses of diaphragm walls of identical cross section were also carried out in which wall installation was either modelled or the wall was wished in place (WIP). The analyses predict ground movements consistent with the experimental observations both in magnitude and trend. The results also show that the maximum horizontal wall deflections and structural loads reduce with increasing panel aspect ratio towards a minimum which is about twice the value computed for WIP analyses. Panel aspect ratios should be larger than about three to take advantage of the three-dimensional effects. The pattern and magnitude of surface vertical displacements obtained from linearly elastic-perfectly plastic analyses, no matter whether three- or two-dimensional, are unrealistic.10aConstitutive relations1 aConti, Riccardo1 ade Sanctis, Luca1 aViggiani, Giulia, M.B. uhttp://hdl.handle.net/1963/693401511nas a2200145 4500008004100000245007100041210006900112260001000181520099600191653006801187100002001255700002901275700002501304856003601329 2012 en d00aTopological sensitivity analysis for high order elliptic operators0 aTopological sensitivity analysis for high order elliptic operato bSISSA3 aThe topological derivative is defined as the first term of the asymptotic expansion of a given shape functional with respect to a small parameter that measures the size of a singular domain perturbation. It has applications in many different fields such as shape and topology optimization, inverse problems, image processing and mechanical modeling including synthesis and/or optimal design of microstructures, fracture mechanics sensitivity analysis and damage evolution modeling. The topological derivative has been fully developed for a wide range of second order differential operators. In this paper we deal with the topological asymptotic expansion of a class of shape functionals associated with elliptic differential operators of order 2m, m>=1. The general structure of the polarization tensor is derived and the concept of degenerate polarization tensor is introduced. We provide full mathematical justifications for the derived formulas, including precise estimates of remainders.10aTopological derivative, Elliptic operators, Polarization tensor1 aAmstutz, Samuel1 aNovotny, Antonio, André1 aVan Goethem, Nicolas uhttp://hdl.handle.net/1963/634300505nas a2200133 4500008004100000245010600041210007000147260004400217300001400261490000700275100001600282700001700298856005600315 2011 eng d00aCluster solutions for the Schrödinger-Poisson-Slater problem around a local minimum of the potential0 aCluster solutions for the SchrödingerPoissonSlater problem aroun bReal Sociedad Matemática Españolac01 a253–2710 v271 aRuiz, David1 aVaira, Giusi uhttps://projecteuclid.org:443/euclid.rmi/129682883400860nas a2200181 4500008004100000022001400041245007500055210007200130300001600202490000700218520024700225653004900472653002400521100002300545700002200568700001700590856007100607 2011 eng d a0362-546X00aInfinitely many positive solutions for a Schrödinger–Poisson system0 aInfinitely many positive solutions for a Schrödinger–Poisson sys a5705 - 57210 v743 a

We are interested in the existence of infinitely many positive solutions of the Schrödinger–Poisson system −Δu+u+V(|x|)ϕu=|u|p−1u,x∈R3,−Δϕ=V(|x|)u2,x∈R3, where V(|x|) is a positive bounded function, 1<p<5 and V(r

10aNon-autonomous Schrödinger–Poisson system10aPerturbation method1 ad’Avenia, Pietro1 aPomponio, Alessio1 aVaira, Giusi uhttp://www.sciencedirect.com/science/article/pii/S0362546X1100351802590nas a2200265 4500008004100000245013200041210006900173260001000242520175000252100001702002700002402019700002002043700001902063700002102082700001802103700003002121700001802151700001702169700001702186700002002203700002202223700002402245700001902269856003602288 2010 en d00aGene expression analysis of the emergence of epileptiform activity after focal injection of kainic acid into mouse hippocampus.0 aGene expression analysis of the emergence of epileptiform activi bWiley3 a

