Seminario de Ecuaciones
Diferenciales

Resumen. In this talk, I will discuss vortex dynamics in the planar Euler equations, focusing on two key aspects. First, I will present a rigorous derivation of leapfrogging quartets of concentrated vortex patches near singular time-periodic relative equilibria of the point vortex system, using KAM theory. In the second part, I will show how to extend these techniques to desingularize time-periodic vortex orbits when the Euler equation is set in a generic bounded simply connected domain. Specifically, we can prove that for a single point vortex, under certain non-degeneracy conditions, it is possible to desingularize most of these trajectories into time-periodic concentrated vortex patches.

Resumen. Let $\Omega$ be a bounded domain in $\mathbb{R}^n$ whose boundary is $C^{1, \gamma}$ for $\gamma \in (0, 1)$. Consider 2D Euler equation for the vorticity or the $n$-D aggregation equation in the case of the initial condition being a positive multiple of the characteristic function of $\Omega$. In this talk we discuss on global in time analyticity of the flow generated by the velocity field which propagates the vorticity or density solution respectively. These results are obtained from a detailed study of the Beurling or Riesz transform, that represents derivatives of the velocity field. The precise estimates obtained for the solutions of an equation satisfied by the Lagrangian flow, are a key point in the development.

Resumen. Establecemos condiciones necesarias para la existencia de soluciones periódicas y subarmónicas de una ecuación diferencial que describe el movimiento de una perla sobre un aro circular que gira sujeto a una velocidad angular constante ω y un forzamiento periódico. Nuestro enfoque implica utilizando métodos de sub y super soluciones y algunas técnicas presentadas por Zanolin-Boscaggin y Ureña. Para más información, haz click aquí.

Resumen. Nonlocal contact problem for two-dimensional linear elliptic equations is stated and investigated. The method of separation of variables is used to find the solution of a stated problem in the case of Poisson’s equation. Then, the more general problem with nonlocal multipoint contact conditions for elliptic equation with variable coefficients is considered, and the iterative method to solve the problem numerically is constructed and investigated. The uniqueness and existence of the regular solution are proved. The iterative method allows reducing the solution of a nonlocal contact problem to the solution of a sequence of classical boundary value problems. The numerical experiment is conducted. The results fully agree with the theoretical conclusions and show the efficiency of the proposed iterative procedure.

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Resumen. The Stokes-transport system describes the evolution of a density scalar advected by a velocity field that satisfies the Stokes equation with gravitational forcing. This system of PDEs belongs to the class of active scalar equations and models the dynamics of highly viscous fluids with heterogeneous density. In this talk, we will combine a Lagrangian approach with harmonic analysis techniques to find global solutions for this system in 2D, with a focus on the case where the density takes two distinct constant values. This leads to a two-phase problem that models the evolution of two different immiscible fluids in a 2D domain.

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Resumen. We consider the Lane-Emden problem on a smooth bounded planar domain. We find nodal concentrating solutions, for large values of p, where both the positive and negative part blow up, the latter with a non-radial profile. Up to our knowledge, this is the first result concerning concentration with such a pattern for planar elliptic problems. Joint work with I. Ianni and A. Pistoia (Rome Sapienza).

Resumen. The analysis of differential equations using Lie symmetries is an efficient method that has been used extensively in the literature, especially for nonlinear equations where the systematic tools available for analytical solutions are limited. The study of symmetry groups of differential equations reveals the symmetries of the equations originating from the physical nature of the problem. Determining the symmetry groups and algebras that leave the equations invariant, finding group-invariant exact analytical solutions, and classifying families of equations according to their symmetry algebras are some of the approaches used in this field. In this talk I will present results on Lie symmetry algebras of several equations of mathematical physics, including multidimensional Benney-Roskes system, the Rosenau equation and a higher order Boussinesq equation. Besides the invariance algebras I will mention several exact solutions, some of which have physical importance, like line solitons and lump solutions.

References:

[1] Gönül, Ş., Özemir, C. (2022). Benney–Roskes/Zakharov–Rubenchik system: Lie symmetries and exact solutions. The European Physical Journal Plus, 137(10), 1107.

[2] Gönül, Ş., Özemir, C. (2022). Lie Symmetries and traveling wave solutions of the 3D Benney–Roskes/Zakharov–Rubenchik system. Chaos, Solitons & Fractals, 165, 112807.

[3] Demirci, A., Hasanoğlu, Y., Muslu, G. M., Özemir, C. (2022). On the Rosenau equation: Lie symmetries, periodic solutions and solitary wave dynamics. Wave Motion, 109, 102848.

[4] Hasanoğlu, Y., Özemir, C. (2021). Group classification and exact solutions of a higher-order Boussinesq equation. Nonlinear Dynamics, 104(3), 2599-2611.

Resumen. We prove the existence and uniqueness of Yudovich-type solutions to the Euler equations for a two-dimensional incompressible fluid coupled with the full set of Maxwell’s equations for electromagnetism. This is shown to hold under the condition that the speed of light is sufficiently large (compared to the velocity of the plasma, loosely speaking). Moreover, due to a refined analysis of the dispersive (–dissipative) properties of Maxwell’s equations, involving suitable high–low–frequency cutoff (in terms of the speed of light), the bounds on the solution are shown to be uniform with respect to the speed of light. This matter of fact allows us to derive (a simple version of) the MHD system as the speed of light tends to infinity. In our setting, the convergence of the solution in strong topologies comes with a rate and it relies on a robust stability mechanism of perturbations of the Euler equations in Yudovich’s class combined with a fine study of the behavior (evanescence) of the electric field in the non-relativistic regime. This is based on joint work with Diogo Arsenio from NYUAD.

Resumen. In this talk I will explain a new numerical framework, employing machine learning techniques (physics informed neural networks), to find a smooth self-similar solution (or asymptotically self-similar solution) for different equations in fluid dynamics, such as Euler or Boussinesq. The new numerical framework is shown to be both robust and readily adaptable to several situations. Joint work with Tristan Buckmaster, Gonzalo Cao-Labora, Ching-Yao Lai and Yongji Wang.

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Resumen. We introduce an algebraic method designed to identify a sequence of functions that exponentially converge towards the solution of a second-order ordinary differential equation (ODE) within a naturally defined distance metric. This method is exemplified through its application to various scenarios involving both Boundary Value Problems (BVP) and Initial Value Problems (IVP). By exploring its properties, advantages, limitations, and suggesting practical algorithmic enhancements, we will provide a comprehensive overview of the approach.

Resumen. Cell-cell adhesion plays a vital role in the development and maintenance of multicellular organisms. In cancer, one of its functions is the regulation of cell migration. In this talk, a versatile multiscale approach to modelling a moving self-adhesive cell population will be presented that combines a careful microscopic description of a deterministic adhesion-driven motion component with an efficient mesoscopic representation of a stochastic velocity-jump process. This approach gives rise to mesoscopic models in the form of kinetic transport equations featuring multiple non-localities. Subsequent upscaling produces general classes of equations with non-local adhesion and myopic diffusion. Our simulations show how the combination of these two motion effects can unfold. Cell-cell adhesion relies on binding of the cell adhesion molecules, such as, e.g. cadherins. Our approach lends itself conveniently to capturing this microscopic effect. On the macroscale, this results in an additional non-linear integral equation of a novel type that is coupled to the cell density equation.

Resumen. In this talk we will introduce two new fractional operators. The first one is based on the classical formula that writes the usual Laplacian as the sum of the eigenvalues of the Hessian. The second one comes from looking at the classical fractional Laplacian as the mean value (in the sphere) of the 1-dimensional fractional Laplacians in lines with directions in the sphere. To obtain this second new fractional operator we just replace the mean value by the mid-range of 1-dimensional fractional Laplacians with directions in the sphere. For these two new fractional operators we prove a comparison principle for viscosity sub and supersolutions and then we obtain existence and uniqueness for the Dirichlet problem. Finally, we prove that for the first operator we recover the classical Laplacian in the limit as $s$ goes to 1. Joint work with Jorge Ruiz-Cases (UAM, Madrid).

Resumen. We prove some multiplicity results for Neumann-type boundary value problems associated with a Hamiltonian system. Such a system can be seen as the weak coupling of two systems, the first of which has some periodicity properties in the Hamiltonian function, the second one presenting the existence of a well-ordered pair of lower/upper solutions.

Resumen. The Muskat problem models the movement of two immiscible and incompressible fluids in a porous medium. The problem can be reduced to an evolution equation for the free interface, that satisfies a quasilinear and nonlocal partial differential equation. We will first review the main features of the problem and describe previous results, and we will conclude with two recent results where the regularity of the interface is critical with respect to the natural scaling of the equation: a rising bubble and an interface with corners.

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Resumen. In this work we consider a time-periodic and random version of the Arnold-Beltrami-Childress (ABC) flow. We are concerned with two main subjects. On the one hand, we study the mixing problem of a passive tracer in the three dimensional torus by the action of the ABC vector field. On the other hand, we examine the effect of the ABC flows on the growth of a magnetic field described by the kinematic dynamo equations. To investigate these questions, we analyse the ABC flow as a random dynamical system and examine the ergodic properties of some associated Markov chains. This work settles that the random ABC vector field is an example of a space-time smooth universal exponential mixer in the three dimensional torus, and moreover, that it is an example of a nondissipative kinematic fast dynamo. This is a joint work with M. Coti Zelati (Imperial College London).

Resumen. In this talk, we will first introduce some recent results about patch solutions for active scaler equation, including two dimensional Euler equation. Then we will present how to establish the existence of simply connected V-states (rotating patches) close to Rankine vortices for active scalar equations with completely monotone kernels. This result allows to unify various results on this topic related to geophysical flows. A key ingredient is a new factorization formula for the spectrum using a universal function which is independent of the model. This function admits sesveral interesting properties allowing to track the spectrum distribution. This talk is based on a joint work with Taoufik Hmidi and Liutang Xue.

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Resumen. We are interested in qualitative properties of solutions of the semilinear elliptic problems with zero Dirichlet boundary conditions in bounded and smooth domains. We want to focus on the relations between the shape of the solutions and the one of the domain. In particular we investigate the number of critical points of the solutions with respect to the convexity of the domain. Both the cases of positive and sign-changing solutions are treated. Works in collaboration with L. Battaglia (Università di Roma Tre) and M. Grossi (Università La Sapienza).

Resumen. We are concerned with the spectrum of a linearized Liouville-type problem. Among other things, we characterize the case in which the first eigenvalue is zero. To this end, we refine the Alexandrov-Bol inequality suitable for our problem and characterize its equality case. As a consequence, we obtain new pointwise information on the first eigenfunction associated to the zero eigenvalue.

Resumen. Cancer is a very complex phenomenon that involves many different scales and situations. In this talk, we consider two principal models describing the evolution of a tumor colony and we derive a new asymptotic model for tumor growth. We discuss both the case with free boundary and infinite depth, and the case with two free boundaries. We focus on the case of a single phase tumor colony taking into account chemotactic effects in an early stage where there is no necrotic inner region. Thus, our models are valid for the case of multilayer avascular tumors with very little access to both nutrients and inhibitors or the case where the amount of nutrients and inhibitors is very similar to the amount consumed by the multilayer tumor cells. Besides deriving the model, we also prove a well-posedness result in a particular case and discuss some ongoing projects. This talk is based on joint works with R. Granero-Belinchón and Alejandro Ortega.

