El Seminario de Ecuaciones Diferenciales es una actividad organizada por miembros de los departamentos de Análisis Matemático y Matemática Aplicada de la Universidad de Granada. Se celebra habitualmente en el IEMath-GR y se reúne aproximadamente una vez cada dos semanas. Aquí puedes encontrar las charlas programadas y una lista de las que se han celebrado ya.

Preguntas y suscripción. Para preguntas, sugerencias de conferenciantes, o (de)suscripción a la lista de correo por favor contacta con María Medina o Sebastian Throm.

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Próxima charla

  1. Lunes 11 de febrero de 2019, 13:10.
    Slawomir Rybicki (Universidad de Torun)
    Symmetric Lyapunov Center Theorem.
    Seminario de la primera planta, IEMath-GR.
    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.

Charlas anteriores

  1. Lunes 28 de enero de 2019, 13:10.
    Pablo Ochoa (Universidad Nacional de Cuyo-CONICET)
    Fractional elliptic problems with nonlinear gradient sources and measures.
    Seminario de la primera planta, IEMath-GR.
    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.
  2. Lunes 17 de diciembre de 2018, 13:10.
    Sebastian Throm (Universidad de Granada)
    Dynamics of power networks.
    Seminario de la primera planta, IEMath-GR.
    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.
  3. Lunes 3 de diciembre de 2018, 13:10.
    Anna Gołębiewska (Universidad de Toruń)
    Degree for equivariant gradient operators.
    Seminario de la primera planta, IEMath-GR.
    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.
  4. Lunes 26 de noviembre de 2018, 13:10.
    Markus Schmidtchen (Imperial College London)
    Pattern Formation in Cross-Interaction Systems.
    Seminario de la primera planta, IEMath-GR.
    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.
  5. Lunes 12 de noviembre de 2018, 13:10.
    María Medina (Universidad de Granada)
    A first example of nondegenerate solution for the Yamabe problem with maximal rank.
    Seminario de la primera planta, IEMath-GR.
    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.
  6. Lunes 15 de octubre de 2018, 13:10.
    Bastien Polizzi (Université de Lyon 1)
    Mixtures models for phototrophic biofilms and gut microbiota ecology.
    Seminario de la primera planta, IEMath-GR.
    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.
  7. Lunes 24 de septiembre de 2018, 13:10.
    Wen-Xin Qin (Soochow University)
    Birkhoff and Non-Birkhoff Solutions for Monotone Recurrence Relations.
    Seminario de la primera planta, IEMath-GR.
    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.
  8. Viernes 8 de junio de 2018, 12:00.
    Vajiheh Vafaei (University of Tabriz, Tabriz-Iran)
    Fractional Calculus.
    Seminario de la segunda planta, IEMath-GR.
    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 in- terdisciplinary fields can be accurately modelled by fractional-order differential equations, such as viscoelastic systems, dielectric polariza- tion, quantitative finance, nonlinear oscillation of earthquakes, robotic manipulating systems, muscular blood vessel model, hydrologic models and so on. In this talk, I will discuss about 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.
  9. Lunes 4 de junio de 2018, 13:00.
    Lucio Boccardo ("Sapienza" Università di Roma)
    Existence and properties of saddle points of some integral functionals defined in $W_{0}^{1,2}(\Omega) \times W_{0}^{1,2}(\Omega)$.
    Sala de Conferencias de la Facultad de Ciencias.
    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}
  10. Viernes 25 de mayo de 2018, 12:00.
    José Miguel Mendoza Aranda (Universidade Federal de São Carlos)
    Local Coercivity for semilinear elliptic problems.
    Seminario de la primera planta, IEMath-GR.
    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.
  11. Viernes 18 de mayo de 2018, 12:00.
    Umida Baltaeva (National University of Uzbekistan)
    Boundary value problems for a third-order loaded differential equation with the parabolic-hyperbolic operator.
    Seminario de la primera planta, IEMath-GR.
    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.
  12. Lunes 14 de mayo de 2018, 13:10.
    Pilar Guerrero (University College London)
    Coarse-graining and hybrid methods for efficient simulation of stochastic multi-scale models of tumour growth.
    Seminario de la primera planta, IEMath-GR.
    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.
  13. Viernes 11 de mayo de 2018, 12:00.
    Faruk Güngör (Istanbul Technical University)
    Superintegrability with higher order Painlevé transcendent potentials.
    Seminario de la segunda planta, IEMath-GR.
    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 adnitting higher-order potentials. The main emphasis will be on the emergence of the Painlevé transcendent potentials in the quantum case.
  14. Viernes 20 de abril de 2018, 12:00.
    David Poyato (Universidad de Granada)
    El modelo de Kuramoto con pesos singulares: sincronización, clustering y modelos macroscópicos.
    Seminario de la segunda planta, IEMath-GR.

