Analysis and differential equations

Dia 21 de Setembro (quinta-feira), às 13h30, sala 6.2.33

Positive powers of the Laplacian: from hypersingular integrals to boundary value problems

Alberto Saldaña (Karlsruher Institut für Technologie)

Abstract: We revisit the classical concepts of Green Function and Poisson kernel for the Laplacian as a way of constructing explicit solutions to boundary value problems. We mention how these ideas extend to the polyharmonic operator and to the fractional Laplacian for powers $ 0<s<1 $, where some similarities appear, but also surprising differences. Finally, we show that Green Functions, Poisson kernels, and other boundary kernels can be extended to find explicit solutions and representation formulas for any positive power of the Laplacian, including the higher-order regime, where the operator can be represented as a hypersingular integral with finite differences.

 

Dia 26 de Outubro (quinta-feira), às 13h30, sala 6.2.33

Choquard type equations with Hardy-Littlewood-Sobolev critical growth

Daniele Cassani (Università degli Studi dell'Insubria and RISM)

Abstract: We are concerned with existence and concentration of ground state solutions to a class of nonlocal Schrödinger equations. We consider the case in which the nonlinearity exhibits critical growth, namely in presence of the Hardy-Littlewood-Sobolev lower, as well as upper critical exponents. Assuming mild conditions on the nonlinearity and by using variational methods, we obtain existence and non-existence results. The limiting case of dimension two is also addressed. (Joint works in collaboration with: J. Zhang; J. van Schaftingen and J. Zhang; C. Tarsi and M. Yang).

 

Dia 9 de Novembro (quinta-feira), às 13h30, sala 6.2.33

Sign-changing solutions of the nolinear heat equation with positive initial value

Fred Weissler (Institut Galilée - Université Paris 13)

Abstract: We consider the nonlinear heat equation $ u_t - \Delta u = | u |^\alpha u $ on $ \mathbb{R}^N $, where $ \alpha > 0 $. It is well known that the Cauchy problem is locally well-posed in a variety of spaces. For instance, for every $ \alpha > 0 $, it is well-posed in the space $ C_0 ( \mathbb{R}^N ) $ of continuous functions that converge  to $ 0 $ at infinity. It is also well-posed in $ L^p ( \mathbb{R}^N ) $ for $ p > 1 $, $ p > \frac{N \alpha}{2} $, but not well-posed in $ L^p $ for $ 1 \leq p < \frac{N \alpha}{2} $ if $ \alpha > \frac{2}{N} $. In particular, for such $ p $ there exist positive initial values $ u_0 \in L^p $ for which there is no local in time positive solution. Also, if one considers the initial value $ u_0 (x) = c | x |^{-\frac{2}{\alpha}} $ for all $ x \in \mathbb{R}^N \setminus \{ 0 \} $, with $ c > 0 $, it is known that if $ c $ is small, there exists a global in time (positive) solution with $ u_0 $ as initial value, and in fact this solution is self-similar. On the other hand, if $ c $ is large, there is no local in time positive solution, self-similar or otherwise. We prove that in the range $ 0 < \alpha < \frac{4}{N - 2} $, for every $ c > 0 $, there exists infinitely many self-similar solutions to the Cauchy problem with initial value $ u_0 (x) = c | x |^{-\frac{2}{\alpha}} $. Of course, this solutions are all sign-changing if $ c $ is sufficiently large. Also, in the range $ \frac{2}{N} < \alpha < \frac{4}{N - 2} $, we prove the existence of local in time sign-changing solutions  for a class of nonnegative initial values $ u_0 \in L^p $, for $ 1 \leq p < \frac{N \alpha}{2} $, for which no local in time positive solution exists.

This is joint work with T. Cazenave, F. Dickstein and I. Naumkin.

 

 


 

Dia 12 de Setembro (terça-feira), às 13:30, sala 6.2.33

The regularity of solutions to the elasto-plastic problem and to variational problems of fast growth

Arrigo Cellina (Università di Milano-Bicocca)

Abstract: We consider the problem of the higher differentiability of solutions to some variational problems, characterized by the fast growth of the Lagrangian with respect to the variable gradient. In particular,  the case of the elasto-plastic problem, where the Lagrangian is an extended-valued function, will be discussed.

 

Dia 11 de Setembro (segunda-feira), às 13:30, sala 6.2.33

On $s$-harmonic functions on cones

Susanna Terracini (Università di Torino)

Abstract: We deal with functions satisfying $$ \begin{cases} (-\Delta)^s u_s=0 & \mathrm{in} \quad C, \\ u_s=0 & \mathrm{in} \quad \mathbb{R}^n\setminus C, \end{cases} $$ where $ s \in (0,1) $ and $C$ is a given cone on $ \mathbb{R}^n $ with vertex at zero. We are mainly concerned with the case when $s$ approaches $1$. These functions are involved in the study of the nodal regions in the case of optimal partitions and other free boundary problems and play a crucial role in the extension of the Alt-Caffarelli-Friedman monotonicity formula to the case of fractional diffusions. This is a joint work with Giorgio Tortone and Stefano Vita.

 

Dia 13 de Julho (quinta-feira), às 13:30, sala 6.2.33

Stability and attractivity for Nicholson systems with time-dependant delays

Diogo Caetano (Mestrando em Matemática - FCUL)

Abstract: In the first part of this seminar, we introduce some general concepts and results from the theory of delay differential equations (DDEs), concerning existence and uniqueness of solutions and stability for linear autonomous DDEs. As an illustration, stability properties of the scalar Nicholson's  equation $$ N'(t)=-dN(t)+\beta N(t-\tau)e^{-N(t-\tau)}\quad (d,\beta,\tau>0) $$ are presented.

Next, we concentrate our study on Nicholson-type systems with constant coefficients and multiple time-varying delays of the form $$ N_i'(t)=-d_iN_i(t)+\sum_{j=1,j\ne i}^n a_{ij}N_j(t)+\sum_{k=1}^m \beta_{ik}    N_i(t-\tau_{ik}(t))e^{-c_iN_i(t-\tau_{ik}(t))},\ i=1,\dots,n,\ t\ge 0, $$ where $d_i,c_i>0$, $a_{ij}, \beta_{ik}\ge 0$ with $\beta_i:=\sum_{k=1}^m \beta_{ik} >0$, $\tau_{ik}:[0,\infty)\to [0,\infty)$  are continuous and bounded, for $i,j=1,\dots,n,k=1,\dots,m$. Motivated by the works [1, 3], original results about the global attractivity of the positive equilibrium (when it exists) are established. First, sufficient conditions on the coefficients are given  for  the existence and absolute global exponential stability of a unique positive equilibrium $N^*$. On the other hand, upper bounds on the   size of the delay functions are  established,  for $N^*$ to be a global attractor of all positive solutions, improving results in [3], where a very restrictive and artificial hypothesis was imposed. In this latter case and contrarily to the situation in [3], our criterion does not require the a priori explicit  knowledge of the equilibrium.

