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. Para preguntas o sugerencias de conferenciantes por favor contacta con José Cañizo o Rafael López Soriano.

Suscripción. Normalmente enviamos dos anuncios antes de cada charla. Para recibirlos en tu email (o dejar de recibirlos) por favor escribe a Rafael López Soriano.

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Próxima charla

  1. 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.

Charlas anteriores

  1. 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.
  2. 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
  3. 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
  4. 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.
  5. 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.
  6. 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.
  7. 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.
  8. 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.
    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.

  9. 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 $f\in\ L^2(\Omega)$.
  10. 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.
    Resumen.

    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.

  11. 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.
  12. 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 $\Omega$.

    Referencias

    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.
  13. Jueves 1 de diciembre de 2016, 11:00.
    Jinyeong Park (Universidad de Granada)
    Emergence of synchronization in the Kuramoto model.
    Sala de Conferencias FisyMat.
  14. 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.
  15. 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).
  16. 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.
  17. 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.
  18. 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 $u$.
  19. 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).

    Referencias

    [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.

  20. 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.
  21. 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.
  22. 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.
  23. 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.
  24. 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.
  25. 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.
  26. 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.
  27. 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.
  28. 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.
  29. 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.
  30. 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.
  31. 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.
  32. 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.
  33. 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.
  34. 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).
  35. 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.
  36. 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.
    Resumen.

    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.

    Referencias

    [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.

  37. 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).
  38. 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.
  39. 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).
  40. 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.
  41. 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.
  42. 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.
  43. 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.