    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.

  15. Lunes 16 de abril de 2018, 12:00.
    Antonio Tineo (Universidad de Los Andes)
    Un sistema depredador presa tridiagonal.
    Seminario de la primera planta, IEMath-GR.
    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.
  16. Viernes 6 de abril de 2018, 12:00.
    Teresa E. Pérez (Universidad de Granada)
    Clásicos ortogonales.
    Seminario de la primera planta, IEMath-GR.
    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.
  17. Jueves 22 de febrero de 2018, 12:00.
    Pedro J. Martínez Aparicio (Universidad Politécnica de Cartagena)
    Ecuaciones elípticas casilineales con singularidad en el término de orden inferior.
  18. Lunes 12 de febrero de 2018, 11:00.
    Cristian Bereanu (Universidad de Bucarest)
    A variational approach for the Neumann problem in some FLRW spacetimes.
    Seminario de la primera planta, IEMath-GR.
    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.
  19. Viernes 9 de febrero de 2018, 12:00.
    Óscar Sánchez Romero (Universidad de Granada)
    Modelado de dispersión mediante limitadores de flujo no lineales.
    Seminario de la segunda planta, IEMath-GR.
    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.
  20. Domingo 28 de enero de 2018, 13:00.
    Salvador López (Universidad de Granada)
    Resultados óptimos de existencia y unicidad de solución para ecuaciones casilineales singulares.
    Seminario de la primera planta, IEMath-GR.
    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.
  21. Jueves 11 de enero de 2018, 10:00.
    José M. Mazón (Universidad de Valencia)
    Las desigualdades de Kurdyka-Lojasiewicz-Simon para flujos gradiente en espacios métricos..
    Sala de conferencias de la Facultad de Ciencias.
    Dentro del ciclo "One Day Partial Differential Equations" organizado por David Arcoya y José Carmona
    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.
  22. Jueves 11 de enero de 2018, 12:00.
    Denis Bonheure (Université Libre de Bruxelles)
    From nonlinear electrodynamics to series of p-Laplacians and regularity theory for non-uniformly elliptic operators.
    Sala de conferencias de la Facultad de Ciencias.
    Dentro del ciclo "One Day Partial Differential Equations" organizado por David Arcoya y José Carmona
    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.
  23. Lunes 11 de diciembre de 2017, 13:00.
    Juan Casado (Universidad de Sevilla)
    Un problema elíptico no lineal con un segundo miembro singular que puede cambiar de signo.
    Sala de conferencias de la Facultad de Ciencias.
    El horario de esta charla se ha retrasado debido a problemas en el transporte causados por la lluvia.
    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.
  24. Lunes 27 de noviembre de 2017, 13:10.
    Jo Evans (University of Cambridge)
    Convergence to equilibrium in relative entropy for the linear relaxation Boltzmann equation.
    Seminario de la primera planta, IEMath-GR.
    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.
  25. Lunes 20 de noviembre de 2017, 13:10.
    Eduardo García Juárez (Universidad de Sevilla)
    Regularidad global para interfases de fluidos incompresibles.
    Seminario de la primera planta, IEMath-GR.
    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).