[1] T. Faria, G. Röst, Persistence, permanence and global stability for an $n$-dimensional Nicholson system,  J. Dyn. Diff. Equ., 26 (2014), 723–744.

[2] J. Hofbauer, An index theorem for dissipative systems. Rocky Mountain J. Math., 20 (1990), 1017–1031.

[3] R. Jia, Z. Long, M. Yang, Delay-dependent criteria on the global attractivity of Nicholson's blowflies model with patch structure, Math. Meth. Appl. Sci., 40 (2017), 4222–4232.

Remark: This is also a seminar of the Master Programme in Mathematics of the DM, FCUL, and will be presented in Portuguese.

 

Dia 29 de Junho (quinta-feira), às 13h30, sala 6.2.33

Quantization of energy and weakly turbulent profiles for some dissipative wave equations

Alain Haraux (Labo. J.-L. Lions and Univ. Pierre et Marie Curie)

Abstract: We consider a second order equation with a linear “elastic” part and a nonlinear damping term involving the spatial integral of the square of the velocity: $ u'' \left( t \right) + \left| u' \left( t \right) \right|^2 u' \left( t \right) + A u \left( t \right) = 0 $. We investigate the asymptotic behavior of solutions, after rescaling them suitably in order to take account of the decay rate and bound their energy away from zero. It turns out that solutions with finitely many non-trivial Fourier components are asymptotic to solutions of the linear equation without damping, and exhibit some sort of equipartition of the energy among the components. Solutions with infinitely many Fourier components tend to zero weakly but not strongly. We show also that the limit of the energy of solutions depends only on the number of their non-trivial Fourier components. The idea of the proofs is based on comparison with a simplified model devised through an averaging procedure.

 

Dia 22 de Junho (quinta-feira), às 13h30, sala 6.2.33

On a quadratic Schrödinger System

Filipe Oliveira (Centro de Matemática e Aplicações - FCT - Universidade Nova de Lisboa)

Abstract: In this talk we will consider the quadratic Schrödinger system $$ \begin{cases} i u_t + Δ_{\gamma_1} u + \bar{u} v = 0 & \\ 2 i v_t + Δ_{\gamma_2} u - β v + \frac{1}{2} u^2 = 0, & t \in \mathbb{R}, x \in \mathbb{R}^d \times \mathbb{R} \end{cases} $$ in dimensions $ 1 \leq d \leq 4 $ and for $ \gamma_1, \gamma_2 > 0 $, the so-called elliptic-elliptic case. This system arises as a model for the interaction of waves propagating in a certain class of dispersive media, said $ \chi^2 $ dispersive media. We will show the formation of singularities and blow-up in the $ L^2 $-(super)critical case. Furthermore, we derive several stability results concerning the ground state solutions of this system. This is a joint work with Ádan Corcho (UFRJ), Simão Correia (FCUL) and Jorge D. Silva (IST-UL).

 

Dia 20 de Junho (terça-feira), às 13h30, sala 6.2.33

SIR-Network model: epidemics dynamics in a city & climate variations

Stefanella Boatto (INRIA-CentraleSupelec & Universidade Federal de Rio de Janeiro)

Abstract: The SIR-network mode deals with the propagation of disease epidemics in highly populated cities. The nodes, or vertices, are the city’s neighborhoods, in which the local populations are assumed to be well-mixed. The directed edges represent the fractions of people moving from their neighborhoods of residence to those of daily activities. First, we present some fundamental properties of the basic reproduction number (Ro) for this model. In particular, we focus on how Ro depends upon the geometry and the heterogeneity (different infection rates in each vertex) of the network. This allows us to conclude whether an epidemic outbreak can be expected or not. Second, we submit the SIR-network model to data fitting, using data collected during the 2008 Rio de Janeiro dengue fever epidemic. Important conclusions are drawn from the fitted parameters, and we show that improved results are found when a time-dependent infection parameter is introduced. This work is in collaboration with Lucas Stolerman (UFRJ/IMPA, Brazil) and Daniel Coombs (UBC, Canada).

Finally we can show some recent results, in collaboration with Catherine Bonnet (INRIA-CentraleSupelec), Bernard Cazelles (UPMC) and  Frederic Mazenc (INRIA-CentraleSupelec), about existence and approximation of attractors in the case of a periodic or bounded time dependent infectious parameter.

 

Dia 8 de Junho (quinta-feira), às 13h30, sala 6.2.33

Non-local quantities and their application

Marco Caroccia (CNA - Carnegie Melon University - Pittsburgh - USA)

Abstract: Non-local quantities and fractional Sobolev norms has received an incresing amount of attention in the last decade due to the interesting property they present and their various application in different fields, from graph theory to PDEs. In the first part of the talk I will introduce these topics by starting with a general overview on such quantities and by talking about the state of arts. Then in the second part of my seminar I will talk about two results obtained in collaboration with Dejan Slepcev and Matteo Rinaldi, involving a non-local version of Mumford-Shah functional for graph and the non-local perimeter.

 

Dia 1 de Junho (quinta-feira), às 14h15, sala 6.2.33

Topological derivative for two-scale topology optimization

Àlex Ferrer (CIMNE - Universitat Politècnica de Catalunya)

Abstract: The aim of the talk is to address multiscale topology optimization problems. For this purpose, the concept of topological derivative in conjunction with the computational homogenization method will be considered and the following techniques will be presented:
• A precise treatment of the interface elements to reduce the numerical instabilities and the time-consuming computations that appear when using the slerp algorithm.
• A closed formula of the anisotropic topological derivative by solving analytically the exterior elastic problem. Complex variable theory and symbolic computation is considered.
• A reduction technique to mitigate the high computational cost of the twoscale topology optimization problem. In addition, the proposed algorithm is modified in order to obtain manufacturable optimal designs.
Finally, two-scale topology optimization examples will be presented in order to display the potential of the methodology.

