Juan Pablo Agnelli  (FaMAF, UNC)

Numerical solutions to minimal/maximal operators iterating obstacle problems.

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Gaston Beltritti  (Universidad Nacional de Rio Cuarto)

Aproximación no-local en tiempo y espacio de EDP's

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Pablo Blanc  (Departamento de Matemática, FCEyN - UBA)

Juegos del tipo Tug-of-War y EDPs

Tug-of-War es un juego estocastico de suma cero para dos jugadores. Inicialmente una ficha es colocada en un dominio acotado. En cada turno se lanza una moneda para decidir que jugador puede mover la ficha un paso de longitud epsilon. El juego termina cuando la ficha alcanza el borde del dominio, allí una función dada determina cuanto debe pagarle el jugador II al jugador I. En esta charla dicutiremos algunas versiones del juego y como se relacionan con las EDPs.

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Referencias
1. Peres, Yuval; Oded Schramm; Scott Sheffield; David Wilson. "Tug-of-war and the infinity Laplacian." Journal of the American Mathematical Society 22, no. 1 (2009): 167-210.

2. Manfredi, Juan J., Mikko Parviainen; Julio D. Rossi. "On the definition and properties of p-harmonious functions." Annali della Scuola Normale Superiore di Pisa-Classe di Scienze-Serie V 11.2 (2012): 215.

3. Blanc, Pablo; Juan P. Pinasco; Julio D. Rossi. “Maximal operators for the p-Laplacian family.” Pacific Journal of Mathematics 287 (2017), no. 2, 257–295

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João Vitor Da Silva (Departamento de Matemática, FCEyN - UBA)

 Regularity issues in nonlinear free boundary problems
 

In this talk we will study regularity issues for some nonlinear free boundary problems with p-Laplacian type structure. Some of the topics we will treat are related to dead core problems (elliptic and parabolic
scenery) and their asymptotic behavior as p diverges. Such themes of research are mathematically interesting because play a fundamental role in applied sciences since they appear in a number of chemical, physical and biological modeling processes. Throughout the talk we will focus our attention in establishing sharp and improved regularity estimates to weak solutions along free boundary points. The mathematical devices for approaching such regularity issues are based on refined techniques imported from theory of Nonlinear Analysis and PDEs.

The results presented in this lecture are joint works with Julio D. Rossi and Ariel M. Salort (Universidad de Buenos Aires), Analia Silva (Universidad Nacional de San Luis) and Pablo Ochoa (Universidad Nacional
de  Cuyo - Mendoza).

Leandro del Pezzo (Universidad Torcuato Di Tella)

A Liouville theorem for indefinite fractional diffusion equations

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Julián Fernández Bonder (Departamento de Matemática, FCEyN - UBA)

Fractional order Orlicz-Sobolev spaces

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In this talk I will define the fractional order Orlicz-Sobolev spaces, and prove its convergence to the classical Orlicz-Sobolev spaces when the fractional parameter $s\to 1$ in the spirit of the celebrated result of Bourgain-BrezisMironescu. We then deduce some consequences such as $\Gamma-$convergence of the modulars and convergence of solutions for some fractional versions of the $\Delta_g$ operator as the fractional parameter $s\to 1$.

Tomás Godoy (FaMAF- CIEM)

Existencia de soluciones no negativas para problemas elípticos singulares 

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Alberto Grünbaum (University of California, Berkeley)

"How often should Peter and Louis be simultaneously happy?"

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The proportion of time that one player beats the house in a fair game has been know since the days of Paul Levy. The case of two independent  players is open. I will review the case of one player by using the Feynman-Kac formula and display my failed attempts to do something similar in the case of two players.

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Maria Medina (Pontificia Universidad Católica de Chile)

Effect of Boundary Conditions on a Mixed Fractional Problem

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Pablo Ochoa (Universidad Nacional de Cuyo)

Distributional and viscosity solutions for a class of quasilinear equations with fractional diffusions

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Sergio Oliva

Human mobility as diffusion in epidemic models

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Vector-borne diseases attract increasing attention in research because of their wide spread and potential to invade new world areas. We present some ways to introduce space models to the well known SIR and SIS models. We point out some knew results when dealing with non-local diffusion. We use dengue data in the state of Rio de Janeiro to point out the usefulness of such results. Finally, since in vector-borne diseases, the human host´s epidemics often acts on a much slower time scales than the one of the mosquitoes transmitting as a vector, we present a invariant manifold approach to the non-local slow dynamic for human and fast local dynamic for the vectors.

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References

(1)Dirk Brockmann and Dirk Helbing. The hidden geometry of complex, network-driven contagion phenomena. Science, 342(6164):1337–1342, 2013.

(2)Dirk Brockmann, Lars Hufnagel and Theo Geisel. The scaling laws of human travel. Nature, 439(7075):462–465, 2006.

(3)Dirk Brockmann, Vincent David and Alejandro Morales Gallardo. Human mobility and spatial disease dynamics. Reviews of nonlinear dynamics and complexity, 2:1–24, 2009.