Resumen. Systems which couple a Schrödinger and a Poisson equation arise in several physical contexts such as quantum mechanics and electromagnetism. When considered in the whole space and in the limiting case for the Sobolev embeddings, one can deal with exponentially growing nonlinearities, but the application of standard variational tools is spoiled by the fact that the Riesz kernel of the Poisson equation is logarithmic, hence unbounded from above and below. In this framework, I will present some existence results about local SP-systems in the case of zero mass, and nonlocal SP-systems, by means of a variational approximating procedure for auxiliary Choquard equations, where the logarithmic Riesz kernel is uniformly approximated by polynomial kernels. This talk is based on joint works with Daniele Cassani (University of Insubria - Varese) and Zhisu Liu (China University of Geosciences - Wuhan)

Resumen. In this talk I will review several results in the modelling of grid cells. The activity generated by an ensemble of neurons is affected by various noise sources. It is a well-recognised challenge to understand the effects of noise on the stability of such networks. We demonstrate that the patterns of activity generated by networks of grid cells emerge from the instability of homogeneous activity for small levels of noise. This is carried out by upscaling a noisy grid cell model to a system of partial differential equations in order to analyse the robustness of network activity patterns with respect to noise. This is rigorously achieved by mean-field type arguments. Inhomogeneous network patterns are numerically understood as branches bifurcating from unstable homogeneous states for small noise levels. We prove that there is a phase transition occurring as the level of noise decreases. Our numerical study also indicates the presence of hysteresis phenomena close to the precise critical noise value. This talk is a summary of four papers/preprints in collaboration with A. Clini, H. Holden, P. Roux and S. Solem.

Resumen. We look at a cross-diffusion system consisting of two Fokker-Planck equations where the gradient of the density for each species acts as a potential for the other one. It is a degenerate system in the sense that it loses its parabolic behavior on some part of the domain. However, the system is also the gradient flow for the Wasserstein distance of a functional which is not lower semicontinuous. We then compute the convexification of the integrand (thus the lower semi continuous envelope of the functional) and provide an existence result in a suitable sense for the gradient flow of the corresponding relaxed functional. This is a joint work with R. Ducasse (Paris) and F. Santambrogio (Lyon).

Resumen. En esta charla abordaremos algunos resultados matemáticos para fluidos no-newtonianos donde el tensor de viscosidad es impar. Pese a que dichos fluidos aparecen en diversas aplicaciones físicas ya desde los años 90, la literatura matemática sobre ellos es escasa. En particular, nosotros presentaremos un resultado de well-posedness así como un modelo asintótico que captura la principal dinámica para el caso de frontera libre.

Resumen. We introduce and study a new model for the progression of Alzheimer's disease incorporating the interactions of Aβ-monomers, oligomers, microglial cells and interleukins with neurons through different mechanisms such as protein polymerization, inflammation processes and neural stress reactions. In order to understand the complete interactions between these elements, we study a spatially-homogeneous simplified model that allows to determine the effect of key parameters such as degradation rates in the asymptotic behavior of the system and the stability of equilibriums. We observe that inflammation appears to be a crucial factor in the initiation and progression of Alzheimer's disease through a phenomenon of hysteresis, which means that there exists a critical threshold of initial concentration of interleukins that determines if the disease persists or not in the long term. These results give perspectives on possible anti-inflammatory treatments that could be applied to mitigate the progression of Alzheimer's disease. We also present numerical simulations that allow to observe the effect of initial inflammation and concentration of monomers in our model.

Resumen. In this talk, we are interested in the well-posedness of certain systems of PDEs arising in models of fluid mechanics and which present a hyperbolic structure. After a short overview of previous results available for the incompressible Euler equations, both in its homogeneous and non-homogeneous versions, we focus on the special case of a system which describes the dynamics of an incompressible fluid having variable density and presenting non-dissipative viscosity effects. Examples of such fluids arise both in quantum and classical hydrodynamics. At the level of the mathematical model, the non-dissipative nature of the viscosity is encoded by an odd term, dubbed precisely odd viscosity tensor. As the odd viscosity term involves higher order space derivatives of the velocity field and of the density, it is responsible for an apparent loss of regularity in the classical a priori estimates. In this talk, we will show how to circumvent such a loss of derivatives by introducing a set of good unknowns, which allow to highlight a hyperbolic structure underlying the system of equations. As a consequence, we will establish a well-posedness result in the framework of Sobolev spaces of high enough regularity. The talk is based on a joint work with Rafael Granero-Belinchón (Universidad de Cantabria) and Stefano Scrobogna (Università degli Studi di Trieste).

Resumen. Starting from a paper by Henrot, Philippin and Prébet (1999), an elementary argument was developed to prove radial symmetry of the domain and the solution of some overdetermined problems related to elliptic equations. I call such problems constrained because the boundary condition depends on the distance to a prescribed point, and therefore it is not invariant under translations. The argument, based on the comparison principle, has been subsequently applied by Henrot and Philippin (2003), Ciraolo and Vezzoni (2015), and by the speaker and some collaborators (Cadeddu, Ciraolo, Mascia, Mebrate, Pisanu, Servadei) to a number of problems related to different operators (possibly nonlinear or nonlocal) posed in the Euclidean space as well as on a Riemannian manifold, and sometimes governed by nonradial shapes. In this talk I will try to present an updated survey.

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Resumen. En mecánica, las restricciones en las configuraciones de un sistema se denominan "holónomas". Un ejemplo sencillo es la longitud constante del péndulo. Sistemas mecánicos con restricciones en las velocidades que no pueden reducirse a restricciones en las posiciones se llaman "no-holónomas". Un ejemplo clásico es una esfera que rueda sin resbalar en una mesa. El reto en el estudio de los sistemas mecánicos no-holónomos aparece debido a que las ecuaciones de movimiento no poseen una estructura Hamiltoniana. En su lugar, la dinámica es descrita en términos de un corchete de funciones que no satisface la identidad de Jacobi. Hablamos entonces de una "corchete casi-Poisson". La pérdida de la identidad de Jacobi da lugar a fenómenos que no son posibles en los sistemas Hamiltonianos clásicos. Algunas preguntas abiertas en el área de mecánica no-holónoma incluyen determinar condiciones para la existencia de una medida conservada y de existencia de equilibrios asintóticos, relación entre simetrías y leyes de conservación, reducción e integrabilidad. En la primera parte de la charla presentaré una introducción básica a los sistemas no-holónomos rica en ejemplos y después me concentraré en el problema de existencia de medidas invariantes suaves para estos sistemas.

Resumen. In this talk I will discuss some aspects on the leapfrogging phenomenon in the vortex dynamics for Euler equations in the plane subject to a linear shear flow. We show the existence of non rigid time periodic solutions in the local frame which is translating uniformly. We use some techniques borrowed from KAM theory. This is a joint work with Zineb Hassainia and Nader Masmoudi.

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Resumen. In this talk we will discuss a class of alignment models with self-propulsion and Rayleigh-type friction forces, which describes the collective behavior of agents with individual characteristic parameters. We describe the long time dynamics via a method which allows to reduce analysis from the multidimensional system to a simpler family of two-dimensional systems parametrized by a proper Grassmannian. With this method we demonstrate exponential alignment for a large (and sharp) class of initial velocity configurations confined to a sector of opening less than $\pi$. In the case when characteristic parameters remain frozen, the system governs dynamics of opinions for a set of players with constant convictions. Viewed as a dynamical non-cooperative game, the system is shown to possess a unique stable Nash equilibrium, which represents a settlement of opinions most agreeable to all agents. Such an agreement is furthermore shown to be a global attractor for any set of initial opinions.

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Resumen. In this talk I will speak about periodic solutions with a fixed period of the Lorentz force equation with the Kepler electric potential. The set of T-periodic solutions is the set of Szulkin critical point of the action functional associated to the Poincare lagrangian. Using the Ekeland variational principle and the Lusternik - Schnirelman category we prove that the action functional has infinitely many critical points, so the Lorentz force equation has infinitely many T-periodic solutions with a fixed period T.

Resumen. Se tratará de ilustrar las ideas geométricas y analíticas subyacentes a la regularidad de superficies minimales, consideradas como fronteras de conjuntos de perímetro finito, contenidas en el trabajo de L. Caffarelli y A. Córdoba (An elementary regularity theory of minimal surfaces). Sea D un conjunto abierto y acotado del espacio euclídeo y E un boreliano de perímetro finito. Existe entonces un conjunto M de perímetro mínimo entre todos los que coinciden con E fuera del dominio D. La parte más intrincada de esta teoría consiste en demostrar que S, la parte de la frontera de M contenida en D, es una superficie diferenciable que satisface la ecuación de las superficies mínimas excepto, quizás, por un conjunto de singularidades cuya dimensión de Hausdorff es pequeña.

Resumen. The theory of viscosity solutions was developed to deal with Hamilton-Jacobi type problems in the late 80s and 90s by Crandall, Lions, and others. The surname "viscosity" comes from their construction through the vanishing viscosity method. In many scenearios have well-posedness and select the physical solution in many settings. Furthermore, their stability properties makes them suitable to study different approximations: finite differences and asymptotic limits as $t \to \infty$. The aim of this talk is to introduce this notion of solution and show its usefulness to study the time limit of Aggregation-Diffusion equations. The talk presents joint work with: J.A. Carrillo, A. Fernández-Jiménez, and J.L. Vázquez.

Resumen. We will discuss several recent results for aggregation-diffusion equations related to partial concentration of the density of particles. Nonlinear diffusions with homogeneous kernels will be reviewed quickly in the case of degenerate diffusions to have a full picture of the problem. Most of the talk will be devoted to discuss the less explored case of fast diffusion with homogeneous kernels with positive powers. We will first concentrate in the case of stationary solutions by looking at minimisers of the associated free energy showing that the minimiser must consist of a regular smooth solution with singularity at the origin plus possibly a partial concentration of the mass at the origin. We will give necessary conditions for this partial mass concentration to and not to happen. We will then look at the related evolution problem and show that for a given confinement potential this concentration happens in infinite time under certain conditions. We will briefly discuss the latest developments when we introduce the aggregation term. This talk is based on a series of works in collaboration with M. Delgadino, J. Dolbeault, A. Fernández, R. Frank, D. Gómez-Castro, F. Hoffmann, M. Lewin, and J. L. Vázquez.

Resumen. We consider the inhomogeneous incompressible Euler equations describing two fluids with different constant densities under the influence of gravity. Initially the fluids are supposed to be at rest and separated by a flat horizontal interface with the heavier fluid being on top of the lighter one. Due to gravity this configuration is unstable, the two fluids begin to mix in a more and more turbulent way. This is called the Rayleigh-Taylor instability. In the talk we will see how admissible solutions to the Euler equations reflecting a turbulent mixing of the two fluids in a quadratically growing zone can be constructed. Furthermore, if time allows, we will discuss an arising selection problem for the averaged motion of solutions. This is based on joint works with József Kolumbán, László Székelyhidi, Jonas Hirsch.