 

Dia 1 de Junho (quinta-feira), às 13h30, sala 6.2.33

Free Boundary Problems for Viscous Fluids

Vsevolod Solonnikov (Steklov Mathematics Institute - St.Petersburg)

Abstract: The communication is concerned with the problem governing the non-stationary motion of two immiscible fluids (both incompressible or incompressible and compressible), contained in a bounded vessel and separated with a free interface. The motion is described by the system of two Navier-Stokes equations completed by initial and boundary conditions at the exterior boundary and at the free interface that is given at the initial instant $t=0$. It is proved that the problem is uniquely solvable in the Sobolev spaces of functions locally in time or in the infinite time interval $t>0$, provided that the initial data are close to the rest state: the velocity vector fields of both fluids vanish, the pressure and the density of the compressible fluid are constant, the free boundary is a sphere. As $t\to\infty$, the solution tends to the equilibrium state. The resuls are obtained in collaboration with I.V.Denisova.

 

Dia 11 de Maio (quinta-feira), às 13h30, sala 6.2.33

Rubén Figueroa (Universidade de Santiago de Compostela)

Degree theory for discontinuous operators with applications to Ordinary Differential Equations

Abstract: The classical Leray-Schauder's degree is a very powerful tool in order to guarantee the existence of fixed points for suitable continuous operators in Banach spaces. As well-known, the solutions of a large class of boundary value problems can be written in terms of fixed points of continuous operators and so degree theory becomes very useful to deal with this kind of problems. However, Leray-Schauder's degree and the classical fixed point theorems fail when the corresponding mapping is discontinuous. In this talk we will develop a new degree theory for a certain class of operators that need not be continuous and we will show how this can be applied to Ordinary Differential Equations with discontinuous nonlinearities.

 

Dia 4 de Maio (quinta-feira), às 13h30, sala 6.2.38

Francesca Dalbono (Università degli Studi di Palermo)

Multiplicity results for a class of asymptotically linear systems of second-order ordinary differential equations

Abstract: We study multiplicity of solutions to an asymptotically linear Dirichlet problem associated with a planar system of second order ordinary differential equations. The multiplicity result is expressed in term of the Maslov indexes of the linearizations at zero and infinity: the gap between the Maslov indexes provides a lower estimate on the number of solutions. The proof is developed in the framework of the shooting methods and it is based on the concepts of phase angles and moments of verticality.

 

Dia 27 de Abril (quinta-feira), às 13h30, sala 6.2.33

Carlos Rocha (CAMGSD - Instituto Superior Técnico)

Evolution Processes generated by Semilinear Parabolic Equations

Abstract: Evolution Processes generated by Semilinear Parabolic Equations We consider small nonautonomous perturbations of autonomous evolution processes generated by certain scalar semilinear parabolic differential equations. Extending the notion of Morse-Smale dynamical system to the nonautonomous framework we show that under generic assumptions the above evolution processes are Morse-Smale. This is a joint work with Radoslaw Czaja and Waldyr Oliva.

 

Dia 6 de Abril (quinta-feira), às 13h30, sala 6.2.33

Ilaria Lucardesi (École des Mines de Nancy et Institut Élie Cartan de Lorraine)

On two functionals involving the maximum of the torsion function

Abstract: The two most studied elliptic PDEs are probably the torsion problem, also known as St-Venant problem, and the Dirichlet eigenvalue problem. For these classical problems, many estimates and qualitative properties have been obtained, see for example works by Pólya, Szegö, Schiffer, Payne, Hersch, Bandle, and many others.

In this seminar I present some recent results about upper and lower bounds of two shape functionals involving the maximum of the torsion function: I consider the ratio $T(\Omega)\lambda_1(\Omega)/|\Omega|$ and the product $M(\Omega)\lambda_1(\Omega)$, where $\Omega$ is bounded open set with finite Lebesgue measure $|\Omega|$, $T(\Omega)$ denotes the torsion, and $\lambda_1(\Omega)$ the first Dirichlet eigenvalue. Particular attention is devoted to the subclass of convex sets.

This is a joint work with A. Henrot and G. Philippin.

 

Dia 31 de Março (sexta-feira), às 11h00, sala 6.2.38

(em co-organização com o Grupo de Física Matemática da Universidade de Lisboa)

Michael Röckner (Bielefeld University)

Global solutions to random 3D vorticity equations for small initial data

Abstract: One proves the existence and uniqueness in $\big( L^p (\mathbb{R}^3) \big)^3 $, $ \frac {3}{2} < p < 2 $, of a global mild solution to random vorticity equations associated to stochastic 3D Navier-Stokes equations with linear multiplicative Gaussian noise of convolution type, for sufficiently small initial vorticity. This resembles some earlier deterministic results of T. Kato and are obtained by treating the equation in vorticity form and reducing the latter to a random nonlinear parabolic equation. The solution has maximal regularity in the spatial variables and is weakly continuous in $ \big( L^3 \cap L^{\frac {3 p} {4 p - 6}} \big)^3 $ with respect to the time variable. Furthermore, we obtain the pathwise continuous dependence of solutions with respect to the initial data.

This is joint work with Viorel Barbu.

 

Dia 24 de Março (sexta-feira), às 15h00, sala 6.2.38

Ana Margarida Ribeiro (CMA - FCT, Universidade Nova de Lisboa)

Characterization of Sobolev Spaces Through Functionals without Derivatives Dependence

Abstract: Extending previous works due to Bourgain, Brézis, and Mironescu [J. Anal. Math. 87 (2002)] and Leoni and Spector  [J. Funct. Anal. 261, 10 (2011)], we provide new characterizations of Sobolev spaces in terms of functionals involving difference quotients. These characterizations have their origin on the study of the limit behavior of the Gagliardo semi-norms taken by Bourgain, Brézis, and Mironescu and may have some applications to imaging problems. Our results include classical higher-order Sobolev spaces as well as the variable exponent case. More general spaces as Musielak-Orlicz spaces will also be discussed. This talk is a consequence of a joint work with C. Kreisbeck and R. Ferreira [Nonlinear Analysis: Theory, Methods & Applications (2015)] and with P. Hästö [Communications in Contemporary Mathematics (2016 online ready)].