We report gene profiling data on genomic processes underlying the progression towards recurrent seizures after injection of kainic acid (KA) into the mouse hippocampus. Focal injection enabled us to separate the effects of proepileptic stimuli initiated by KA injection. Both the injected and contralateral hippocampus participated in the status epilepticus. However, neuronal death induced by KA treatment was restricted to the injected hippocampus, although there was some contralateral axonal degeneration. We profiled gene expression changes in dorsal and ventral regions of both the injected and contralateral hippocampus. Changes were detected in the expression of 1526 transcripts in samples from three time-points: (i) during the KA-induced status epilepticus, (ii) at 2 weeks, before recurrent seizures emerged, and (iii) at 6 months after seizures emerged. Grouping genes with similar spatio-temporal changes revealed an early transcriptional response, strong immune, cell death and growth responses at 2 weeks and an activation of immune and extracellular matrix genes persisting at 6 months. Immunostaining for proteins coded by genes identified from array studies provided evidence for gliogenesis and suggested that the proteoglycan biglycan is synthesized by astrocytes and contributes to a glial scar. Gene changes at 6 months after KA injection were largely restricted to tissue from the injection site. This suggests that either recurrent seizures might depend on maintained processes including immune responses and changes in extracellular matrix proteins near the injection site or alternatively might result from processes, such as growth, distant from the injection site and terminated while seizures are maintained.

1 aMotti, Dario1 aLe Duigou, Caroline1 aChemaly, Nicole1 aWittner, Lucia1 aLazarevic, Dejan1 aKrmac, Helena1 aMarstrand, Troels, Torben1 aValen, Eivind1 aSanges, Remo1 aStupka, Elia1 aSandelin, Albin1 aCherubini, Enrico1 aGustincich, Stefano1 aMiles, Richard uhttp://hdl.handle.net/1963/448000526nas a2200133 4500008004100000245007800041210007200119260001900191300001400210490000800224100002100232700001700253856012200270 2010 eng d00aPositive solutions for some non-autonomous Schrödinger–Poisson systems0 aPositive solutions for some nonautonomous Schrödinger–Poisson sy bAcademic Press a521–5430 v2481 aCerami, Giovanna1 aVaira, Giusi uhttps://www.math.sissa.it/publication/positive-solutions-some-non-autonomous-schr%C3%B6dinger%E2%80%93poisson-systems00857nas a2200133 4500008004100000245010700041210007200148300001200220490000700232520040100239100002000640700001700660856004600677 2009 eng d00aSolutions of the Schrödinger–Poisson problem concentrating on spheres, part I: necessary conditions0 aSolutions of the Schrödinger–Poisson problem concentrating on sp a707-7200 v193 a

In this paper we study a coupled nonlinear Schrödinger–Poisson problem with radial functions. This system has been introduced as a model describing standing waves for the nonlinear Schrödinger equations in the presence of the electrostatic field. We provide necessary conditions for concentration on sphere for the solutions of this kind of problem extending the results already known.

1 aIanni, Isabella1 aVaira, Giusi uhttps://doi.org/10.1142/S021820250900358900915nas a2200145 4500008004100000245009900041210007000140260003700210300001400247490000600261520034300267100002000610700001700630856012200647 2008 eng d00aOn concentration of positive bound states for the Schrödinger-Poisson problem with potentials0 aconcentration of positive bound states for the SchrödingerPoisso bAdvanced Nonlinear Studies, Inc. a573–5950 v83 a

We study the existence of semiclassical states for a nonlinear Schrödinger-Poisson system that concentrate near critical points of the external potential and of the density charge function. We use a perturbation scheme in a variational setting, extending the results in [1]. We also discuss necessary conditions for concentration.