(4)Felipe Rocha, Maíra Aguiar, MAx Souza and Nico Stollenwerk. Time-scale separation and centre manifold analysis describing vector-bonre disease dynamics. Int. . of Computer Mathematics, 90, n0. 10, 2015-2125, 2013.

(5)Anibal Rodriguez-Bernal, Silvia Sastre-Gomez. Linear non-local diffusion problems in metric measure spaces. Proc. Royal Society of Edinburgh, 146A, 833-863, 2016.

(6)Laura Forero Vega. Análise da dinamica de uma rede para a dengue. Dissertacão de Mestrado, IME-USP, 2017.

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Sebastián Pauletti

Fixed point method to minimize shape functionals

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Humberto Ramos Quoirin

Positivity issues in sublinear indefinite problems
 

I will discuss the existence of positive solutions for a class of elliptic problems with indefinite and sublinear nonlinearities. These ones have no maximum principle structure, therefore proving the existence of a positive solution is a nontrivial issue. I will show a continuity argument that enables to recover a positivity property in some cases, as well as some existence and uniqueness results.

This talk is based on a joint work with U.Kaufmann and K. Umezu.

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Diego Rial (Departamento de Matemática, FCEyN - UBA)

Solitones ópticos en cristales líquidos nematicos: modelos con efectos de saturación

 

 

Estudiamos un sistema acoplado bidimensional, donde una de las variable evoluciona según una ecuación de Schroedinger y la otra satisface una ecuación elíptica no lineal. Este sistema modela la propagación de una haz láser en un cristal líquido nemático. La ecuación elíptica no lineal describe la respuesta del ángulo director al campo eléctrico del láser. Obtenemos resultados sobre el buen planteo y la existencia de ondas solitarias. Este trabajo generaliza los resultados para el sistema donde la ecuación elíptica para el campo director es lineal. El análisis del problema elíptico no lineal muestra la existencia de una rama global de soluciones con ángulos de director que permanecen acotados para el campo eléctrico arbitrario. Se prueba también la inexistencia de ondas viajeras pequeñas.

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Antonella Ritorto (Departamento de Matemática, FCEyN - UBA)

Un problema de partición óptima para el laplaciano fraccionario

 

Probaremos existencia de solución para un problema de partición óptima, en el que la ecuación involucrada está dada en términos del laplaciano fraccionario. Analizaremos la transición de las ecuaciones no locales a la local, obteniendo una noción de convergencia de las formas óptimas.

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Constanza Sanchez de la Vega (Departamento de Matemática, FCEyN - UBA)

Optimal Control of 1D Non Linear Schrödinger Equation

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This talk is concerned with the optimal control of a 1D cubic nonlinear Schrödinger equation that describes the propagation of optical pulses. We consider the noise on an optical transmission systems as a control variable and study the existence of a minimum norm control such that the pulse is degraded at the end of the transmission (integral restriction on the state). We also give first order necessary conditions for an optimal solution. 

Trabajo en colaboración con Diego Rial.

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Analia Silva (Universidad Nacional de San Luis)

El rol de los agentes testarudos en los modelos de formación de opinión.
 

La manera en que la población  alcanza el acuerdo o no, sobre un determinado tema ha captado el reciente interes de modelos físicos y matemáticos. En esta charla daremos una breve descripción de este tipo modelos y mostraremos un trabajo reciente en el que cada individuo se considera caracterizado por dos parámetros: uno que expresa su calidad de lider y otro que representa su grado de  testarudez. Los resultados presentados son parte de un trabajo conjunto con : J.P. Pinasco, N. Saintier y M. Perez Llanos.

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Juan Spedaletti (Universidad Nacional de San Luis)

Continuity results with respect to domain perturbation for the fractional p-laplacian

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In this work we give sufficient conditions on the approximating domains in order to obtain the continuity of solutions for the fractional p-􀀀laplacian. These conditions are given in terms of the fractional capacity of the approximating domains

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Cristina Turner (FaMAF - CIEM)

Adjoint method for a tumour invasion PDE-constrained optimization problem using FEM 

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In this talk we present a method for estimating unknown parameters that appear on a non-linear reaction-diffusion model of cancer invasion. This model considers that tumor-induced alteration of micro-enviromental pH provides a mechanism for cancer invasion. A coupled system reaction-diffusion describing this model is given by three partial differential equations for the non dimensional spatial distribution and temporal evolution of the density of normal tissue, the neoplastic tissue growth and the excess concentration of H+ ions. Each of the model parameters has a corresponding biological interpretation. For instance, the growth rate of neoplastic tissue, the diffusion coefficient. After solving the forward problem properly, we use the model for the estimation of parameters by fitting the numerical solution with real data, obtained via in vitro experiments and medical imaging. We define an appropriate functional to compare both the real data and the numerical solution. We use the adjoint method for the minimization of this functional and Finite element method to solve both the direct and inverse problem, computing the posterior error in both problem. Moreover, we show some ideas about the possibilities of a therapeutic methodology in order to treat the tumor.

Raúl Vidal 

Un Problema de difusión no-local en el grupo de Heisenberg

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