 

Dia 16 de Março (quinta-feira), às 13h30, sala 6.2.33

Paolo Gidoni (CMAF-CIO - Universidade de Lisboa)

Rate-independent systems and anisotropic dry friction.

Abstract: In this talk we discuss how the framework and techniques of rate-independent systems can be applied to models involving dry friction. After a quick overview on the motility of bio-inspired crawlers, we will focus on the genesis of an anisotropy in friction when the interaction is mediated by bristle-like elements. We illustrate a convergence result, showing the rate-independent nature of the limit of a family of systems characterized by a vanishing viscosity and a wiggly perturbation in the energy, scaling to zero. We then apply the result to some simple mechanical models, that exemplify the interaction of a bristle with a surface having small fluctuations, and discuss the effect of the geometry and elasticity of the bristle on the friction coefficients.

 

Dia 23 de Fevereiro (quinta-feira), às 13h30, sala 6.2.33

Carlos N. Rautenberg (Humboldt Universität zu Berlin)
 
Shape optimization for Navier-Stokes with mixed boundary conditions

Abstract: A shape optimization problem of a pipe for the stationary Navier-Stokes equations in an industrial application is considered.  The Navier-Stokes system is provided with mixed boundary conditions: Non-homogeneous Dirichlet-type on one part of the boundary (inlet and walls) and “do nothing” on the remaining part (outlet).  The shape design of the duct is focused on two objectives: 1) To obtain an almost-uniform outflow, and 2) To reduce the total pressure drop of the flow.  The well-posedness of the optimization problem and existence of solutions to the Navier-Stokes system are studied, and a continuous approach to the derivation of the shape gradient is presented. Details on the derivation of the numerical descent scheme and its implementation are given together with several numerical tests.

 

Dia 9 de Fevereiro (quinta-feira), às 13h30, sala 6.2.33

Alberto Saldaña (CAMGSD - Instituto Superior Técnico)

On maximum principles for higher-order fractional Laplacians

Abstract: Maximum principles are one of the most powerful tools in the analysis of linear and nonlinear elliptic partial differential equations. In particular, they guarantee that a solution of an equation inherits the sign of the data of the problem, this can be used, for example, to show symmetry properties of solutions, regularity, a priori bounds, existence and nonexistence of solutions. It is well known that higher-order operators do not satisfy in general maximum principles.  In this talk we focus on positivity preserving properties for higher-order fractional powers of the Laplacian. We discuss counterexamples and some positive results.  The aim of this talk is to be introductory and as much self-contained as possible.

 

Dia 2 de Fevereiro (quinta-feira), às 14:30, sala 6.2.33

Patrícia Gonçalves (Mathematics Department, IST, University of Lisbon)

Phase transition for the heat equation with boundary conditions

Resumo/Abstract: In this talk I will present a toy model for the heat conduction, which consists of a stochastic dynamics in contact with stochastic reservoirs.

In this model, particles evolve on the set of sites {1, 2, …, N - 1} to which we call the bulk, according to the following dynamics. Each particle waits a random clock, which is exponentially distributed, and after a ring of the clock it jumps to one of its nearest neighbors with probability 1⁄2. At the reservoirs, particles can enter or leave the system at a rate which is slower with respect to the rates in the bulk. The parameter that rules the boundary rates is θ ∈ R.

Random Clocks

The main purpose of the talk is to analyse the macroscopic PDE’s governing the space-time evolution of the density of particles for each regime of θ and to discuss recent results for the case in which particles can give long jumps.

Joint work with C. Bernardin (U. Nice) and B.Oviedo (U. Nice).

 

Dia 2 de Fevereiro (quinta-feira), às 13h30, sala 6.2.33

Dietmar Hoemberg (Institut für Mathematik - Technische Universität Berlin)

Mathematical aspects of multi-frequency induction heating

Summary: Induction hardening is a modern method for the heat treatment of steel parts. A well directed heating by electromagnetic waves and subsequent quenching of the workpiece increases the hardness of the surface layer. The process is very fast and energy efficient and plays an important role in modern manufacturing facilities in many industrial application areas. Although the original process is quite old, recent years have seen an important progress due to a new technology which allows to work simultaneously with several frequencies in one induction coil.

In the first part of my talk, a model for multifrequency induction hardening of steel parts is presented. It consists of a system of partial differential equations including Maxwell's equations and the heat equation. We show that the coupled system admits a unique weak solution. In the second part of the talk I will discuss the numerical approximation of the problem. It turns out to be quite intricate since one has to cope with different time scales for heat diffusion and the Maxwell system. Moreover, owing to the skin effect only the boundary layers of the component are heated by induced eddy currents, hence we also have to consider different spatial scales. We present a numerical algorithm based on adaptive edge-finite elements for the Maxwell system, which allows to treat these difficulties. We show some 3D simulations and conclude with results of an experimental validation in an industrial setting.

 

Dia 26 de Janeiro (quinta-feira), às 13h30, sala 6.2.33

Simão Correia (CMAF-CIO - Faculdade de Ciências da Universidade de Lisboa)

Some new qualitative results on the nonlinear Schrödinger equation

Abstract: In this seminar, we shall consider the nonlinear Schrödinger equation on Rd,

i ut + Δu + λ |u|σ u = 0

with an initial condition at t = 0. This is already a classical equation, with a vast literature regarding the behaviour of the solutions to this problem.  We shall discuss two new subjects: the extension of the H1 local well-posedness result to a larger functional space; a concept of finite speed of propagation, which we shall call finite speed of disturbance. The first topic relies deeply on a simple functional transform, the plane wave transform. This transform is of independent mathematical interest, with connections to the Fourier transform, the Laplace transform and the convolution of functions. The second topic relies on a first integral of the equation and, using finite speed of disturbance, we shall prove new global well-posedness results.

 

Dia 19 de Janeiro (quinta-feira), às 13h30, sala 6.2.33

Riccardo Scala (CMAF-CIO - Faculdade de Ciências da Universidade de Lisboa)

Confinement of dislocations inside a crystal via gamma convergence

Abstract: We study screw dislocations in an isotropic crystal undergoing antiplane shear. In the framework of linear elasticity, by fixing a suitable boundary condition for the strain (prescribed non-vanishing boundary integral), we manage to confine the dislocations inside the material. More precisely, in the presence of an external strain with circulation equal to n times the lattice spacing, it is energetically convenient to have n distinct dislocations lying inside the crystal. The novelty of introducing a Dirichlet boundary condition for the tangential strain is crucial to the confinement: it is well known that, if Neumann boundary conditions are imposed, the dislocations tend to migrate to the boundary. The results are achieved using PDE techniques and Gamma-convergence theory, in the framework of the so-called core radius approach.

