Juan Pablo Agnelli  (FaMAF, UNC)

Numerical solutions to minimal/maximal operators iterating obstacle problems.

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Gaston Beltritti  

Non-local approach in time and space of PDE's

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Pablo Blanc  (Departamento de Matemática, FCEyN - UBA)

Tug-of-War games and PDEs

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Tug-of-War is a two-player zero-sum stochastic game. At an initial time, a token is placed on an bounded domain. At each turn a fair coin is tossed to decide which player is allow to move the token an epsilon-step. The game ends when the token hits the boundary, where a given function determines how much Player II must pay to Player I. In this talk we will discuss some versions of this game and how they are related to PDEs.

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References:
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

     
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

Existence of nonnegative solutions for singular elliptic problems

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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 quailinear equations with fractional diffusions

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Sergio Oliva

Human mobility as diffusion in epidemic models

 

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

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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)

Optical solitons in nematic liquid crystals: models with saturation effects

 

We study a 2-D system that couples a Schr ̈odinger evolution equa- tion to a nonlinear elliptic equation and models the propagation of a laser beam in a nematic liquid crystal. The nonlinear elliptic equa- tion describes the response of the director angle to the laser beam electric field. We obtain results on well-posedness and solitary wave solutions of this system, generalizing results for a well-studied simpler system with a linear elliptic equation for the director field. The analy- sis of the nonlinear elliptic problem shows the existence of an isolated global branch of solutions with director angles that remain bounded for arbitrary electric field. The results on the director equation are also used to show local and global existence, as well as decay for ini- tial conditions with sufficiently small L2 −norm. For sufficiently large L2−norm we show the existence of energy minimizing optical solitons with radial, positive and monotone profiles.

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Antonella Ritorto (Departamento de Matemática, FCEyN - UBA)

Optimal partition problem for the fractional Laplacian

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We prove existence of solution for an optimal partition problem, where the equation involved is given in terms of the fractional Laplacian. We analyze the transition from non-local equations to local ones, so that we obtain a notion of convergence for the optimal shapes.

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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. 

This is a joint work with Diego Rial.

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Analia Silva (Universidad Nacional de San Luis)

Opinion formation model with conviction

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Understanding the way a human population reaches an agreement on a given subject or on the opposite way, why multiple opinions survive and oppose themselves is a long standing subject in sociology. In recent years, opinion formation, have attracted a considerable attention from the physic and the mathematic community.
In this talk we explain a model of opinion formation, where each agent has a priori, certain power of
conviction and also certain willingness of changing his/her own opinion. This is a joint work with  M. Perez LLanos, J.P. Pinasco and N. Saintier.

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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 

A Nonlocal Diffusion Problem in the Heisenberg Group

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