Resumen. We construct multiple solutions to the Liouville type equation $$ (-\Delta)^{\frac12} u = k(x) e^u, \quad \textup{ in } \mathbb{\mathbb{R}} $$ More precisely, for $k$ of the form $k(x) = 1+\epsilon\kappa(x)$ with $\epsilon \in (0,1)$ small and $\kappa \in C^{1,\alpha}(\mathbb{R}) \cap L^{\infty}(\mathbb{R})$ for some $\alpha > 0$, we prove the existence of multiple solutions to the above equation bifurcating from the so-called Aubin-Talenti bubbles. These solutions provide examples of flat metrics in the half-plane with prescribed geodesic curvature $k(x)$ on its boundary. Moreover, they imply the existence of multiple ground state soliton solutions for the Calogero-Moser derivative NLS. The talk is based on joint works with L. Battaglia (Roma), M. Cozzi (Milano) and A. Pistoia (Roma).

Resumen. Para grafos con peso hay su ciente literatura sobre el corte de Cheeger y el problema de Cheeger, pero para los grafos métricos hay pocos resultados sobre estos problemas. Nuestro objetivo es estudiar el corte Cheeger y el problema de Cheeger en grafos métricos. Para eso, necesitamos introducir los conceptos de variación total y perímetro en grafos métricos, de tal forma que tengan en cuenta los saltos en los vértices de las funciones de la variación acotada. También necesitamos una fórmula de integración por partes. Además, estudiamos el problema del valores propios para operador 1-laplacio en grafos métricos, mediante el cual damos un método para resolver el problema de corte óptimo de Cheeger.

Resumen. Our aim is to study a nonlocal Cahn-Hilliard model (CHE) in the framework of random walk spaces, which includes as particular cases, the CHE on locally finite weighted connected graphs, the CHE determined by finite Markov chains or the Cahn-Hilliard Equations driven by convolution integrable kernels. We consider different transitions for the phase and the chemical potential, and a large class of potentials including obstacle ones. We prove existence and uniqueness of solutions in L1 of the Cahn-Hilliard Equation. We also show that the Cahn-Hilliard equation is the gradient flow of the Ginzburg-Landau free energy functional on an appropriate Hilbert space. We finally study the asymptotic behaviour of the solutions. Joint work with J. Toledo

Resumen. Nos proponemos estudiar algunas relaciones entre el grado de Brouwer de un campo vectorial y la dinámica del flujo inducido. Estamos interesados, en particular, en las propiedades dinámicas y topológicas de los conjuntos invariantes aislados y de sus variedades inestables. Estudiamos también relaciones análogas para el índice de un campo vectorial y obtenemos de este modo nuevas formas del teorema de Poincaré-Hopf. También obtenemos algunas consecuencias relativas a los teoremas antipodales de Borsuk y de Hirsch. Como aplicación, calculamos el grado de Brouwer y el índice de campos vectoriales en algunas situaciones topologicamente relevantes, obtenemos criterios para la detección de orbitas conectantes en las descomposiciones atractor-repulsor de los conjuntos invariantes aislados y calculamos el grado de Brouwer del campo vectorial de las ecuaciones de Lorenz en bloques aislantes del conjunto extraño. Estos resultados han sido obtenidos en colaboración con Héctor Barge.

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Resumen. In this preliminary work we will concern about the "Finite Morse Index" scenario (instead of the usual stability one) of the following long-standing conjecture: "Let u be a (compactly supported) weak stable solution of the general non-linear Poisson equation and assume that the non-linearity is positive, non-decreasing, convex, and superlinear at +∞, and let n<10. Then u is bounded." Recently, Cabré, Figalli, Ros-Oton and Serra end up a complete proof in the classical stability setting: W^{1,2}-stable solutions are universally bounded for n<10 (and therefore smooth by classical elliptic regularity theory); namely they are bounded in terms only of their L^1 norm, with a constant that is independent of the non-linearity. This conjecture is in a sense equivalent to another problem stated before by Brezis: Is it possible to prove that the extremal solution of the so-called Gelfand problem is smooth at least in low dimensions?. We will take this as the starting point to reproduce our new results recovering the existing results by S. Villegas in the (semi-)stable case. Using some estimates on the "size" of the local stability behavior of finite Morse Index solutions we provide a uniform a priori bound and some pointwise estimates of that solutions that partially answer positively the long standing conjecture in this more general setting.

Resumen. We propose a numerical method for a Vlasov-Poisson-Fokker-Planck model and prove quantitative results ensuring that it is Asymptotic-Preserving for the linearized model in both the macroscopic and the long time regime. We illustrate these results with various numerical experiments in which we observe, among others, transition phase between macroscopic and long time behavior as well as oscillations and instability phenomena.

Resumen. We consider the Gauss equation that governs the minimal immersion of a closed surface of genus greater than two in a hyperbolic three manifold for which the second fundamental form is prescribed. The PDE admits two solutions and we will analyze the behavior of such solutions when the norm of the second fundamental form is small.

Resumen. We provide a qualitative explanation of the co-orbital motion of two small moons orbiting a planet. The two small bodies revolve about the planet in nearly circular orbits with almost equal radii. The system is modelled as a planar three-body problem whose Hamiltonian is expanded as a perturbation of two uncoupled Kepler problems. A combination of averaging, normal form, symplectic scaling and Hamiltonian reduction theories and the application of a KAM theorem for high-order degenerate systems allows us to establish the existence of quasi-periodic motions and KAM 4-tori related to the co-orbital motion of the moons. By conveniently selecting a suitable region of the reduced phase space (which is the Cartesian product of a two-dimensional sphere and one sheet of a two-sheet hyperboloid of revolution), we are able to establish the existence of these quasi-periodic motions that are valid for any value of an action variable, related to the angular momenta of the two moons. This is a joint work with Josep M. Cors and Patricia Yanguas.

Resumen. We will present the derivation of a nucleation boundary condition to the Lifshitz-Slyozov equation, the well-posedness of the Cauchy problem as well as the long-time behaviour of the solutions.

Resumen. We present results on a stochastic version of a well-known kinetic nucleation model for phase transition phenomena. In the Becker-Döring model, aggregates grow or shrink by addition or removal of one-by-one particle at a time. Under certain conditions, very large aggregates emerge and are interpreted as a phase transition. We study stationary and quasi-stationary properties of the stochastic Becker-Döring model in the limit of infinite total number of particles, and compare with results from the deterministic nucleation theory. Our findings are largely inspired from recent results from stochastic chemical reaction network theory.

Resumen. In this talk, we present the existence of nontrivial simply connected domains of the sphere such that the overdetermined elliptic problem admits a positive solution by a local bifurcation argument. This shows in particular that Serrin's theorem for overdetermined problems in the Euclidean space cannot be generalized to the sphere even for simply connected domains.

Resumen. The existence of a local curve of corotating vortex pairs was proven by Hmidi and Mateu via a desingularization of a pair of point vortices. In this talk, we will present a global continuation of these local curves. That is, we consider solutions which are more than a mere perturbation of trivial solution. Indeed, while the local analysis relies on the study of the linear equation at the trivial solution, the global analysis requires on a deeper understanding of topological properties of the nonlinear problem. For our proof, we adapt the powerful analytic global bifurcation theorem due to Buffoni and Toland, to allow for the singularity at the bifurcation point. This is a collaboration with Susanna V. Haziot.

Resumen. Motivated by a popular kinetic model by Saintillan and Shelley for the dynamics of suspensions of active elongated particles, we study phase-mixing and enhanced dissipation on the sphere. In particular, we show that, up to log errors, the phase mixing estimate persists until the enhanced dissipation takes over. This is proved by combining an optimized hypocoercive approach with the vector field method. Joint work with Michele Coti Zelati and David Gérard-Varet (https://arxiv.org/abs/2207.08431).

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Resumen. We study compactness properties of metrics of prescribed fractional $Q$-curvature of order 3 in $\R^3$. We use an approach inspired from conformal geometry, regarding a metric on a subset of $\R^3$ as the restriction of a metric on $\R^4_+$ with vanishing fourth-order $Q$-curvature. In particular, in analogy with a 4-dimensional result of Adimurthi, Robert and Struwe, we prove that a sequence of such metrics with uniformly bounded fractional $Q$-curvature can blow up on a large set (roughly, the zero set of the trace of a nonpositive biharmonic function $\Phi$ in $\R^4_+$), and we also construct examples of such behaviour. Towards this result, an intermediate step of independent interest is the construction of general Poisson-type representation formulas (also for higher dimension).

This is a work done in collaboration with María del Mar González, Ali Hyder and Luca Martinazzi.

Resumen. We analyse the existence and the properties of conformal metrics on $R^n$ with prescribed constant Q-curvature and possibly a singularity. Our approach is new also in the local case, as it uses Campanato space estimates instead of pointwise estimates (much more difficult to obtain). We mention applications to the Moser-Trudinger and the mean-field equation.

Resumen. Kinetic equations play an important role in multiscale modeling hierarchy. It serves as a basic building block that connects the microscopic particle models and macroscopic continuum models. Numerically approximating kinetic equations presents several difficulties: 1) high dimensionality (the equation is in phase space); 2) nonlinearity and stiffness of the collision/interaction terms; 3) positivity of the solution (the unknown is a probability density function); 4) consistency to the limiting fluid models; etc. I will start with a brief overview of the kinetic equations including the Boltzmann equation and the Fokker-Planck equation, and then discuss in particular our recent effort of constructing efficient and robust numerical methods for these equations, overcoming some of the aforementioned difficulties.

Resumen. Elastic materials are those that deform under the action of an applied force and recover their original configuration when the load stops acting. When the elastic potential energy can be modelled as a variational principle we call then hyperplastic materials, and it is the natural way to model large deformation in materials under the action of very big loads. In this case, deformations are minimizers of the variational principle given by the potential energy. In this talk, we first recall the classical existence theory in hyperelasticity, in which the central requirement is polyconvexity of the integrand in the variational principle. Main ingredient for obtaining the existence result is the weak continuity of the deformation gradient, which is itself a very remarkable compensated compactness result. Then, we propose a fractional model for hyperelasticity by replacing the gradient of the deformation by its Riesz fractional gradient. Functional space for this model will be a Bessel potential space, which is a fractional space in between the Lebesgue and Sobolev spaces. We show well-posedness of this new model by proving a nonlocal Piola identity, that yields to the weak continuity of the Riesz fractional gradient. One remarkable and fortunate feature of this new model is that it allows for singularities, such a fracture or cavitation, to happen in the optimal deformations. This was forbidden in the classical models on Sobolev spaces.

The talk will be intended for a broad mathematical audience.

This is a joint work with J. Cueto (UCLM) and C. Mora-Corral (UAM). (Nonlocal hyperelasticity and polyconvexity in fractional spaces, arXiv:1812.05848, 2019).

Resumen. We study the McKean–Vlasov equation on the torus which is obtained as the mean field limit of a system of interacting diffusion processes enclosed in a periodic box. We focus our attention on the stationary problem - under certain assumptions on the interaction potential, we show that the system exhibits multiple equilibria which arise from the uniform state through continuous bifurcations. We then attempt to classify continuous and discontinuous phase transitions for this system and show that at the point of discontinuous transition the free energy possesses a mountain pass point. Finally, we comment on further work generalising these results to equations with porous medium-type diffusion. Joint work with José A. Carrillo, Greg Pavliotis, and André Schlichting.