 

2016

 

Dia 15 de Dezembro (quinta-feira), às 14h30, sala 6.2.33

Maurizio Garrione (University of Milano Bicocca)

A notion of instability for a nonlinear beam modeling a suspension bridge

Abstract: We consider the non-linear beam equation

utt + uxxxx + f(u) = g(x, t),

complemented with Navier boundary conditions. Motivated by the famous Tacoma Narrows Bridge collapse in 1940, we introduce a suitable notion of instability wishing to embody a sudden energy transfer between two different modes of oscillation. We then show some experiments and numerically discuss the appearence of this kind of instability for the two nonlinearities f(u) = u3 and f(u) = max{u, 0}.

Joint work with F. Gazzola (Polytechnic of Milan)

 

Dia 15 de Dezembro (quinta-feira), às 13h30, sala 6.2.33

Philippe G. LeFloch (University of Paris 6 and CNRS)

Global Solutions of the Einstein-Klein-Gordon system

Abstract: I will discuss the global evolution problem for self-gravitating massive matter in the context of Einstein's theory and, more generally, of the f(R)-theory of gravity. In collaboration with Yue Ma (Xian), I have investigated the global existence problem for the Einstein-Klein-Gordon system and established that Minkowski spacetime is globally nonlinearly stable in presence of massive fields. The method proposed by Christodoulou and Klainerman and the more recent proof in wave gauge by Lindblad and Rodnianski only cover vacuum spacetimes or massless fields. Analyzing the time decay of massive waves requires a completely new approach, the Hyperboloidal Foliation Method, which is based on a foliation by asymptotically hyperboloidal hypersurfaces and on investigating the algebraic structure of the Einstein-Klein-Gordon system. Blog: https://philippelefloch.org/

 

5 de Dezembro, 13h30, sala 6.2.33

Edgard Pimentel (Universidade Federal de São Carlos)

A priori Sobolev regularity for fully nonlinear parabolic equations

Abstract: In this talk, we present sharp Sobolev estimates for (viscosity) solutions of fully nonlinear parabolic equations, under minimal, asymptotic, assumptions on the governing operator. Our argument unfolds by importing improved regularity from a limiting configuration - the recession function - along a path that touches our problem of interest. This machinery allows us, among other things, to impose conditions solely on the associated recession operator; from a heuristic viewpoint, integral regularity would be set by the behavior of the original operator at the infinity of S(d). We conclude the talk discussing further implications of our main result; these include a comment on the so-called Escauriaza's exponent in the parabolic setting, a priori regularity in p-BMO spaces, and applications to the Kähler-Ricci flow.

This is joint work with Ricardo Castillo (UFSCar).

 

27 de Outubro, 13h30, sala 6.2.33

Nikolay Tzvetkov (Univ. de Cergy-Pontoise)

Quasi-invariant gaussian measures for the nonlinear wave equation

Abstract:

We will show that a natural class of gaussian measures living on Sobolev spaces of varying regularity are quasi-invariant under the flow of the two dimensional cubic defocusing wave equation. For that purpose, we introduce renormalised energies and we establish the associated energy estimates.

This is a joint work with Tadahiro Oh (Edinburg University).

 

29 de Setembro, 14H30, sala 6.2.33

Darya Apushkinskaya (Saarland University, Saarbrücken, Germany)

When the Boundary Point Principle does not hold?

Abstract

 

29 de Setembro, 13H30, sala 6.2.33

Nazarov A. I.  (St. Petersburg Dept of Steklov Institute and St. Petersburg State University)

Dirichlet and Neumann problems for parabolic non-divergence equations with main coefficients measurable in time

Abstract

 

15 de Setembro 13h30, na sala 6.2.33

Lisa Santos (Universidade do Minho)

Problems with the operator $\nabla\times(\alpha\nabla\times\cdot)$

Abstract

 

12 de Setembro 13h30, sala 6.2.33

Antonio Iannizzotto (Università degli Studi di Cagliari, Italy)

Sign-changing Solutions of the Fractional Heat Equation

Abstract

 

21 de Julho, 13:30, sala 6.2.33

Eric Carlen (Rutgers University)

Fermion hypercontractivity and quantum convolution inequalities

Abstract: The fermionic oscillator semigroup is a natural quantum analog of the classical Mehler semigroup, which is the semigroup generated by the bosonic number operator in its standard representation as an opertator on functions on Euclidean space with a Gaussian reference measure. The Mehler semigroup plays an important role in the proof of important inequalities that govern classical information theory.  There is a very close analogy between the classical Mehler semigroup and its fermionic analog which was borne out in the proof of Gross's conjecture that that the fermionic semigroup should have the same optimal hypercontractivity properties as its classical cousin. The optimal fermion hypercontractivity inequality can be viewed as a quantum convolution inequality. We present some recent results developing this perspective, which are relevant to questions concerning the entropy power inequality in quantum information theory, and which  are joint work with Elliott Lieb and Jan Maas.

 

19 de Julho, 13h30, sala 6.2.38

Amit Einav (University of Cambridge)

Recent Advances in Kinetic Theory and their connection to Nonlinear PDEs, Functional Inequalities and Applied Probability

Abstract: Kinetic theory is the field of mathematics that deals with systems of many objects. In recent years, this field has seen an awakening and renewed interest, and has been the focus of attention of many prominent mathematicians. In this talk we will discuss recent advances in the field, mainly in relation to the Boltzmann-Nordheim equation and Kac’s Model, and tie them to the fields of Nonlinear PDEs, Functional Inequalities and Applied Probability.

 

14 de julho 13:30, sala 6.2.33

Elisa Sovrano (Universidade de Udine)

Remarks on the Ambrosetti-Prodi periodic problem

 

14 de julho 15:00, sala 6.2.33

Raquel Filipe (estudante 2º ciclo)

What if they swim?