  26. Lunes 23 de octubre de 2017, 13:10.
    Salvador Villegas Barranco (Universidad de Granada)
    Acotación de las soluciones extremales en dimensión 4.
    Seminario de la primera planta, IEMath-GR.
    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 $N=6$.
  27. Lunes 16 de octubre de 2017, 13:10.
    Juan Campos Rodríguez (Universidad de Granada)
    Ecuaciones diferenciales lineales en un mundo oscilante.
    Seminario de la primera planta, IEMath-GR.
    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.
  28. Martes 19 de septiembre de 2017, 13:10.
    Björn Gebhard (Justus-Liebig-Universität Giessen)
    The $N$-vortex problem -- Periodic solutions consisting of clusters.
    Seminario de la primera planta, IEMath-GR.
    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 explicitely 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.
  29. Martes 27 de junio de 2017, 13:10.
    Pedro Torres Villarroya (Universidad de Granada)
    Los operadores de curvatura como fuente de problemas en Análisis No Lineal.
    Seminario de la primera planta, IEMath-GR.
    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.
  30. Martes 16 de mayo de 2017, 13:10.
    Maria Schonbek (University of California Santa Cruz)
    Decaimiento de soluciones de las ecuaciones de Stokes con "drift".
    Seminario de la primera planta, IEMath-GR.
    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.
  31. Martes 9 de mayo de 2017, 13:10.
    Manuel Pájaro Diéguez (Instituto de Investigaciones Marinas, CSIC)
    Estudio de la convergencia asintótica de ecuaciones integro diferenciales usadas en el modelado de redes de regulación genética.
    Seminario de la primera planta, IEMath-GR.
    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
  32. Jueves 20 de abril de 2017, 13:10.
    Manuel J. Castro Díaz (Universidad de Málaga)
    Approximate Osher-Solomon Schemes for hyperbolic systems.
    Seminario de la primera planta, IEMath-GR.
    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
  33. Martes 18 de abril de 2017, 13:10.
    Meirong Zhang (Tsinghua University)
    Solutions and eigenvalues of measure differential equations.
    Seminario de la primera planta, IEMath-GR.
    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 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.
  34. Martes 4 de abril de 2017, 13:10.
    Stefano Marò (ICMAT Madrid)
    Aubry-Mather theory for conformally symplectic systems.
    Seminario de la primera planta, IEMath-GR.
    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.
  35. Martes 21 de marzo de 2017, 13:10.
    Andrea Sfecci (Università Politecnica delle Marche (Ancona))
    Dynamics of radial elliptic PDEs: the Fowler transformation.
    Seminario de la primera planta, IEMath-GR.
    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 application are possible in presence of Hardy potentials and for $p$-Laplace equation.
  36. Jueves 23 de febrero de 2017, 13:10.
    Stefano Iula (Universität Basel)
    Fractional Moser-Trudinger type inequalities in one dimension.
    Seminario de la primera planta, IEMath-GR.
    (Notar el día inusual)
    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.
  37. Martes 7 de febrero de 2017, 13:10.
    David Ruiz (Universidad de Granada)
    Some results on overdetermined elliptic problems.
    Seminario de la primera planta, IEMath-GR.

    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.

  38. Martes 17 de enero de 2017, 13:10.
    José Miguel Mendoza Aranda (Universidade Federal de São Carlos)
    Existence of solutions for a nonhomogeneous semilinear elliptic equation.
    Seminario de la primera planta, IEMath-GR.
    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 $f\in\ L^2(\Omega)$.
  39. Martes 10 de enero de 2017, 13:10.
    André Schlichting (Institute for Applied Mathematics, University of Bonn)
    Variational formulation and limits of evolution equations possesing a gradient structure.
    Seminario de la primera planta, IEMath-GR.

    In this talk, we consider evolution equations possesing 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.