Resumen. In this talk we will prove that three different 2-soliton solutions of the sine-Gordon equation (SG) are orbitally stable in the natural energy space of the problem [4]. We will prove this result without using the inverse scattering technique for the equation nor the steepest descent method, which allows us to work in the very large energy space $H^1(\R) \times L^2(\R)$. The three families which we will study are called 2-kink, kink-antikink and breather of SG, described by Lamb [3]. To prove this result we will use a well-chosen Bäcklund transformation which allow us to reduce the stability question of these families to the zero solution case, in the same spirit as the result of Alejo and Muñoz for the case of the modified Kortweg-de Vries equation [1]. However, we will see that SG presents several new difficulties that we will have to solve appropriately. Possible connections to asymptotic stability results will also be discussed. This work is in colaboration with C. Muñoz and improves in several directions the results in [2].

References

[1] M.A. Alejo, and C. Muñoz, , Anal. and PDE. 8 (2015), no. 3, 629–674.

[2] M.A. Alejo, C. Muñoz, and J. M. Palacios, On the Variational Structure of Breather Solutions I: Sine-Gordon case, J. Math. Anal. Appl. Vol.453/2 (2017) pp. 1111–1138.

[3] G.L. Lamb, Elements of Soliton Theory, Pure Appl. Math., Wiley, New York, 1980.

[4] C. Muñoz, and J.M. Palacios, Stability of the 2-soliton solutions of the SG equation in the energy space, to appear in Ann. IHP C, Analyse Nonlineaire.

Resumen. En esta charla analizaremos el siguiente problema de contorno $(P_\lambda)$ en un dominio acotado $\Omega \subseteq \R^N$:

$$ \begin{equation} \begin{cases} -\Delta u = \lambda u + \mu(x) \frac{|\nabla u|^q}{u^\alpha} + f(x) \qquad & \text{in $\Omega$} \\ u > 0 \qquad & \text{in $\Omega$} \\ u = 0 \qquad & \text{on $\partial \Omega$} \end{cases} \end{equation} $$

donde $f, \mu \colon \Omega \to [0, \infty)$ son funciones acotadas, $q \in (1, 2)$, $\alpha \in [0, 1]$ y $\lambda \in \R$.

Mostraremos que, para $\lambda \leq 0$, existe una única solución de $(P_\lambda)$. Por contra, para $\lambda > 0$, la naturaleza del problema depende de $\alpha$. En efecto, veremos que si $\alpha \in [0, q−1)$, se puede probar un resultado de existencia y multiplicidad de solución de $(P_\lambda)$ para $\lambda > 0$ suficientemente pequeño. Por contra, si $\alpha \in [q − 1, 1]$, existe una única solución de $(P_\lambda)$ para $\lambda > 0$. Los resultados que se presentarán han sido obtenidos recientemente en colaboración con José Carmona (Universidad de Almería), Tommaso Leonori ("Sapienza" Università di Roma I) y Pedro J. Martínez-Aparicio (Universidad de Almería).

Resumen. A lo largo de esta charla consideraremos la ecuación de Hénon no local

$$ \begin{equation} (-\Delta)^s u = |x|^\alpha u^p, \quad \R^N \end{equation} $$ donde $(-\Delta)^s$ representa el Laplaciano fraccionario, $0 < s < 1$, $−2s < α$, $p > 1$ y $N > 2s$. Se presentará un resultado de tipo Liouville en el que se pruebe la no existencia de soluciones positivas en el rango optimo de la nolinearidad, esto es, cuando

$$ \begin{equation} 1 < p < p^*_{\alpha, s} := \frac{N + 2 \alpha + 2s}{N - 2 s} \end{equation} $$ Además demostraremos que, en el caso crítico $p = p^*_{\alpha, s}$, incluso para $\alpha > 0$, existen soluciones radialmente simétricas con decaimiento rápido, es decir soluciones tipo "bubble".

Los resultados presentados en esta charla han sido obtenidos en colaboración con Alexander Quaas (Universidad Tecnica Federico Santa Marıa, Valparaıso, Chile).

Resumen. In this talk, we deal with the following problem (BI)

$$ \begin{equation} \tag{BI} \left\{ \begin{aligned} &-\operatorname{div}\left( \frac{\nabla \phi }{\sqrt{1 - |\nabla \phi|^2}} \right) = \rho, \qquad x \in \R^N,\\ &\lim_{|x|\to \infty} \phi(x) = 0 \end{aligned} \right. \end{equation} $$ The equation in (BI) appears for instance in the Born-Infeld nonlinear electromagnetic theory: in the electrostatic case it corresponds to the Gauss law in the classical Maxwell theory and so φ is the electric potential and ρ is an assigned extended charge density.

In the first part of the talk, we discuss existence, uniqueness and regularity of the solution of (BI). In the second part, instead, we deal with existence of equilibrium measures $\rho^*$, namely distributions that produce least-energy potentials among all the possible charge distributions, and properties of the corresponding equilibrium potentials $\phi_{\rho^*}$ for (BI).

The results have been obtained in joint works with Denis Bonheure (Universite libre de Bruxelles, Belgium), Pietro d’Avenia (Politecnico di Bari, Italy) and Wolfgang Reichel (Karlsruher Institut für Technologie, Germany).

Resumen. We analyze how the behavior of a chaotic dynamic system changes when it is subjected to a coupling. We consider several types of couplings and several free dynamics, analyzing in each situation the evolution of the behavior as a function of the force-coupling constant. We define and delimit behavior windows. If one dynamic system, instead of being coupled to only one other, is connected to several, we are faced with a network of dynamic systems. We have added to this situation some of the results obtained, namely those concerning full synchronization.

Resumen. Las ecuaciones diferenciales de dimensión baja y la topología del plano son galaxias paralelas, los arcos de traslación viajan entre ellas.

Resumen. Let $U : \Omega \subset \R^N \to \R$ be a potential of the class $C^2$ defined on an open subset $\Omega \subset \R^N$ and let $q_0 \in \Omega$ be an isolated critical point of $U$ i.e. $U'(q_0)=0.$ \\ The Lyapunov center theorem gives sufficient conditions for the existence of non-stationary periodic solutions of the system $(*) \:\: \ddot q(t)=-U'(q(t))$ in any neighborhood of $q_0.$ The aim of may talk is to present the Lyapunov center theorem for symmetric potentials. More precisely speaking, assume additionally that $\Omega \subset \R^N$ is an open and invariant subset of an orthogonal representation $\R^N$ of a compact Lie group $\Gamma$ and that the potential $U : \Omega \to \R$ is $\Gamma$-invariant i.e. it is constant on the orbits of the group $\Gamma.$ Since the orbit of critical points $\Gamma(q_0)=\{\gamma q_0 : \gamma \in \Gamma\} \subset U'^{-1}(0)$ is $\Gamma$-homeomorphic to $\Gamma / \Gamma_{q_0}$ ($\Gamma_{q_0}=\{\gamma q_0=q_0 : \gamma \in \Gamma \}$ is the stabilizer of $q_0$), the critical points of the potential $U$ usually are not isolated in $U'^{-1}(0)$ and therefore we can not apply the Lyapunov center theorem to the study of non-stationary periodic solutions of the system $(*)$. Assume that the orbit $\Gamma(q_0)$ is isolated in $U'^{-1}(0)$. We will formulate sufficient conditions for the existence of periodic orbits of solutions of the equation $(*)$ in any neighborhood of the orbit $\Gamma(q_0).$ Moreover, we will estimate the minimal period of these solutions. The basic idea of the proof is to apply the infinite-dimensional generalization of the $(\Gamma \times S^1)$-equivariant Conley index theory.

Resumen. In this talk, we will deal with the study of existence and regularity of appropriate weak solutions for non-local quasi-linear problems involving measures: $$ \left\{ \begin{aligned} (-\Delta)^{\alpha} u(x) &= g(x, |\nabla u|) + \sigma \nu \qquad &\text{in } \Omega, \\ u(x) &= \rho \mu \qquad &\text{in } \mathbb{R}^N \setminus \Omega, \end{aligned} \right. $$ where $\rho, \sigma \geq 0$, $\mu$ and $\nu$ are suitable Radon measures, $g \colon \Omega \times [0, \infty) \to [0, \infty)$ is a continuous function fulfilling certain growth conditions (to be presented a posteriori) and $\Omega \subseteq \mathbb{R}^N$ is a $C^2$ bounded domain. We shall discuss different regimes where a solution may be defined and we will extend the presentation to Dirichlet problems like (1) with the addition of measures concentrated on the boundary ∂Ω. This is a joint work with Analia Silva from Universidad Nacional de San Luis and Joao Vitor da Silva from Universidad de Buenos Aires.

Resumen. Caused by the development of renewable energies in recent years, power networks experience a considerable change from few major generators to smart grids of small producers. A fundamental question for such systems is whether they achieve synchronisation to a common frequency on maybe even very large network structures. In this talk, we study this problem for a second-order Kuramoto-type rotator model. More precisely, we will present a theory of continuum limit for this model, both for deterministic and random networks. Based on these limits, the stability properties of synchronised states and their dependence on the topological properties of the underlying network are examined.

Resumen. The aim of my talk is to present the concept of the degree for equivariant gradient maps and its infinite dimensional version, namely the degree for invariant strongly indefinite functionals. These degrees generalize the idea of the Brouwer and the Leray-Schauder degree to the situation of the map defined on a representation of a compact Lie group. This appears in a natural way for example in the problem of looking for periodic solutions of an autonomous hamiltonian system. I would like to show how the degree for equivariant maps can be used to prove the existence of such solutions.

Resumen. Multi-agent systems in nature oftentimes exhibit emergent behaviour, i.e. the formation of patterns in the absence of a leader or external stimuli such as light or food sources. We present a non-local two-species cross-interaction system of partial differential equations with cross-diffusion and explore its long-time behaviour. We observe a rich zoology of behaviours exhibiting phenomena such as mixing and/or segregation of both species and the formation of travelling pulses. One of the most fascinating real world applications of this model are zebrafish with their black and yellow pigment cells whose interspecific and intraspecific interactions lead to the characteristic stripe pattern formation.

Resumen. In this talk we will construct a sequence of nondegenerate nodal nonradial solutions to the critical Yamabe problem $$-\Delta u=\frac{n(n-2)}{4}|u|^{\frac{4}{n-2}}u,\qquad u\in\mathcal{D}^{1,2}(\mathbb{R}^n),$$ which provides the first example in the literature of a solution with maximal rank. This is a joint work with M. Musso and J. Wei that can be found at arxiv.org/pdf/1712.00326.pdf.

Resumen. The framework of mixture theory allows describing complex systems of heterogeneous fluids at the mesoscale which is an intermediary scale between micro and macro. This formalism generalises Euler equations and uses partial differential equations to model multicomponent fluids. Therefore, mixture theory is particularly well adapted to describe complex biological ecosystems such as photosynthetic microalgae biofilm and gut microbiota ecology. The purpose of the talk will be to introduce mixture theory formalisms and then present these two applications. In both cases, the context and issues will be specified. Eventually, the numerical schemes and simulations will be commented.