Abstract: Over the last years, there has been a sustained interest in the collective dynamics of micro-organisms. These organisms are able to convert energy from the surrounding environment into directional movement. They are therefore called active particles, in contrast to the passive particles (e.g., colloids) which erratic motion (Brownian) in solution only results from multiple collisions with atoms and/or molecules of the surrounding medium. Given the lack of ability to describe deterministically the interaction of these particles with their surrounding medium, their individual dynamics is typically described by phenomenological models where the equations of motion are described by stochastic differential equations. It will be discussed some differences and similarities between the dynamics of passive and active particles. It will be presented some numerical results obtained by integrating the equations of motion of each type of particles. It will also be discussed some interesting applications.

 

7 de julho, 13:30, sala 6.2.33

Samuel Amstutz (Université d'Avignon)

Topology optimization and minimal partitions using a gradient-free perimeter approximation

Abstract: I will present an original functional dedicated to the Gamma-convergence approximation of the relative perimeter of a set. Compared to standard perimeter approximations it has mainly two specific features: on one hand it does not involve the density gradient, hence it can be directly applied to characteristic functions, and on the other hand it can be formulated as the minimum of an auxiliary unconstrained quadratic problem, allowing the use of rather efficient alternating minimizations algorithms. 
I will describe the main mathematical properties of this functional and show several examples of applications in topology optimization, where it is incorporated in homogenization or topological gradient methods. Then, I will present natural extensions to minimal partition problems with applications in image classification and deblurring. Finally, I will discuss recent developments towards more general interface energies.

 

7 de julho, 15:00, sala 6.2.33

Manuel Zamora (Universidad del Bío-Bío, Chile)

Periodic solutions to second order indefinite singular equations

 

30 de junho 13:30, sala 6.2.33

Jorge Fragoso (estudante, 2º Ciclo)

Uma criterização intrínseca para a bijectividade de operadores de Hilbert relacionados com os sistemas de Friedrichs

Abstract: Neste seminário do Mestrado de Matemática serão expostos resultados recentes, de A. Ern, J-L. Guermond e G. Caplain <Comm. P. D.E. 32 (2007). 317-341> sobre uma nova abordagem Hibertiana à teoria de Friedrich para os sistemas simétricos positivos que visa caracterizar as condições de fronteira admissíveis. Inclui-se a apresentação de aplicações a problemas clássicos nos limites para equações com derivadas parciais.

 

2 de Junho 13h30, sala 6.2.33

Elvira Zappale (Universidade de Salerno)

Equilibrium and Euler-Lagrange equation for hyperelastic materials

Abstract: By means of duality we prove existence and uniqueness of equilibrium for energies described by integral functionals which fail to be convex. This analysis is motivated by some physical models of elastic materials (cf. for istance [2, 4]) and the techniques generalize the methods first introduced in [5, 1]. A suitable Euler Lagrange equation characterizing the minimizers is derived.

 

Joint work with G. Carita and G. Pisante

 

References:

[1] AWI, R. & GANGBO, W., A polyconvex integrand; Euler-Lagrange Equations and Uniqueness of Equilibrium ARMA 214 (2014), no. 1, 143–182.

[2] BEATTY, M., Topics in finite elasticity: hyperelasticity of rubber materials, elastomers and biological tissues, with examples, Appl. Mech. Rv. Vol 40, no. 12 (1987) 1700-1734.

[3] CARITA G., PISANTE G., & ZAPPALE E. In preparation.

[4] CONTI S.,& DOLZMANN G. On the Theory of Relaxation in Nonlinear Elasticity with Constraints on the Determinant, ARMA, (2014), doi:10.1007/s00205-014-0835-9.

[5] GANGBO, W. & VAN DER PUTTEN, R. Uniqueness of equilibrium configurations in solid crystals. SIAM Journal on Math. Anal. 32, (3) (2000) 465-492.

 

7 de abril, 14:15,  sala 6.2.38

E. Zaouche

(Ecole Normale Supérieure, Algerie)

Lp -Continuity of Solutions to Parabolic Free Boundary Problems

Abstract: We consider a class of parabolic free boundary problems. We establish some properties of the solutions, including L-infinity -regularity in time and a monotonicity property, from which we deduce strong Lp-continuity in t

 

7 de abril, 13:30  sala 6.2.38

Riccardo Scala
(Weierstrass Institute (WIAS), Berlin)

Analytic and geometric properties of functions with dislocations singularity

Abstract: We study the nature of the singularities of the strain fields due to the  presence of dislocations in a crystal. We prove and collect some measure  theoretic properties of such singularities. Among them, we give the  explicit description of the boundary of the graph defined by deformation  fields, which, in the presence of dislocations, are well-defined as  torus-valued maps. Using such description we are able to deal and solve  some variational problems involving such maps, as, for instance,  problems of minimization of energies depending on  the elastic strain  and the dislocation density as well.

 

16 de Março, 11H00, sala 6.2.33

Vladimir I. Man'ko (Lebedev Physical Institute, Moscow)

Contractions and deformations: Quantum-to-classical and classical-to-quantum

Abstract:

The construction of structure constants of associative products of functions in terms of pairs of operator families acting in a Hilbert space and called quantizer and dequantizer operators, respectively, is demonstrated. The Moyal product of Weyl symbols of the operators, which are the functions in the phase space, is reconsidered in terms of the presented construction. The invertible map of the quantum-state density operators onto the probability distributions (tomographic map) expressed in terms of the tomographic quantizer-dequantizer is given, and the kernel of the associative products of the operator tomographic symbols is obtained in an explicit form. The procedure for obtaining new kernels of star-products based on nonlinear transforms of the fiducial kernel is developed, and the example of Gronewald kernel of Moyal associative product of functions in the phase space is considered as another quantization version. The large mass limit of Moyal product is discussed and hybrid quantum-classical description of light and heavy particle systems is formulated. The quantum-to-classical transition is discussed.

 

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17 de Março, 13H30, sala 6.2.33

Margarita A. Man'ko and Vladimir I. Man'ko (Lebedev Physical Institute, Moscow)

Correlations in qudits as a resource for quantum technologies

Abstract:

New entropic inequalities are discussed for single qudit systems (spin states). In particular, the subadditivity condition known for bipartite quantum systems and the strong subadditivity condition known for tripartite quantum systems are shown to exist for the states of noncomposite quantum systems. A new entropy -energy uncertainty relation with a bound determined by the partition function is discussed for an arbitrary set of qudits (set of spin particles). Bell inequality and its violation known, e.g., for two qubits (two spin-1/2 particles) are shown to exist for a single spin j=3/2 particle. Discussion of quantum correlations (hidden correlations) is presented. Application to artificial atom states in superconducting circuits based on Josephson junctions is considered. The hidden quantum correlations are discussed for possible applications in quantum technologies.