  40. Martes 13 de diciembre de 2016, 13:10.
    Juan Calvo Yagüe (Universidad de Granada)
    Mathematical models in Developmental Biology.
    Seminario de la primera planta, IEMath-GR.
    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.
  41. Lunes 5 de diciembre de 2016, 13:10.
    Andrés Mauricio Salazar Rojas (Pontificia Universidad Javeriana Cali–Colombia)
    Curvatura en placas empotradas.
    Seminario de la primera planta, IEMath-GR.
    Resumen. Si se aplica una distribución de fuerzas $f$ sobre una placa sujeta por su borde esta experimenta una deflexión $u$, que se entiende como la altura de cada punto de la placa con respecto de la posición inicial. Dicha deflexión se puede modelar mediante el siguiente problema elı́ptico de cuarto orden: $$ \begin{cases} \Delta^2 u = f \quad & \text{en $\Omega$,}\\ u = \partial_\nu u = 0 \quad & \text{en $\partial \Omega$}, \end{cases} $$ donde $\Delta^2 \equiv \Delta(\Delta)$ es el operador biarmónico, $\Omega$ es un dominio planar, $\partial\Omega$ es su frontera y $\partial_\nu u$ denota la derivada de la función $u$ en la dirección de la normal exterior a la curva $\partial\Omega$. El problema de existencia, unicidad y regularidad de soluciones del problema (1) está resuelto en el caso en que $f$ sea una función real analı́tica [4].

    A diferencia de los problemas elı́pticos de segundo orden, no existe una clara relación entre el signo de $f$ y el signo de $u$, esto como una consecuencia del principio del máximo. Más aún se pueden encontrar dominios $\Omega$, elı́pticos de gran excentricidad, en donde $u$ cambia de signo y presenta mı́nimos y máximos locales al interior de $\Omega$, aunque $f$ sea una función no negativa y no nula en $\Omega$ [6]. En algunos dominios como la bola [2] y ciertos tipos de limaçones [3], $u$ preserva el signo del dato $f$. Dominios con esta propiedad serán conocidos a lo largo de la presentación como dominios PPS, esto es:

    Definición (Propiedad de Preservar Signo (P P S)). Diremos que en el problema (1) el dominio $\Omega$ es PPS, si $f \geq 0$ ($f \leq 0$) en $\Omega$ implica que $u \geq 0$, ($u \leq 0$) en $\Omega$.

    La expresión para la curvatura de la curva de nivel de una función real $w \in C^2(\Omega)$ viene dada por: $$ k(x) = \frac{H_\omega(x) \theta(x) \cdot \theta(x)}{|\nabla u(x)|}, $$ en donde $H_\omega$ corresponde con la matriz Hessiana de $\omega$ y $\theta(x)$ es la dirección tangente a la curva de nivel en $x$. Note que si $\omega$ es la solución del problema (1) la condición $\partial_\nu w \big\vert_{\partial\Omega} = 0$ implica que la función curvatura no esta definida en la frontera $\partial\Omega$.

    El objetivo de la charla es probar que la función curvatura (2) de las curvas del nivel de la solución $u$ del problema (1) se puede extender de manera continua a la frontera $\partial\Omega$ en el caso en que $\Omega$ sea cierto tipo de dominios PPS y $f$ sea una función real analı́tica en $\Omega$.