1 aIanni, Isabella1 aVaira, Giusi uhttps://www.math.sissa.it/publication/concentration-positive-bound-states-schr%C3%B6dinger-poisson-problem-potentials00746nas a2200145 4500008004300000245004200043210004200085260002800127520032600155100002000481700001900501700001800520700002600538856003600564 2008 en_Ud 00aNoncommutative families of instantons0 aNoncommutative families of instantons bOxford University Press3 aWe construct $\\\\theta$-deformations of the classical groups SL(2,H) and Sp(2). Coacting on the basic instanton on a noncommutative four-sphere $S^4_\\\\theta$, we construct a noncommutative family of instantons of charge 1. The family is parametrized by the quantum quotient of $SL_\\\\theta(2,H)$ by $Sp_\\\\theta(2)$.1 aLandi, Giovanni1 aPagani, Chiara1 aReina, Cesare1 avan Suijlekom, Walter uhttp://hdl.handle.net/1963/341700666nas a2200157 4500008004100000022001300041245006000054210005700114300001200171490000600183520016900189100002400358700001400382700002200396856009000418 2008 eng d a1534039200aOn periodic elliptic equations with gradient dependence0 aperiodic elliptic equations with gradient dependence a601-6150 v73 aWe construct entire solutions of Δu = f(x, u, ∇u) which are superpositions of odd, periodic functions and linear ones, with prescribed integer or rational slope.1 aBerti, Massimiliano1 aMatzeu, M1 aValdinoci, Enrico uhttps://www.math.sissa.it/publication/periodic-elliptic-equations-gradient-dependence01006nas a2200157 4500008004100000245003400041210002700075260001300102520058600115100002200701700002000723700002000743700002600763700002300789856003600812 2005 en d00aThe Dirac operator on SU_q(2)0 aDirac operator on SUq2 bSpringer3 aWe construct a 3^+ summable spectral triple (A(SU_q(2)),H,D) over the quantum group SU_q(2) which is equivariant with respect to a left and a right action of U_q(su(2)). The geometry is isospectral to the classical case since the spectrum of the operator D is the same as that of the usual Dirac operator on the 3-dimensional round sphere. The presence of an equivariant real structure J demands a modification in the axiomatic framework of spectral geometry, whereby the commutant and first-order properties need be satisfied only modulo infinitesimals of arbitrary high order.1 aDabrowski, Ludwik1 aLandi, Giovanni1 aSitarz, Andrzej1 avan Suijlekom, Walter1 aVarilly, Joseph C. uhttp://hdl.handle.net/1963/442500687nas a2200145 4500008004300000245003900043210003300082520027900115100002600394700002200420700002000442700002000462700002300482856003600505 2005 en_Ud 00aThe local index formula for SUq(2)0 alocal index formula for SUq23 aWe discuss the local index formula of Connes-Moscovici for the isospectral noncommutative geometry that we have recently constructed on quantum SU(2). We work out the cosphere bundle and the dimension spectrum as well as the local cyclic cocycles yielding the index formula.1 avan Suijlekom, Walter1 aDabrowski, Ludwik1 aLandi, Giovanni1 aSitarz, Andrzej1 aVarilly, Joseph C. uhttp://hdl.handle.net/1963/171300942nas a2200109 4500008004300000245005300043210005300096520060100149100002000750700002600770856003600796 2005 en_Ud 00aPrincipal fibrations from noncommutative spheres0 aPrincipal fibrations from noncommutative spheres3 aWe construct noncommutative principal fibrations S_\\\\theta^7 \\\\to S_\\\\theta^4 which are deformations of the classical SU(2) Hopf fibration over the four sphere. We realize the noncommutative vector bundles associated to the irreducible representations of SU(2) as modules of coequivariant maps and construct corresponding projections. The index of Dirac operators with coefficients in the associated bundles is computed with the Connes-Moscovici local index formula. The algebra inclusion $A(S_\\\\theta^4) \\\\into A(S_\\\\theta^7)$ is an example of a not trivial quantum principal bundle.1 aLandi, Giovanni1 avan Suijlekom, Walter uhttp://hdl.handle.net/1963/228401171nas a2200133 4500008004300000245008600043210006900129260003700198520070300235100002400938700001700962700002200979856003601001 2004 en_Ud 00aPeriodic orbits close to elliptic tori and applications to the three-body problem0 aPeriodic orbits close to elliptic tori and applications to the t bScuola Normale Superiore di Pisa3 aWe prove, under suitable non-resonance and non-degeneracy ``twist\\\'\\\' conditions, a Birkhoff-Lewis type result showing the existence of infinitely many periodic solutions, with larger and larger minimal period, accumulating onto elliptic invariant tori (of Hamiltonian systems). We prove the applicability of this result to the spatial planetary three-body problem in the small eccentricity-inclination regime. Furthermore, we find other periodic orbits under some restrictions on the period and the masses of the ``planets\\\'\\\'. The proofs are based on averaging theory, KAM theory and variational methods. (Supported by M.U.R.S.T. Variational Methods and Nonlinear Differential Equations.)1 aBerti, Massimiliano1 aBiasco, Luca1 