 

M.A. Man'ko and V.I. Man'ko, Entropy, Vol. 17, p. 2876 (2015).

M.A. Man'ko and V.I. Man'ko, Journal of Russian Laser Research, Vol. 37, p. 1 (2016).

 

10 de dezembro, 13:30, sala 6.2.33

Simão Correia

CMAF-CIO

The Hyperbolic Nonlinear Schrödinger Equation

 

26 de Novembro, 13h30, sala 6.2.33

Alain Haraux

Laboratoire J L Lions, Université Pierre et Marie Curie

Decay rates of the solutions to some second order evolution problems

Abstract:

We report on recent results on decay estimates of solutions to some evolution equations of the general form
u´´ (t) + u´(t) + Au(t) + f(u(t)) = 0
where H is a real Hilbert space, A is a nonnegative self-adjoint linear operator on H with dense domain, and f is a nonlinearity tangent to 0 at the origin.
 

20 de Novembro, 15h00, sala 6.2.33

 

Pedro Areias

Universidade de Évora

Phase-field analysis of finite-strain plates and shells including element subdivision
 

Abstract: With the theme of fracture of finite-strain plates and shells based on a phase-field model of crack regularization, we introduce a new staggered algorithm for elastic and elasto-plastic materials. To account for correct fracture behavior in bending, two independent phase-fields are used, corresponding to the lower and upper faces of the shell. This allows realistic behavior in bending-dominated problems and is sharply shown in classical beam and plate problems. Finite strain behavior for both elastic and elasto-plastic constitutive laws is made compatible with the phase-field model by use of a consistent updated-Lagrangian algorithm. To guarantee sufficient resolution in the definition of the crack paths, a local remeshing algorithm based on the phase-field values at the lower and upper shell faces is introduced. In this local remeshing algorithm, two stages are used: edge-based element subdivision and node repositioning. Five representative numerical examples are shown, consisting of a bi-clamped beam, two versions of a square plate, the Keesecker pressurized cylinder problem, the Hexcan problem and the Muscat-Fenech and Atkins plate.

 

 

5 de novembro, 16:30, sala 6.2.44

Anca-Maria Toader

CMAFCIO

Identification of material and shape based on eigenvalues and traces of eigenmodes​

 

Abstract:
We present an inverse method, both analytical and numerical for detecting the shape of unknown holes inside an elastic body. Free vibrations of this body are considered. Physical measurements on the eigenvalues and on the traces of the eigenmodes on a certain part of the exterior boundary are used as input data in the process of recovering the shape(s) of the hole(s).

 

5 de novembro, 13:30, sala 6.2.33

 

Rémi Carles CNRS

Université Montpellier

On semiclassical limit of nonlinear quantum scattering
 

Abstract: We consider the cubic Schrödinger equation with a short-range external potential, in a semi-classical scaling. We show that for fixed Planck constant, a complete scattering theory is available, showing that both the potential and the nonlinearity are asymptotically negligible for large time. Then, for data under the form of coherent state, we show that a scattering theory is also available for the approximate envelope of the propagated coherent state, which is given by a nonlinear equation. In the semi-classical limit, these two scattering operators can be compared in terms of classical scattering theory, thanks to a uniform in time error estimate. We infer a large time decoupling phenomenon in the case of finitely many initial coherent states.

 

15 de Outubro,  16:30, sala 6.2.44

Cristian Barbarosie

CMAFCIO

Teoria da homonegeização e tratamento numérico de EDPs em domínios periódicos

Resumo : A teoria da homogeneização dá um enquadramento rigoroso à descrição de propriedades mecânicas de corpos com estrutura microscópica, como materiais compósitos ou cristais. Neste seminário será apresentada uma breve introdução à teoria da homogeneização e será discutida a abordagem numérica de problemas associados, nomeadamente resolução numérica de equações com derivadas parciais sujeitas a condições de periodicidade. Será descrito o código existente, em 2D, e serão apontadas as possibilidades de generalização para três dimensões.

 

5 de out 15:45 sala 6.2.33

Estabilidade global num modelo de doenças infecciosas com compartimentos

Luís Sampaio (bolseiro Incentivo/MAT/UI0209/2014)

Resumo: O cálculo de soluções para sistemas de EDOs não lineares é frequentemente muito difícil. No entanto, há métodos sistemáticos para estudar propriedades como a estabilidade dos equilíbrios do sistema, de forma a obter informação sobre o comportamento das suas soluções. Num problema de doenças infecciosas o estudo da estabilidade assume especial importância, nomeadamente perceber se o sistema fica livre de doença ou se esta se torna endémica. Nesta exposição será apresentado um modelo com compartimentos, o qual será estudado com métodos de estabilidade de Lyapunov e persistência.

 

6 de out 15:45 sala 6.2.33

Uma condição de Landesman-Lazer no problema periódico para a equação de Liénard

Bernardo Fernandes (bolseiro Incentivo/MAT/UI0209/2014)

Resumo: Nesta exposição mostrar-se-á como um conceito criado por E.M Landesman e Alan C. Lazer se aplica a problemas envolvendo certos tipos de E.D.Os, com o objectivo de obter existência de soluções periódicas.  Pretende-se dar uma motivação para as condições utilizadas, e ao mesmo tempo destacar a importância do uso de Teoremas de Ponto Fixo neste tipo de problemas.

 

1 de Outubro, 13h30sala 6.2.33

Riccardo Scala

Universitá di Pavia

A weak formulation for a dynamic contact process with unilateral constraint

Abstract:
We study a rate-independent delamination process of an interface between two elastic bodies. The evolution of the displacement u, whose constitutive equation is an hyperbolic PDE (wave-type), interacts with the adhesive broking its macromolecular links and weakening its effect. The process is supplemented with a unilateral constraint on the jump of the displacement at the interface, in order to avoid interpenetration of matter. The constraint arises in the wave equation for u as a term which might provide instantaneous reactions, and thus it is not a Sobolev function but turns out to be a non regular distribution (actually, a measure concentrated on negligible subset of the time-space). This requires a very weak formulation of the problem, of which we prove existence of solutions.