    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.
  42. Jueves 1 de diciembre de 2016, 11:00.
    Jinyeong Park (Universidad de Granada)
    Emergence of synchronization in the Kuramoto model.
    Sala de Conferencias FisyMat.
  43. Miércoles 30 de noviembre de 2016, 12:00.
    Enrique Fernández Nieto (Universidad de Sevilla)
    Modelos bifásicos en simulación de avalanchas aéreas y submarinas.
    Sala de Conferencias FisyMat.
  44. Martes 15 de noviembre de 2016, 13:10.
    Xian Liao (Universität Bonn)
    Global regularity of the density patch problem for two dimensional inhomogeneous incompressible flow.
    Seminario de la primera planta, IEMath-GR.
    Resumen. 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 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).
  45. Martes 8 de noviembre de 2016, 13:10.
    Cristian Bereanu (University of Bucarest)
    Periodic solutions for some singular perturbations with weight of the relativistic acceleration.
    Seminario de la primera planta, IEMath-GR.
    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 joit work with Manuel Zamora.
  46. Martes 25 de octubre de 2016, 13:00.
    Faruk Güngör (Istanbul Technical University)
    Construction of Heat Kernels by Lie Symmetry Group Methods.
    Seminario de la primera planta, IEMath-GR.
    (Día y hora no definitivos)
    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.
  47. Jueves 20 de octubre de 2016, 13:00.
    Francisco Odair Vieira de Paiva (Universidade Federal de São Carlos)
    Generalized Nehari manifold and semilinear Schrödinger equation.
    Seminario de Matemáticas, (junto al Dep. de Análisis Matemático, primera planta, edificio de matemáticas, Facultad de Ciencias).
    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 $u$.
  48. Martes 11 de octubre de 2016, 13:00.
    Begoña Barrios Barrera (Universidad de la Laguna)
    Monotonicity of solutions for some nonlocal elliptic problems in half-spaces.
    Seminario de la primera planta, IEMath-GR.
    Resumen. Along this talk we will consider classical solutions of the semilinear fractional problem $$\left\{ \begin{array}{ll} (-\Delta)^s u = f(u) & \hbox{in }\R^N_+,\\[0.35pc] \ \ u=0 & \hbox{on } \partial \R^N_+, \end{array} \right.$$ 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 $$ \frac{\partial u}{\partial x_N}>0 \quad \hbox{in } \R^N_+. $$ 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 $f(0)<0$ (see for instance [1,2,3]).

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


    [1] H. Berestycki, L. Caffarelli, L. Nirenberg, Further qualitative properties for elliptic equations in unbounded domains. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997), 69--94.

    [2] Cortázar, M. Elgueta, J. García-Melián, Nonnegative solutions of semilinear elliptic equations in half-spaces, J. Math. Pures Appl. (2016), in press.

    [3] A. Farina, B. Sciunzi, Qualitative properties and classification of nonnegative solutions to $-\Delta u = f(u)$ in unbounded domains when $f(0) < 0$, Rev. Mat. Iberoam. (2016), in press.