Resumen. For variational monotone recurrence relations we know from the Aubry-Mather theory the existence and properties of foliation or lamination consisting of Birkhoff solutions. In this talk, we discuss for the general monotone recurrence relations the existence of Birkhoff solutions and implications of non-Birkhoff solutions. In particular, we show that a solution with bounded action implies the existence of a Birkhoff solution and the rotation set contains an interval with end points being the Farey neighbours of p/q provided there is a non-Birkhoff (p,q) periodic solution.

Resumen. As a branch of mathematics, fractional calculus is a generalization of differentiation and integration to arbitrary (non-integer) orders. The concept of fractional-order calculus can be traced to the early work of Leibniz and L'Hospital in 1695, but it has attracted lots of attention from physicians and engineers in recent years. Many systems in interdisciplinary fields can be accurately modelled by fractional-order differential equations, such as viscoelastic systems, dielectric polarization, quantitative finance, nonlinear oscillation of earthquakes, robotic manipulating systems, muscular blood vessel model, hydrologic models and so on. In this talk, I will discuss Riemann-Liouville differential and integral operators and Caputo fractional derivative. Then, I will talk about fractional-order dynamical systems defined by differential operators of Caputo type.

<Resumen. Let \( \Omega \) be a bounded, open subset of \( \R^{N} \), with \( N > 2 \). Let us define, for \( (v,\psi) \) in \( W_{0}^{1,2}(\Omega) \times W_{0}^{1,2}(\Omega) \), \begin{equation} \label{j} J(v,\psi ) = \frac12 \int_{\Omega} \,A(x)\,\nabla v\,\nabla v - \frac{1}{2}\int_{\Omega}\,M(x)\,\nabla\psi\,\nabla\psi + \int_{\Omega} v\,E(x) \nabla\psi - \int_{\Omega} f(x)\,v\,. \end{equation} where \( A(x) \), \( M(x) \) are symmetric measurable matrices such that \begin{equation} \label{al} \begin{cases} A(x)\,\xi\,\xi \geq \alpha|\xi|^2\,, \qquad |A(x)| \leq \beta\,, \\ M(x)\,\xi\,\xi \geq \alpha|\xi|^2\,, \qquad |M(x)| \leq \beta\,, \end{cases} \end{equation} for almost every \( x \) in \( \Omega \), for every \( \xi \) in \( \R^{N} \), with \( 0 < \alpha \leq \beta \), and \begin{equation}\label{fm} f\in L^{m}(\Omega)\,,\ m\geq 2_{*} =\frac{2N}{N+2}, \end{equation} \begin{equation} \label{e} E\in(L^{N}(\Omega))^N. \end{equation} We study the existence of saddle points of the functional \( J \) defined above both in the regular case, i.e., if \( E \) belongs to \( (L^{N}(\Omega))^{N} \) and in the singular one, i.e., if \( E \) belongs to \( (L^{2}(\Omega))^{N} \). The second problem concerns the functional \begin{equation} \label{i} I(v,\psi ) = \frac12\int_{\Omega}\,A(x)\,\nabla v\,\nabla v - \frac{1}{2}\int_{\Omega}\,M(x)\,\nabla\psi\,\nabla\psi + \int_{\Omega}|v|^r\psi - \int_{\Omega} f(x)\,v\,. \end{equation}

Resumen. For a bounded domain $\Omega$, a bounded Carathéodory function $g$ in $\Omega\times\mathbb{R}$, $p>1$, a nonnegative integrable function $h$ in $\Omega$ which is strictly positive in a set of positive measure and a continuous function $a$ which is superlinear with polynomial growth we prove that, contrarily with the case $h\equiv 0$, there exists a solution of the semilinear elliptic problem \begin{equation}\label{pa} \left \{ \begin{array}{rcll} -\Delta u & = & \lambda u +g(x,u)- h(x) a(u) +f, & \mbox{in } \Omega \\ u & = & 0, & \mbox{on } \partial\Omega,\\ \end{array} \right. \end{equation} for every $\lambda\in\mathbb{R}$ and $f\in\ L^2(\Omega)$. And also give results of existence and multiplicity of similar problems, such that fractional laplacian problem, homogeneous problem and a concave perturbation of the above problem.

Resumen. I will discuss the boundary value problems for the loaded differential equations associated with nonlocal boundary value problems for classical partial differential equations. In our investigations we formulate the main boundary value problems (such as the Tricomi, Darboux) and their generalizations, and well-posed new boundary value problems for the linear loaded differential and integro-differential equations of the third order, with the classic and mixed operators.

Resumen. The development of hybrid methodologies is of current interest in both multi-scale modelling and stochastic reaction-diffusion systems regarding their applications to biology. We formulate a hybrid method for stochastic multi-scale models of cells populations that extends the remit of existing hybrid methods for reaction-diffusion systems. Such method is developed for a stochastic multi-scale model of tumour growth, i.e. population-dynamical models which account for the effects of intrinsic noise affecting both the number of cells and the intracellular dynamics. In order to formulate this method, we develop a coarse-grained approximation for both the full stochastic model and its mean-field limit. Such approximation involves averaging out the age-structure (which accounts for the multi-scale nature of the model) by assuming that the age distribution of the population settles onto equilibrium very fast. We than couple the coarse-grained mean-field model to the full stochastic multi-scale model. By doing so, within the mean-field region, we are neglecting noise in both cell numbers (population) and their birth rates (structure). This implies that, in addition to the issues that arise in stochastic-reaction diffusion systems, we need to account for the age-structure of the population when attempting to couple both descriptions. We exploit our coarse-graining model so that, within the mean-field region, the age-distribution is in equilibrium and we know its explicit form. This allows us to couple both domains consistently, as upon transference of cells from the mean-field to the stochastic region, we sample the equilibrium age distribution. Furthermore, our method allows us to investigate the effects of intracellular noise, i.e. fluctuations of the birth rate, on collective properties such as travelling wave velocity. We show that the combination of population and birth-rate noise gives rise to large fluctuations of the birth rate in the region at the leading edge of front, which cannot be accounted for by the coarse-grained model. Such fluctuations have non-tivial effects on the wave velocity. Beyond the development of a new hybrid method, we thus conclude that birth-rate fluctuations are central to a quantitatively accurate description of invasive phenomena such as tumour growth.

Resumen. On the last decade, there has been an increasing interest on the superintegrability in classical and quantum Mechanics. In this talk, I will start by briefly reviewing some basic notions of superintegrability and then present some recent results on the classification of superintegrable systems admitting higher-order potentials. The main emphasis will be on the emergence of the Painlevé transcendent potentials in the quantum case.

Resumen. Desde que Kuramoto propusiera su modelo para osciladores acoplados, la sincronización ha recibido gran atención desde distintos puntos de vista: biología, química, neurociencia, ... Dicho fenómeno es el comportamiento emergente natural de un conjunto de individuos que interaccionan mediante reglas periódicas. Estos patrones se observan en diversos sistemas biológicos complejos como el relampagueo de luciérnagas, los latidos de células cardíacas o los disparos sinápticos de neuronas en el cerebro. En este último ambiente, el aprendizaje de Hebb da una explicación de cuáles son los mecanismos de adaptación de las conexiones sinápticas de la neuronas.

En esta charla exploramos el régimen de aprendizaje rápido hacia una función de adaptación con singularidades acoplado con el modelo de Kuramoto. Para dicho modelo singular de $N$ osciladores acoplados, comenzaremos estudiando el buen planteamiento (donde aparecen problemas de concentración, clustering, no unicidad, etc.), extendiendo el concepto de solución en sentido de Filippov. Posteriormente, caracterizaremos el fenómeno de clustering en subgrupos y daremos estimaciones de las tasas de sincronización. Concluimos presentando el modelo macroscópico de tipo Vlasov-McKean asociado, que se corresponde con la contraparte singular del modelo de Kuramoto-Sakaguchi, y comparando con sistemas relacionados como Cucker-Smale.

[1] J. Park, D. Poyato, J. Soler, Hebbian learning and clustering in Kuramoto models with singular weighted coupling, (2018), preprint.

[2] J. Park, D. Poyato, J. Soler, Eulerian hydrodynamics for Kuramoto models with Hebbian singular coupling, (2018), in preparation.

Resumen. Consideramos $n$ especies biológicas las cuales habitan en un segmento de recta, lo cual permite dar un orden a ese conjunto de especies. Luego podemos hablar de términos como "la primera especie" o de "especies vecinas", sin ambigüedad. Si cada especie interactúa solo con sus vecinas, decimos que el sistema es tridiagonal. Un tal sistema se llamará depredador-presa si cada dos especies vecinas es un subsistema depredsor-presa ordinario. Probaremos la existencia de un único equilibrio saturado del sistema. Tal equilibrio es un atractor local si todas sus coordenadas son positivas. Conjetura. El equilibrio saturado es un atractor global. Esta conjetura está íntimamente ligada a la de Marcus-Yamabe.

Resumen. En 1929, S. Bochner caracterizó, salvo cambio de variable afín, las familias de polinomios ortogonales que son funciones propias de un operador diferencial de segundo orden con coeficientes polinomiales independientes del grado. Estas familias se reducen a cuatro tipos (Hermite, Laguerre, Jacobi y Bessel) y se corresponden con las formas canónicas que puede adoptar el coeficiente del término de mayor orden en la ecuación diferencial. Estas familias se caracterizan por sus propiedades diferenciales, encuentran numerosas aplicaciones y han sido ampliamente estudiadas en la literatura. En esta charla, después de analizar los polinomios ortogonales clásicos en una variable, realizaremos una introducción a los polinomios ortogonales clásicos en dos variables como funciones propias de operadores diferenciales en derivadas parciales de segundo orden con coeficientes polinomiales.

Resumen. La clásica desigualdad de Lojasiewicz y sus extensiones debidas a Simony Kurdyka han tenido un considerable impacto en el estudio del comportamiento asintótico para flujos gradiente en espacios de Hilbert. Nuestro objetivo es adaptar estas clásicas desigualdades al marco general de flujos gradiente en espacios métricos. Mostramos que la validez de la desigualdad de Kurdyka- Lojasiewicz implica la convergencia al equilibrio y la validez de la desigualdad de Lojasiewicz-Simonnos permite obtener estimaciones del decaimiento y en algunos casos el tiempo finito de extinción. Los métodos de entropía han provado ser muy útiles en el estudio del comportamiento asintótico de mucha ecuaciones en derivadas parciales. Dicho método se basa en la desigualdad de producción de entropía (Desigualdad log-Sobolev) y en la desigualdad de transporte de entropía (Desigualdad de Talagrand). Demostramos que para funcionales de energía geodésicamente convexos dichas desigualdades son equivalentes y coinciden con la desigualdad de Lojasiewicz-Simon. Finalmente aplicamos nuestros resultados abstractos a ejemplos concretos en espacios de Hilbert y en espacios de medidas de probabilidad con la distancia de Wasserstein. Por ejemplo, para el funcional de energía asociado con ecuaciones doblemente no lineales obtenemos la equivalencia entre las desigualdades de Lojasiewicz-Simon,log-Sobolev y Talagrand; estudiando también estimaciones del decaimiento de sus soluciones.