 

30 de setembro, 15:00, sala 6.2.33 (seminário conjunto com o Grupo de Física Matemática)

Christian Léonard 

Université Paris Ouest

When do random walks share the same bridges?

Abstract:
The natural analogue of Hamilton's least action principle in presence of randomness is the generalized Schrödinger problem where the Lagrangian action is replaced by the relative entropy with respect to some reference path measure. The role of the action minimizing paths between two prescribed endpoints  is played by the bridges of the reference path measure and the solutions of Schrödinger problem are mixtures of these bridges. Therefore, the family of all the bridges of the reference measure encodes its whole "Lagrangian dynamics" and searching for a criterion for two path measures to solve the same Schrödinger problem, i.e. to be driven by the same source of randomness and the same force field, amounts to answer the question of our title. The answer is given in the special case of random walks. This is a joint work with Giovanni Conforti.

 

 

17 september 13:30 Room 6.2.38

On some fourth order problems associated to Engineering

Sanjiban Santra

CIMAT, Guanajato, Mexico

Abstract: We consider a fourth order traveling wave equation associated to the Suspension Bridge Problem (SBP). This equations are modeled by the traveling wave behavior on the Narrows Tacoma and the Golden Gate bridge. We prove existence of homoclinic solutions when the wave speed is small. We will also discuss the associated  fourth order Liouville theorem to the problem and possible link with the De Giorgi's conjecture.  This is an attempt to prove the McKenna-Walter conjecture which is open for the last two decades.

 

 

Sala 6.2.38 do C6, 23 de julho, 13:30

 

Nodal solutions for supercritical Laplace equations

Francesca Dalbono 

Università degli Studi di Palermo
 

Abstract

 

 

Sala B3-01 do Instituto para a Investigação Interdisciplinar, 16 de Julho

 

Surface diffusion and mean curvature motion

A. Novick-Cohen (Technion IIT)

Mean curvature motion as well as surface diffusion constitute geometric interfacial motions which have received considerable attention. However in many applications a complex combination of coupled surfaces appear whose evolution may be described by coupling these two types of motion. In my lecture, a variety of physical problems will be described which may be reasonably
modeled by such motions. While some these problems appear to require an anisotropic formulation, often an isotropic formulation is helpful to consider. A panoply of analytic and numerical results will be presented, in addition to some supporting experimental evidence.

 

Sala 6.2.38, 9 de Julho, 13:30 

La evolución del soporte en operadores con limitador de flujo

Juan Campos (Univ. Granada). 

 

2 julho 13:30 - Sala B3-01 do Instituto para a Investigação Interdisciplinar

Pierre Vuillermot

UMR-CNRS 7502, Institut Élie Cartan de Lorraine, Nancy

Parabolic problems and Bernstein processes: some recent results

 

Abstract: In this talk I will discuss some recent results regarding the connections that exist between certain linear parabolic partial differential equations and Bernstein stochastic processes. In particular, I will show how to reconstruct certain stationary and non-stationary Ornstein-Uhlenbeck processes from a PDE point of view, as well as certain well-known and not so well-known Markovian bridges. I will also introduce a new class of stationary, non-Markovian processes which I will relate to the so-called periodic Ornstein-Uhlenbeck process, and touch upon some potential applications to statistical mechanics,
the random evolution of loops and general pattern theory.

       

 

 

25 jun às 13:30 - Sala B3-01 do Instituto para a Investigação Interdisciplinar

Paulo Amorim
Instituto de Matemática, Universidade Federal do Rio de Janeiro

A Continuous Model of Ant Foraging with Pheromones and Trail Formation

Resumo:
We propose and analyze a PDE model of ant foraging behavior. Ant foraging is among the most interesting behaviors in the animal kingdom, and a prime example of individuals following simple behavioral rules based on local information producing complex, organized and ``intelligent' strategies at the population level. One of its main aspects is the widespread use of pheromones, which are chemical compounds laid by the ants used to attract other ants to a food source. We consider a continuous description of a population of ants and simulate numerically the foraging behavior using a system of PDEs of chemotaxis type. We show that, numerically, this system accurately reproduces observed foraging behavior, such as trail formation, optimization of routes, and efficient removal of food sources. Furthermore, in collaboration with R. Alonso (PUC-RJ) and Th. Goudon (INRIA-Nice) we present a mathematical analysis of a simplified version of the model. The talk is accessible to students and anyone interested in applications of Mathematics.

 

 

28/05/2015 13.30h sala B3-01 Instituto para a Investigação Interdisciplinar

 

Jose Angel Cid

Departamento de Matematicas, Universidad de Vigo,

Pabellon 3, Campus de Ourense, 32004.

 

Recent contributions to a periodic boundary value problem displaying the pumping efect

 

19/05/2015 14h  sala A2-25 Instituto para a Investigação Interdisciplinar

AMIT EINAV (Department of Pure Mathematics and Mathematical Statistics, Cambridge)

On the Boltzmann-Nordheim Equation for Bosonic Gas.

 

Abstract: One of the most influential equations in the kinetic theory of gases is the so-called Boltzmann equation, describing the time evolution of the probability density of a particle in a classical dilute gas. The irrefutable appearance of Quantum Mechanics, however, required a modification to this celebrated kinetic equation, resulting in the Boltzmann-Nordheim equation. 

In this presentation we will discuss a newly found local Cauchy Theory for a general solution to the spatially homogeneous bosonic Boltzmann-Nordheim equation in any dimension d \geq 3. Interestingly enough, the locality of this result is quite sharp due to the so-called Bose-Einstein condensation. 

The methods used to achieve this theory are similar to those available for the classical Boltzmann equation, yet are entangled with L^infty control that dominates the difference between the classical and quantum kinetic equation.

Time permitting we will discuss some details about the existence of a global solution to the equation.

This is a joint work with Marc Briant.

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09/04/2015 (quinta-feira), às 13:30h, na Sala B3-01 (Inst Inv Interdisciplinar)

E. Zaouche (École Normale Supérieure, 16050 Kouba, Algiers, Algeria)

On the Existence of a Solution of a Class of Non-Stationary Free Boundary Problems

Abstract: We establish existence of a solution for a class of parabolic free boundary problems including the evolutionary dam problem. We use a regularized problem for which we prove existence of a solution by applying the Tychonoff fixed point theorem. Then we pass to the limit to get a solution of our problem. We also give a regularity result of the solutions. Keywords: Free boundary problems, existence, regularity.