  49. Martes 27 de septiembre de 2016, 12:45.
    David Rojas (Universitat Autònoma de Barcelona)
    Analytical tools to study the criticality at the outer boundary of potential centers.
    Seminario de la primera planta, IEMath-GR.
    Día y hora por determinar
    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.
  50. Martes 24 de mayo de 2016, 10:00.
    Andrea Malchiodi (Escuela Normal Superior de Pisa)
    Embedded Willmore tori in three-manifolds with small area constraint.
    Sala de conferencias, IEMath-GR.
    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.
  51. Martes 10 de mayo de 2016, 12:45.
    Willian Cintra da Silva (Universidade Federal do Pará)
    Refuge versus dipersion in the logistic equation.
    Seminario de la primera planta, IEMath-GR.
    Resumen. We analyse 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.
  52. Martes 10 de mayo de 2016, 13:15.
    Ítalo Bruno Mendes Duarte (Universidade Federal do Pará)
    Nonlocal Problem Arising From the Birth-Jump Processes..
    Seminario de la primera planta, IEMath-GR.
    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.
  53. Martes 26 de abril de 2016, 12:45.
    Jesús Rosado (Universidad de Buenos Aires)
    Contagio emocional y comportamiento colectivo.
    Seminario de la primera planta, IEMath-GR.
    Resumen. Repasaremos algunos modelos de comportamiento colectivo y veremos como extender estos para describir otros estados propios de los seres vivos, tanto en el caso discreto como continuo. Estudiaremos como 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.
  54. Martes 19 de abril de 2016, 12:45.
    Pierre Gabriel (Université de Versailles)
    Convergence to the equilibrium for the growth-fragmentation equation.
    Seminario de la primera planta, IEMath-GR.
    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 exists or not an exponential rate of convergence. The proofs rely on semigroup techniques.
  55. Martes 5 de abril de 2016, 12:45.
    Martina Magliocca (Università di Roma "Tor Vergata")
    Existence results for a parabolic problem with nonlinear reaction term of first order.
    Seminario de la primera planta, IEMath-GR.
    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.
  56. Martes 15 de marzo de 2016, 12:45.
    François Hamel (Institut de Mathématiques de Marseille)
    Do positive solutions of elliptic PDEs in convex domains have convex level sets?.
    Seminario de la primera planta, IEMath-GR.
    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.
  57. Martes 8 de marzo de 2016, 12:45.
    Ricardo Roque Enguiça (Instituto Superior de Ingeniería de Lisboa)
    Some considerations on fourth order bvps in bounded intervals.
    Seminario de la primera planta, IEMath-GR.
    Resumen. This talk is divided in two parts. First we analyse 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 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 on the real parameter $M$ for which this operator satisfies a maximum principle. When $M < 0$ we obtain the best estimate by means of the 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 this boundary conditions.
  58. Viernes 19 de febrero de 2016, 12:45.
    Giovany Malcher Figueiredo (Universidade Federal do Pará)
    On ground states of elliptic problems and Nehari's method.
    Seminario del Departamento de Análisis Matemático.
    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 satisfy mild assumptions.
  59. Martes 16 de febrero de 2016, 12:45.
    Francesco Vecil (Université Blaise Pascal, Clermont-Ferrand)
    Implementación en plataforma de altas prestaciones de un resolvedor para MOSFETs de doble puerta.
    Seminario de la primera planta, IEMath-GR.
    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 más pequeño MOSFET utilizado en la práctica. Nuestro objetivo es la simulación 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.
  60. Jueves 11 de febrero de 2016, 12:15.
    Paolo Gidoni (SISSA)
    Twist conditions for a higher dimensional Poincaré-Birkhoff Theorem: an avoiding cones formulation.
    Seminario de la primera planta, IEMath-GR.
    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 extend the twist conditions for the Poincaré-Birkhoff Theorem previously proposed.
  61. Jueves 11 de febrero de 2016, 13:00.
    Alessandro Fonda (Università di Trieste)
    Generalizing the Lusternik-Schnirelmann critical point theorem.
    Seminario de la primera planta, IEMath-GR.
    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.
  62. Martes 2 de febrero de 2016, 12:45.
    Lucio Boccardo ("Sapienza" Università di Roma)
    Efecto regularizador de los términos de orden inferior en problemas elípticos no lineales.
    Seminario de la primera planta, IEMath-GR.
    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.
  63. Jueves 28 de enero de 2016, 13:00.
    Simone Calogero (Chalmers, Göteborgs Universitet)
    Ground states of self-gravitating elastic bodies.
    Sala de Conferencias FisyMat.
    (En colaboración con FisyMat)
    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).
  64. Miércoles 13 de enero de 2016, 12:45.
    Xavier Jarque (Universitat de Barcelona)
    Wandering domains in holomorphic dynamics.
    Seminario de la primera planta, IEMath-GR.
    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.
  65. Martes 15 de diciembre de 2015, 12:45.
    Raúl Emilio Vidal (Universidad de Córdoba)
    Acotaciones de decaimiento para un problema de evolución no local en espacios Orlicz.
    Seminario de la primera planta, IEMath-GR.

    Utilizando metodos de energía, ver [2] y [4], se probarán acotaciones de decaimiento de la forma \begin{equation} \int_{\mathbb{R}^d} \phi(u(x,t)) \,dx \leq C t^{-\mu} \end{equation} para soluciones u acotadas e integrables del problema de evolución no local con una condición inicial no negativa \begin{equation} u_t(x,t) = \int_{\mathbb{R}^d} J(x,y) G( u(y,t) - u(x,t)) (u(y,t) - u(x,t)) \,dy + f(u(x,t)), \end{equation} donde $G$ es una función no negativa e impar, $J$ es un núcleo no negativo y simétrico. $f$ es un función impar que verifica $f(\xi)\xi \leq 0$ para todo $\xi \geq 0$. La funcion $\phi$ y el exponente $\mu$ dependen de $G$ bajo hipótesis adecuadas.