Resumen. By using critical point theory for strongly indefinite functionals, we study the Neumann problem associated to some prescribed mean curvature problems in a FLRW spacetime with one spatial dimension. We assume that the warping function is even and positive and the prescribed mean curvature function is odd and sublinear. Then, we show that our problem has infinitely many solutions. The keypoint is that our problem has a Hamiltonian formulation. The main tool is an abstract result of Clark type for strongly indefinite functionals.

Resumen. En esta charla se van a mostrar algunas de las propiedades cualitativas más características de las soluciones de las ecuaciones de difusión con operadores de flujo limitado en lo referente a su regularidad, existencia de fronteras, tiempo de espera de crecimiento del soporte, existencia de patrones en presencia de términos de reacción, etc... Además, se mostrará una aplicación en procesos de morfogénesis donde la sustitución de la difusión lineal de Fick por uno de estos operadores proporciona resultados más acordes con la evidencia experimental.

Resumen. En esta charla mostraré algunos resultados de existencia y unicidad de solución $u(x)$ de problemas de Dirichlet con términos de orden inferior dependientes del gradiente de $u$ y singulares en $u=0$. Veremos que cuando la no linealidad es 1-homogénea es posible probar resultados óptimos haciendo uso de teoría de valores propios no variacional. Hay que destacar que en las demostraciones se evita la transformación de Cole-Hopf, de manera que estas técnicas permiten trabajar con no linealidades que dependen de la variable $x$, o con dependencia no cuadrática en el gradiente.

La presentación de la charla está disponible aquí.

Resumen. La clásica desigualdad de Lojasiewicz y sus extensiones debidas a Simony Kurdyka han tenido un considerable impacto en el estudio del comportamiento asintótico para flujos gradiente en espacios de Hilbert. Nuestro objetivo es adaptar estas clásicas desigualdades al marco general de flujos gradiente en espacios métricos. Mostramos que la validez de la desigualdad de Kurdyka- Lojasiewicz implica la convergencia al equilibrio y la validez de la desigualdad de Lojasiewicz-Simonnos permite obtener estimaciones del decaimiento y en algunos casos el tiempo finito de extinción. Los métodos de entropía han provado ser muy útiles en el estudio del comportamiento asintótico de mucha ecuaciones en derivadas parciales. Dicho método se basa en la desigualdad de producción de entropía (Desigualdad log-Sobolev) y en la desigualdad de transporte de entropía (Desigualdad de Talagrand). Demostramos que para funcionales de energía geodésicamente convexos dichas desigualdades son equivalentes y coinciden con la desigualdad de Lojasiewicz-Simon. Finalmente aplicamos nuestros resultados abstractos a ejemplos concretos en espacios de Hilbert y en espacios de medidas de probabilidad con la distancia de Wasserstein. Por ejemplo, para el funcional de energía asociado con ecuaciones doblemente no lineales obtenemos la equivalencia entre las desigualdades de Lojasiewicz-Simon, log-Sobolev y Talagrand; estudiando también estimaciones del decaimiento de sus soluciones.

Resumen. In this talk, I will discuss some questions related to the nonlinear theory of electromagnetism formulated by Born and Infeld in 1934. I will focus on the set of PDEs arising from this theory and mainly on the static regime. I will discuss also some links between this theory and the curvature operators in the Euclidean and in the Lorentz-Minkowski space. Finally, I will address the solvability of an approximated model and some related questions.

Resumen. Trabajo en colaboración con François Murat, Universidad Paris VI, Francia. Consideramos una problema de Dirichlet homogéno donde la parte principal está formada por un operador monótono clásico de tipo Leray-Lions y el segundo miembro viene dado por una función no lineal de la incógnita que tiende a más infinito en u=0. En este tipo de problemas es usual tratar con segundos miembros no negativos y buscar soluciones no negativas. En el presente trabajo mostramos que una solución no negativa siempre existe aún cuando el segundo miembro cambie de signo. También obtenemos condiciones para la existencia y no existencia de soluciones que cambian de signo.

Resumen. The tools of hypocoercivity have been used to show convergence to equilibrium for many spatially inhomogeneous kinetic equations in Hilbert spaces. Villani proved similar results showing convergence in relative entropy for a wide class of equations. The linear relaxation Boltzmann equation is not in this class. I will discuss the general methods of hypocoercivity theory, and how they can be used to prove results in relative entropy in particular showing convergence to equilibrium for the linear relaxation Boltzmann equation.

Resumen. Una cuestión fundamental en los problemas de frontera libre en dinámica de fluidos es determinar si la interfase preserva su regularidad inicial al evolucionar en tiempo o por el contrario desarrolla singularidades en tiempo finito. En esta charla mostraremos resultados de propagación de regularidad globalmente en tiempo para ecuaciones parab ́olicas no lineales, concretamente, para parches de temperatura en Boussinesq [1], parches de densidad en Navier-Stokes [2] y el problema de Muskat con salto de viscosidades [3]. En este último resultado se verá como para datos iniciales de talla media, la interfase se hace instantáneamente analítica y se tienen tasas de decaimiento de distintas normas. Como consecuencia, se consigue probar que en el caso inestable (fluido más denso encima) el problema está mal propuesto incluso en espacios de baja regularidad.

[1] F. Gancedo, E. García-Juárez. Global regularity for 2D Boussinesq temperature patches with no diffusion, Ann. PDE, 3: 14. https://doi.org/10.1007/s40818-017-0031-y, (2017).

[2] F. Gancedo, E. García-Juárez. Global regularity of 2D density patches for inhomogeneous Navier-Stokes. Submitted, arXiv:1612.08665, (2016).

[3] F. Gancedo, E. García-Juárez, N. Patel, R. Strain. On the Muskat problem with viscosity jump: Global in time results. Preprint arXiv:1710.11604, (2017).

Resumen. En esta charla establecemos la acotación de la solución extremal $u^\ast$ en dimensión $N=4$ de la ecuación elíptica semilineal $$ -\Delta u=\lambda f(u), $$ en un dominio acotado $\Omega \subset \mathbb{R}^N$, con condición de tipo Dirichlet $u|_{\partial \Omega}=0$, donde $f$ es una función positiva $C^1$, no decreciente y convexa en $[0,\infty)$ tal que $f(s)/s\rightarrow\infty$ cuando $s\rightarrow\infty$. Además, probamos que para $N\geq 5$, la solución extremal $u^*\in W^{2,\frac{N}{N-2}}$. Por tanto, $u^\ast\in $L^\frac{N}{N-4}$, si $N\geq 5$ y $u^*\in H_0^1$, si $N=6$.

Resumen. Sea $\Theta$ un flujo compacto minimal. Es decir para cada elemento del espacio de fases $\theta \in \Theta$ y $t\in \mathbb{R}$ tenemos definida la evolución $\theta \cdot t$ verificando $\theta \cdot 0= \theta $ y $\theta \cdot (t+s))=(\theta \cdot t) \cdot s$. Se va a suponer que dicho flujo actúa sobre los coeficientes de una ecuación lineal mediante unas aplicaciones $A: \Theta \to \mathbb{M}_n(\mathbb{R})$, $b: \Theta \to \mathbb{R}^n$, continuas. Es decir vamos a analizar ecuaciones diferenciales con un parámetro $\theta\in \Theta$ del tipo $$ x'=A(\theta\cdot t)x +b(\theta \cdot t), $$ en particular ecuaciones diferenciales lineales con coeficientes acotados. Se analizará la existencia de solución acotada así como la de solución representable que es un caso particular de solución acotada obtenida como $x(t)=X(\theta \cdot t)$ para una función $X: \Theta \to \mathbb{R}^n$ continua.

Resumen. We will discuss the existence of nonstationary collision-free periodic solutions for the $N$-vortex problem on general domains $\Omega\subset\mathbb{R}^2$. The problem in question is a first-order Hamiltonian system arising as a singular limit (not only) in two-dimensional fluid dynamics. The Hamiltonian contains logarithmic singularities and is except for some special domains not explicitly known. The aim of the talk is to present a superposition idea that combines two types of existing solutions -- stationary solutions on $\Omega$ and rigidly rotating configurations of the whole-plane system. This leads us to periodic solutions consisting of several clusters with arbitrarily many vortices.

Resumen. El objetivo de la charla es repasar algunos desarrollos recientes que tienen como ingrediente común algún tipo de operador de curvatura, dando lugar de forma natural a problemas de contorno para ecuaciones semilineales de tipo elíptico. En primer lugar, explicaremos la base de algunos trabajos orientados a determinar hipersuperficies con curvatura prescrita en espacio-tiempos de Friedman-Lemaitre-Robertson-Walker. Identificaremos problemas abiertos y señalaremos un posible paralelismo con ecuaciones de reacción-difusión con dominio variable en el tiempo. También presentaremos un modelo de crecimiento epitaxial que lleva a una ecuación con bi-laplaciano y operador k-hessiano.

Resumen. Voy a describir el comportamiento de las normas de Lebesgue y Sobolev de las soluciones al problema de Cauchy para un sistema de Stokes con "drift". El drift se supone que tiene divergencia nula, es regular y satisface ciertas condiciones de invariancia (scale invariance). Trabajo en colaboración con G. Seregin.

Resumen. Las redes de regulación genética están formadas por una serie de genes que se transcriben en ARN mensajero que a su vez se traduce en proteínas (dogma central de la biología molecular). Las proteínas producidas también pueden intervenir en la regulación de los genes activándolos o inhibiéndolos. Normalmente, estos sistemas tienen un comportamiento de naturaleza estocástica, debido sobre todo, al bajo número de copias en las especies que intervienen. Habitualmente se usan ecuaciones maestras para su modelado, que son un conjunto de ecuaciones diferenciales que describen la evolución temporal de la probabilidad de que cada especie esté en uno de los posibles estados (que pueden ser infinitos). Debido a la complejidad de su resolución surge la necesidad de utilizar algoritmos de simulaciones estocásticas con un alto coste computacional para obtener su solución. Otra alternativa es derivar modelos resolubles que aproximen las ecuaciones maestras como es el caso de las ecuaciones integro diferenciales (aproximación continua de las ecuaciones maestras). Estas ecuaciones describen la evolución temporal de la función de densidad de probabilidad de la cantidad de proteínas existentes en el sistema. Los modelos integro diferenciales admiten una solución estacionaria analítica para redes en las que solamente interviene un gen expresando un tipo de proteína (1D). Mientras que para casos más complejos, en los que están involucrados más de un gen expresando proteínas (nD), su solución se obtiene numéricamente. Finalmente, se realiza un estudio de su convergencia hacia el estado de equilibrio. En este sentido usando técnicas que entropía relativa, se ha llegado a probar matemáticamente que la convergencia de las ecuaciones integro diferenciales es exponencial en el caso 1D. Además, para el caso general $n$-dimensional hay evidencias numéricas de que esta propiedad se conserva.