aValdinoci, Enrico uhttp://hdl.handle.net/1963/298500352nas a2200097 4500008004100000245006800041210006800109260001800177100002400195856003500219 1996 en d00aSolving Honig generic problem about Volterra integral equations0 aSolving Honig generic problem about Volterra integral equations bSISSA Library1 aVidossich, Giovanni uhttp://hdl.handle.net/1963/94100381nas a2200097 4500008004100000245009600041210006900137260001800206100002400224856003500248 1990 en d00aA general existence theorem for boundary value problems for ordinary differential equations0 ageneral existence theorem for boundary value problems for ordina bSISSA Library1 aVidossich, Giovanni uhttp://hdl.handle.net/1963/63200412nas a2200097 4500008004100000245012700041210006900168260001800237100002400255856003500279 1989 en d00aOn the continuous dependence of solutions of boundary value problems for ordinary differential equations (Revised version)0 acontinuous dependence of solutions of boundary value problems fo bSISSA Library1 aVidossich, Giovanni uhttp://hdl.handle.net/1963/66600394nas a2200097 4500008004100000245010900041210006900150260001800219100002400237856003500261 1989 en d00aOn the continuous dependence of solutions of boundary value problems for ordinary differential equations0 acontinuous dependence of solutions of boundary value problems fo bSISSA Library1 aVidossich, Giovanni uhttp://hdl.handle.net/1963/63300362nas a2200097 4500008004100000245007700041210006900118260001800187100002400205856003500229 1989 en d00aHyperbolic equations as ordinary differential equations in Banach spaces0 aHyperbolic equations as ordinary differential equations in Banac bSISSA Library1 aVidossich, Giovanni uhttp://hdl.handle.net/1963/77300407nas a2200121 4500008004100000245006300041210006000104260001800164100002100182700001900203700002800222856003500250 1989 en d00aA pointwise regularity theory for the two-obstacle problem0 apointwise regularity theory for the twoobstacle problem bSISSA Library1 aDal Maso, Gianni1 aMosco, Umberto1 aVivaldi, Maria Agostina uhttp://hdl.handle.net/1963/64300402nas a2200097 4500008004100000245011700041210006900158260001800227100002400245856003500269 1989 en d00aOn the solvability of boundary value problems for higher order ordinary differential equations (Revised version)0 asolvability of boundary value problems for higher order ordinary bSISSA Library1 aVidossich, Giovanni uhttp://hdl.handle.net/1963/66200384nas a2200097 4500008004100000245009900041210006900140260001800209100002400227856003500251 1989 en d00aOn the solvability of boundary value problems for higher order ordinary differential equations0 asolvability of boundary value problems for higher order ordinary bSISSA Library1 aVidossich, Giovanni uhttp://hdl.handle.net/1963/63100341nas a2200097 4500008004100000245006500041210006000106260001800166100002400184856003500208 1985 en d00aThe two-point boundary value problem from the Cauchy problem0 atwopoint boundary value problem from the Cauchy problem bSISSA Library1 aVidossich, Giovanni uhttp://hdl.handle.net/1963/33200372nas a2200097 4500008004100000245008700041210006900128260001800197100002400215856003500239 1983 en d00aOn the asymptotic behaviour of solutions to Pazy\\\'s class of evolution equations0 aasymptotic behaviour of solutions to Pazys class of evolution eq bSISSA Library1 aVidossich, Giovanni uhttp://hdl.handle.net/1963/27600377nas a2200097 4500008004100000245009200041210006900133260001800202100002400220856003500244 1983 en d00aTowards a theory for periodic solutions to first order ordinary differential equations.0 aTowards a theory for periodic solutions to first order ordinary  bSISSA Library1 aVidossich, Giovanni uhttp://hdl.handle.net/1963/29500379nas a2200097 4500008004100000245009400041210006900135260001800204100002400222856003500246 1982 en d00aA criterion for he existence of maximal solutions of strongly nonlinear elliptic problems0 acriterion for he existence of maximal solutions of strongly nonl bSISSA Library1 aVidossich, Giovanni uhttp://hdl.handle.net/1963/16100349nas a2200097 4500008004100000245007000041210006300111260001800174100002400192856003500216 1982 en d00aOn the obstacle problem for strongly nonlinear elliptic equations0 aobstacle problem for strongly nonlinear elliptic equations bSISSA Library1 aVidossich, Giovanni uhttp://hdl.handle.net/1963/16200357nas a2200097 4500008004100000245007200041210006900113260001800182100002400200856003500224 1982 en d00aThree uniqueness theorems for strongly non-linear elliptic problems0 aThree uniqueness theorems for strongly nonlinear elliptic proble bSISSA Library1 aVidossich, Giovanni uhttp://hdl.handle.net/1963/16700384nas a2200097 4500008003900000245010100039210006900140260001800209100002400227856003500251 0 end00aUniqueness and multiplicity of periodic solutions to first order ordinary differential equations0 aUniqueness and multiplicity of periodic solutions to first order bSISSA Library1 aVidossich, Giovanni uhttp://hdl.handle.net/1963/321