    Notar que $G$ no se supone homogénea.

    Como consecuencia de este resultado podemos dar ademas una estimacion del decaimiento en normas en espacios de Orlicz de las soluciones.

    Por otro lado, si consideramos $G(\xi) = |\xi|^{p-2}$ nuestros resultados generalizan los obtenidos en [4] al no imponer restricciones sobre $p$.

    Trabajo en colaboracion con Uriel Kaufmann y Julio Rossi.


    [1] F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi, and J. Toledo-Merelo. The limit as $p \to \infty$ in a nonlocal $p$−Laplacian evolution equation: a nonlocal approximation of a model for sandpiles. Calculus of Variations and Partial Differential Equations, 35(3), (2009) 279-316.

    [2] F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. J. Toledo-Melero. Nonlocal Diffusion Problems. Amer. Math. Soc. Mathematical Surveys and Monographs 2010. Vol. 165.

    [3] E. Chasseigne, M. Chaves and J. D. Rossi, Asymptotic behavior for nonlocal diffusion equations, Journal de mathématiques pures et appliquées, 86(3), (2006), 271-291.

    [4] L. Ignat, J. D. Rossi, , J. Math. Pures Appl. 92 (2009), 163–187.

    [5] U. Kaufmann, J. D. Rossi and R. E. Vidal, Decay bounds for nonlocal evolution equation in Orlicz spaces, Annals of Functional Analysis, Duke University Press.

  66. Martes 1 de diciembre de 2015, 12:45.
    Antonio J. Ureña (Universidad de Granada)
    El teorema de Poincaré-Birkhoff en muchas dimensiones para sistemas hamiltonianos.
    Seminario de la primera planta, IEMath-GR.
    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).
  67. Martes 17 de noviembre de 2015, 12:45.
    Asun Jiménez Grande (Universidade Federal Fluminense)
    Un problema de Neumann geométrico para la ecuación de Liouville con singularidades en la frontera.
    Seminario de la primera planta, IEMath-GR.
    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.
  68. Martes 3 de noviembre de 2015, 12:30.
    Jean Van Schaftingen (Université Catholique de Louvain La Neuve)
    Nodal solutions for the Choquard equation.
    Seminario de la primera planta, IEMath-GR.
    Resumen. In this talk, we shall consider the Choquard equation, also known as Schrödinger−Newton and Hartree equation. The goal will be to construct the simplest solutions beyond groundstates. In contrast with the nonlinear Schrödinger equation, this equation admits least action odd solutions and least action nodal solutions. The construction are based on a Palais-Smale condition under a strict inequality condition and a new minimax characterization of minimal action nodal solutions. This is joint work with Marco Ghimenti (Pisa) and Vitaly Moroz (Swansea).
  69. Martes 27 de octubre de 2015, 12:30.
    Stéphane Mischler (Université Paris-Dauphine)
    Spectral analysis of semigroups in Banach spaces and Fokker-Planck equations.
    Seminario de la primera planta, IEMath-GR.
    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.
  70. Martes 20 de octubre de 2015, 12:30.
    Berardino Sciunzi (Università della Calabria)
    Qualitative properties of solutions to quasilinear elliptic equations in unbounded domains.
    Seminario de la primera planta, IEMath-GR.
    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.
  71. Jueves 1 de octubre de 2015, 12:30.
    José Luis Bravo (Universidad de Extremadura)
    El problema del centro-foco y las integrales abelianas cero-dimensionales.
    Seminario de la primera planta, IEMath-GR.
  72. Sábado 19 de septiembre de 2015, 12:30.
    Francesco Patacchini (Imperial College London)
    Existence of compactly supported global minimisers for the interaction energy.
    Seminario de la primera planta, IEMath-GR.
    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.