Presentación disponible aquí

Resumen. This talk is concerned with a new kind of Riemann solvers for hyperbolic systems, which can be applied both in the conservative and nonconservative cases. In particular, the proposed schemes constitute a simple version of the classical Osher-Solomon Riemann solver (see [Osher-Solomon 1982]), and extend in some sense the schemes proposed in [Dumbser-Toro 2011]. The viscosity matrix of the numerical flux is constructed as a linear combination of functional evaluations of the Jacobian of the flux at several quadrature points. Some families of functions have been proposed to this end: Chebyshev polynomials and rational-type functions (see Castro-Gallardo-Marquina 2014). The schemes have been tested with different initial value Riemann problems for ideal gas dynamics, magnetohydrodynamics, and multilayer shallow water equations. The numerical tests indicate that the proposed schemes are robust, stable, and accurate with a satisfactory time step restriction, and provide an efficient alternative for approximating time-dependent solutions in which the spectral decomposition is computationally expensive.

Slides available here

Resumen. Measure differential equations (MDE), or differential equations with measures as coefficients, are used to describe jump or discontinuity phenomena. In this talk, by taking the second-order linear MDE as an example, I will first explain how solutions are defined. Then we will introduce some very recent results on the eigenvalue theory. It will be shown that the structure of weighted eigenvalues of MDE may be completely different from that for ODE.

Slides are available here

Resumen. We develop an analogue of Aubry-Mather theory for a class of dissipative systems, namely conformally symplectic systems, and prove the existence of interesting invariant sets, which, in analogy with the conservative case, will be called Aubry and Mather sets. Beside describing their structure and their dynamical significance, we shall analyze their attracting properties, as well as their role in driving the asymptotic dynamics of the system.

Resumen. Existence of entire radial solutions of radial elliptic PDEs can be investigated by the use of invariant manifold theory. By the introduction of the Fowler transformation we can obtain a nonautonomous dynamical system having a saddle-type equilibrium at the origin. The existence of homoclinic trajectories is strictly related to the existence of regular fast decay solutions of the elliptic PDE. Different asymptotic behaviors of the nonlinearity ruling the PDE, provide different dynamics. Some further applications are possible in the presence of Hardy potentials and for $p$-Laplace equation.

Resumen. In this talk we will present a sharp fractional Moser-Trudinger type inequality on an interval $I \subseteq \mathbb{R}$ and prove the existence of critical points of the corresponding functional. Exploiting a technique by Ruf, we will show a fractional Moser-Trudinger inequality on the whole $\mathbb{R}$. We will also discuss some recent results by Parini and Ruf on a fractional Moser-Trudinger type inequality in the setting of Sobolev-Slobodeckij spaces in dimension one, pushing further their analysis by considering the inequality on the whole $\mathbb{R}$ and giving an answer to one of their open questions.

Resumen. In this talk we consider an elliptic semilinear problem under overdetermined boundary conditions: the solution vanishes at the boundary and the normal derivative is constant. These problems appear in many contexts, particularly in the study of free boundaries and obstacle problems. Here the task is to understand for which domains (called extremal domains) we may have a solution. This question has shown a certain parallelism with the theory of constant mean curvature surfaces, and also with the well-known De Giorgi conjecture. The case of bounded extremal domains was completely solved by J. Serrin in 1971, and the ball is the unique such domain. Instead, the case of unbounded domains is far from being completely understood. In this talk we give a rigidity result in dimension 2, and also a construction of a nontrivial extremal domain.

Resumen. For a bounded domain $\Omega$, a bounded Carathéodory function $g$ in $\Omega\times\mathbb{R}$, $p>1$, and a nonnegative measurable function $h$ in $\Omega$ which is strictly positive in a set of positive measure, we prove that, contrarily with the case $h\equiv 0$, there exists a solution of the semilinear elliptic problem $$ \left \{ \begin{array}{rcll} -\Delta u & = & \lambda u +g(x,u)- h |u|^{p-1} u +f, & \mbox{in } \Omega \\ u & = & 0, & \mbox{on } \partial\Omega,\\ \end{array} \right. $$ for every $\lambda\in \mathbb{R}$ and $f\in\ L^2(\Omega)$.

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In this talk, we consider evolution equations possessing a gradient structure, that is they are the gradient flow of an energy functional with respect to some metric. We will introduce a variational framework, which allows to pass to the limit from one gradient structure to another.

In particular, we will apply the method to gradient structures of a discrete coagulation-fragmentation model, the Becker-Döring equation, and its macroscopic limit. We show that the convergence result obtained by Niethammer (J. Nonlinear Sci.) can be extended to prove the convergence not only for solutions of the Becker-Döring equation towards the Lifshitz-Slyozov-Wagner equation of coarsening, but also the convergence of the associated gradient structures.

Furthermore, we will discuss the role of well-prepared initial data for the convergence statement and its relation to the relaxation of solutions of the Becker-Döring equation towards a quasistationary distribution dictated by the monomer concentration on the considered time-scale.

(arXiv: 1607.08735)

Slides are available here.

Resumen. We will introduce some of the mathematical modeling tools that have been used in the field of Developmental Biology, focusing on specific problems in embryogenesis. The use of multiscale models based on a combination of ordinary and partial differential equations is a well-established research paradigm in this area by now. After reviewing some of the past and present contributions, we will discuss both their merits and shortcomings in the light of recent experimental results.

Resumen. If a force distribution $f$ is applied to a plate fixed at its boundary, it experiences a deflection $u$, which is understood as the height of each point on the plate relative to the initial position. This deflection can be modeled by the following fourth-order elliptic problem:

$$\begin{cases} \Delta^2 u = f \quad & \text{en $\Omega$,}\\ u = \partial_\nu u = 0 \quad & \text{en $\partial \Omega$}, \end{cases}$$ where $\Delta^2 \equiv \Delta(\Delta)$ is the biharmonic operator, $\Omega$ is a planar domain, $\partial\Omega$ is its boundary, and $\partial_\nu u$ denotes the derivative of the function $u$ in the direction of the outward normal to the curve $\partial\Omega$. The problem of existence, uniqueness, and regularity of solutions of problem (1) is solved in the case where $f$ is an analytic real function [4].

Unlike second-order elliptic problems, there is no clear relationship between the sign of $f$ and the sign of $u,"""""this as a consequence of the maximum principle. Furthermore, domains $\Omega$, elliptical domains of high eccentricity can be found, where $u$ changes sign and presents local minima and maxima inside $\Omega$, even if $f$ is a non-negative and non-zero function in $\Omega" [6]. In some domains like the ball [2] and certain types of limaçons [3], $u$ preserves the sign of the data $f." Domains with this property will be known throughout the presentation as PPS domains, that is:

Definition (Sign-Preserving Property (PPS)). We will say that in problem (1), the domain $\Omega$ is PPS if $f \geq 0$ ($f \leq 0$) in $\Omega$ implies that $u \geq 0$ ($u \leq 0$) in $\Omega$.

The expression for the curvature of the level curve of a real function $w \in C^2(\Omega)$ is given by:

$$k(x) = \frac{H_\omega(x) \theta(x) \cdot \theta(x)}{|\nabla u(x)|},$$ where $H_\omega$ corresponds to the Hessian matrix of $\omega$ and $\theta(x)$ is the tangent direction to the level curve at $x". Note that if $\omega$ is the solution of problem (1), the condition $\partial_\nu w \big\vert_{\partial\Omega} = 0$ implies that the curvature function is not defined on the boundary $\partial\Omega$.

The objective of the talk is to prove that the curvature function (2) of the level curves of the solution $u$ of problem (1) can be extended continuously to the boundary $\partial\Omega$ in the case where $\Omega$ is a certain type of PPS domain and $f$ is an analytic real function in $\Omega$.

References

1. Arango, J., Gómez, A., & Salazar, A. (2014). Critical points and curvature in circular clamped plates. Electronic Journal of Differential Equations, 2014(218), 1-13.
2. T. Boggio. Sulle funzioni di Green d'ordinem. Rend. Circ. Mat.Palermo, 20:97-135, 1905.
3. A. Dall'Acqua and G. Sweers, The clamped-plate equation for the limacon, Ann. Mat. Pura Appl. (4) 184 (2005), no. 3, 361-374. MR 2164263 (2006i:35066)
4. Filippo Gazzola, Hans-Christoph Grunau and Guido Sweers. Polyharmonic Boundary Value Problems: Positivity Preserving and Nonlinear Higher Order Elliptic Equations in Bounded Domains. Springer, 1 edition, 2010.
5. Lawlor, G. R. (2012). A L'hospital's rule for multivariable functions. arXiv preprint arXiv:1209.0363.
6. H. Shapiro. T. Tegmark. An elementary proof that the biharmonic Green function of an eccentric ellipse changes sign. Soc. Ind. App. Math. Rev, 36:99–101, 1994.

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We will consider the density taking the constant value 2 (resp. 1) inside (resp. outside) a smooth domain. We will first explain how to propagate the regularity of the velocity in such a rough density case. Then we will show that the regularity of the domain can also be persisted. This is a free boundary problem and the analysis relies heavily on the (time-weighted) energy estimates. This is a joint work with Ping Zhang (Chinese Academy).

Resumen. In this talk we will present an existence result for periodic problems associated to singular perturbations of the relativistic acceleration. We will use a new continuation theorem together with recent strategies concerning nonlinearities having an indefinite weight. The main tool is the Leray - Schauder degree. Notice that, due to the weight, there is no a priori estimates in our problem. This is a joint work with Manuel Zamora.

Resumen. Linear parabolic PDEs in 1+1-dimension, in particular Fokker-Planck equations, arise in diverse areas such as diffusion processes, stochastic (Markov) processes, Brownian motion, probability theory, financial mathematics, population genetics, quantum chaos and others. The efficiency of Lie symmetry methods for constructing fundamental solutions (heat kernels) will be shown by way of examples. A new criteria for transformability to canonical forms with four- and six-dimensional finite symmetry groups will be presented. 2+1-dimensional problems will also be discussed.

Resumen. We study the Schrödinger equation $−\Delta u + V(x)u = f (x, u)$ in $\R^N$. We assume that $f$ is superlinear but of subcritical growth and $u → f (x, u)/|u|$ is nondecreasing. We also assume that $V$ and $f$ are periodic in $x_1, . . . , x_N$. We show that these equations have a ground state and that there exist infinitely many solutions if $f$ is odd in $u$.

Resumen. Along this talk we will consider classical solutions of the semilinear fractional problem $(-\Delta)^s u = f(u)$ in $\R^N_+$, where $(-\Delta)^s$, $0 < s < 1$, stands for the fractional Laplacian, $N \ge 2$, $\R^N_+ = \{x=(x',x_N)\in \R^N:\ x_N>0\}$ is the half-space and $f \in C^1$ is a given function. With no additional restriction on the function $f$, we show that bounded, nonnegative, nontrivial classical solutions are indeed positive in $\R^N_+$ and verify $\frac{\partial u}{\partial x_N}>0$ in $\R^N_+$. This is in contrast with previously known results for the local case $s=1$, where nonnegative solutions which are not positive do exist and the monotonicity property above is not known to hold in general even for positive solutions when $f(0)<0$.

This work is joint with L. Del Pezzo (UBA, Argentina), J. García-Melián (ULL) and A. Quaas (Universidad Técnica Federico Santa María, Chile).

Resumen. Consider a continuous family of planar differential systems with a center at $p$. The period function assigns to each periodic orbit in the period annulus its period. The problem of bifurcation of critical periodic orbits has been studied and there are three different situations to consider: bifurcations from the center, bifurcations from the interior of the period annulus and bifurcations from the outer boundary of the period annulus. In this talk we deal with the study of bifurcation of critical periodic orbits from the outer boundary for families of potential systems $X_{\mu}=-y\partial_x+V_{\mu}'(x)\partial_y$ where $\mu$ is a $d$-dimensional parameter. We introduce the notion of criticality as an analogous version of the ciclicity in the framework of limit cycles, and we give general criteria in order to bound the criticality at the outer boundary. That is, the maximum number of critical periodic orbits that can emerge or disappear from the outer boundary of the period annulus as we move the parameter $\mu$. This is a joint work with Francesc Mañosas and Jordi Villadelprat.

Resumen. While there are lots of contributions on Willmore surfaces in the three-dimensional Euclidean space, the literature on curved manifolds is still relatively limited. One of the main aspects of the Willmore problem is the loss of compactness under conformal transformations. We construct embedded Willmore tori in manifolds with a small area constraint by analyzing how the Willmore energy under the action of the Möbius group is affected by the curvature of the ambient manifold. The loss of compactness is then taken care of using minimization arguments or Morse theory.

Resumen. We analyze an elliptic logistic equation with nonlinear diffusion arising in population dynamics. We present results of existence and uniqueness of positive solutions, as well as about the profile of these solutions. Finally, we interpret the results obtained in terms of population dynamics.

Resumen. In this presentation, we will talk about the existence and uniqueness of positive solution for a nonlocal logistic equation arising from the birth-jump processes. We also will talk about the motivation to study this nonlocal equation and about a sub-super solution method to solve this equation.

Resumen. Repasaremos algunos modelos de comportamiento colectivo y veremos cómo extender estos para describir otros estados propios de los seres vivos, tanto en el caso discreto como continuo. Estudiaremos cómo estos parámetros adicionales afectan al comportamiento del grupo y su reacción ante agentes externos, así como la dependencia del comportamiento asintótico de las soluciones con la velocidad de propagación de la información.

Resumen. The growth-fragmentation is a PDE of the transport type with a nonlocal source term. It models "populations" in which the "individuals" grow with a deterministic rate and splits randomly. Such models appear in biology, physics, or telecommunications. This equation, in its linear version, admits a dominant Perron eigenvalue associated to a positive eigenfunction. This provides a particular solution to the equation, which attracts all the others. In this talk we are interested in the speed of the convergence. We prove that, depending on the coefficients, there can exist or not an exponential rate of convergence. The proofs rely on semigroup techniques.

Resumen. We present an existence result for a nonlinear parabolic problem of Cauchy-Dirichlet type having unbounded initial data and lower-order term which behaves as a power of the gradient. In particular, when the gradient growth is assumed to be superlinear, we show that there exists a link between this growth rate and the class of the data which allows one to have an existence result. We also deal with the sublinear growth rate in a particular case.

Resumen. In this talk, I will discuss some geometrical properties of positive solutions of semilinear elliptic partial differential equations in bounded convex domains or convex rings, with Dirichlet-type boundary conditions. A solution is called quasiconcave if its superlevel sets are convex. I will review some classical properties and positive results and I will present the main elementary steps of a counterexample, that is a case of semilinear elliptic equations for which the solutions are not quasiconcave. This talk is based on a joint work with N. Nadirashvili and Y. Sire.

Resumen. This talk is divided into two parts. First, we analyze two fourth-order problems - one with periodic conditions and another with simply supported conditions - allowing the nonlinearity to depend on $x$, $u(x)$, and $u''(x)$. In both cases, the fourth-order operator can be decomposed into two second-order operators, and using maximum principles, it is possible to prove the existence of a solution between lower and upper solutions (eventually in reversed order). The second part deals with the fourth-order operator $u^{(4)} + M u$ coupled with the clamped beam conditions, for which the approach used previously is not possible. We obtain the exact values of the real parameter $M$ for which this operator satisfies a maximum principle. When $M < 0$, we obtain the best estimate by means of spectral theory, and for $M > 0$, we attain the optimal value by studying the oscillation properties of the solutions of the homogeneous equation $u^{(4)} + M u = 0$. By using the method of lower and upper solutions, we deduce the existence of solutions for nonlinear problems (nonlinearity not depending on the derivatives) with these boundary conditions.

Resumen. In this work, we prove some abstract results about the existence of a minimizer for locally Lipschitz functionals, without any assumption of homogeneity, over a set which has its definition inspired in the Nehari manifold. As applications, we present a result of existence of ground state bounded variation solutions of problems involving the 1-Laplacian and the mean-curvature operator, where the nonlinearity satisfies mild assumptions.

Resumen. El MOSFET de doble puerta es un tipo de transistor muy común. La evolución tecnológica ha visto una constante reducción de su tamaño, desde los 10000 nm de los años setenta hasta los 14 nm del MOSFET más pequeño utilizado en la práctica. Nuestro objetivo es la simulación de un dispositivo de 10 nm utilizando un modelo muy preciso, a costa de que sea muy costosa computacionalmente. Para reducir los tiempos de cálculo, se está llevando a cabo una paralelización sobre tarjeta gráfica. Presentaremos los resultados hasta ahora conseguidos.

Resumen. Recently, A. Fonda and A.J. Ureña demonstrated a higher dimensional version of the Poincaré-Birkhoff theorem, proposing three alternative twist conditions. Following the spirit of similar results obtained for Poincaré-Miranda-like fixed point theorems, in this talk I present a new boundary condition, called avoiding cones condition, that unifies and extends the twist conditions for the Poincaré-Birkhoff Theorem previously proposed.

Resumen. We provide a multiplicity result for critical points of a functional defined on the product of a compact manifold and a convex set, by assuming an avoiding rays condition at the boundary of that set. We then extend this result to an infinite-dimensional setting.

Resumen. Se presentan problemas no lineales de Dirichlet en los que la presencia de un término de orden inferior mejora la regularidad de la solución (con respecto a problemas con términos de orden inferior nulos), ya sea en el caso de soluciones de energía finita como en el caso de soluciones de energía infinita.

Resumen. The existence of static, self-gravitating elastic bodies in the non-linear theory of elasticity is established. Equilibrium configurations of self-gravitating elastic bodies for small deformations of the relaxed state have been constructed previously by Being and Schmidt using the implicit function theorem. In this talk I will show how to construct static bodies for deformations with no size restriction. These solutions are obtained as minimizers of the energy functional of the elastic body. Joint work with Tommaso Leonori (Granada).

Resumen. Iteration of holomorphic maps in the complex plane has been an interesting piece of dynamical systems during the last 40 years. The main ideas introduced by Fatou and Julia around 1930 were (almost) forgotten for more than 40 years until some authors were attracted by the Mandelbrot set. Right after, some people start to work on the iteration of transcendental functions. In this talk I will concentrate in the transcendental entire case and the existence (and non-existence) of wandering domains (that is, domains of the Fatou set which are not eventually periodic). From the celebrated Sullivan's Theorem on the non-existence of those domains for rational maps, until recent results by C. Bishop on the existence of wandering domains in Eremenko-Lyubich class (a class of transcendental entire maps). I'll present the main results and partially discuss some of the key ingredients in the arguments of the proofs.

Resumen. Utilizando métodos de energía, se probarán acotaciones de decaimiento para soluciones acotadas e integrables del problema de evolución no local con una condición inicial no negativa. Como consecuencia de este resultado, se dará una estimación del decaimiento en normas en espacios de Orlicz de las soluciones. Trabajo en colaboración con Uriel Kaufmann y Julio Rossi.

Resumen. El teorema de Poincaré-Birkhoff dice que un homeomorfismo del anillo plano que conserve áreas y orientación, y que rote los círculos frontera en sentidos opuestos ha de tener al menos dos puntos fijos. Proponemos una posible generalización de este resultado para cualquier número par de dimensiones. En esta generalización el anillo pasa a ser el producto de un toro por el interior de una esfera embebida, y el conservar áreas y orientación se garantiza imponiendo que la aplicación pueda interpolarse por el flujo de un sistema Hamiltoniano. Esta charla está basada en un trabajo conjunto con A. Fonda (Università degli Studi de Trieste).

Resumen. En esta charla mostraremos cómo clasificar las soluciones a la ecuación de Liouville $\Delta v + 2K e^v =0$ en el semiplano $\R^2_+$ que cumplen las condiciones de Neumann $\frac{\partial v}{\partial t} = c_i e^{v/2} $, $i=1,2$ respectivamente en $\R^+$, $\R^-$. Este problema describe métricas conformes de curvatura constante $K$ en $\R^2_+$ tales que su curvatura geodésica es $-c_1/2$ a lo largo de $\R^+$ y $-c_2/2$ en $\R^-$. Describiremos las técnicas de análisis complejo necesarias para la demostración de los resultados y algunas aplicaciones de los mismos.

Resumen. El teorema de compensación, que fue introducido por Riemann, es un principio en el cálculo de variaciones que permite obtener información cualitativa sobre las soluciones de ciertas ecuaciones en derivadas parciales. Mostraremos cómo se puede utilizar este principio para estudiar la ecuación $-\Delta u = f$ en $\R^n$, en particular cuando $f$ es una medida, y cómo conduce a nuevas propiedades de las soluciones y a una nueva caracterización de los espacios de Morrey metrizados.

Resumen. The aim of the talk is twofold:

1. On the one hand, we aim to revisit the spectral analysis of semigroups in a general Banach space setting. We present some new and more general versions of classical results such as the spectral mapping theorem, (quantified) Weyl's Theorems and the Krein-Rutman Theorem. The results apply to a wide and natural class of generators which split as a dissipative part plus a more regular part. The approach relies on some factorization and summation arguments reminiscent of the Dyson-Phillips series.

2. On the other hand, we motivate and illustrate our abstract theory by evolution PDE applications. We will focus here on the application to the long time convergence to the equilibrium of solutions to classical, discrete and kinetic Fokker-Planck equations.

Resumen. I will discuss some results regarding qualitative properties of solutions to quasilinear elliptic equations in unbounded domains. Monotonicity and symmetry properties of positive solutions generally follow via the moving plane method. In the quasilinear case such technique is related to many technical difficulties caused by the nonlinear degenerate nature of the operator. I will present some new results in the case when the domain is the half space or the whole space.

Resumen.

Resumen. We show the existence of compactly supported global minimisers under almost optimal hypotheses for continuum models of particles interacting through a potential. The main assumption on the potential is that it is catastrophic, or not H-stable, which is the complementary assumption to that in classical results on thermodynamic limits in statistical mechanics. The proof is based on a uniform control on the local mass around each point of the support of a global minimiser, together with an estimate on the size of the “gaps” it may have. The class of potentials for which we prove existence of global minimisers includes power-law potentials and, for some range of parameters, Morse potentials, widely used in applications. This is a joint work with J. A. Cañizo and J. A. Carrillo.