Olivier Ley. Irmar, Insa Rennes Large Time Behavior of Solutions of First-Order Hamilton-Jacobi Equations in The Unbounded Case Sala de Seminarios del 5to piso del CMM Abstract: I will discuss some results about the large time behavior for unbounded solutions of Eikonal type Hamilton-Jacobi in the whole space R^n and illustrate the additional difficulties with respect to the periodic case on an explicit easy control problem. This is a joint work with Guy Barles (Tours), Thi Tuyen Nguyen (Padova) and Thanh Viet Phan (Ho Chi Minh City). http://capde.cl/scientific-activities-2/
Eiji Yanagida. Tokyo Inst. Technology Moving Singularities in Some Parabolic Partial Differential Equations Sala de seminarios 7° piso, CMM Abstract:
For parabolic partial differential equations, we can consider solutions with singularities which move depending on time. Such singularities are removable if they are weak enough, but some equations do possess singular solutions with proper singular profile. In this talk, I will survey recent progress on the existence and removability of singularities in some parabolic equations. I also discuss asymptotic profile of solutions near singular points.
José Manuel Palacios. Universidad de Chile Stability of Sine-Gordon 2-Solitons in The Energy Space Sala de seminarios 5to piso CMM http://capde.cl/scientific-activities-2/
Hanne Van Der Bosch. Cmm, Uchile Spectrum of Dirac Operators Describing Graphene Quantum Dots Sala Multimedia CMM, 6to piso Abstract: Low energy electronic excitations in graphene, a two-dimensional lattice of carbon atoms, are described effectively by a two–dimensional Dirac operator. For a bounded flake of graphene (a quantum dot), the choice of boundary conditions determines various properties of the spectrum. Several of these choices appear in the physics literature on graphene. For a simply connected flake and a family of boundary conditions, we obtain an explicit lower bound on the spectral gap around zero. We can also study the effect of the boundary conditions on eigenvalue sums in the semiclassical limit. This is joint work with Rafael Benguria, Søren Fournais and Edgardo Stockmeyer. http://capde.cl/scientific-activities-2/
Mateo Rizzi. Universidad de Chile Cahn-Hilliard Equation and Willmore Surfaces CMM, U. de Chile, 6to piso, sala multimedia Abstract:
There are many results in the literature concerning the relation between the Allen-Cahn equation and minimal surfaces. In this talk I will present some analogue results for the Cahn-Hilliard equation, that is a fourth order PDE which is related to the Willmore energy through some Gamma-convergence results. I will present results in dimension 2 and 3, concerning the construction of solutions, and I will briefly discuss some qualitative properties.
Mauricio Romo. Ias, Princeton Derived Categories, Gauge Theories and Analytic Continuation Sala 2 Abstract: In this talk I will introduce the concept of Gauged Linear Sigma Model (GLSM), a class of quantum field theories, in mathematical terms. I will show how the GLSM can be used to conjecture derived equivalences (of categories such as coherent sheaves on projective spaces and a matrix factorizations) and how this translates into analytic properties of physical quantities. http://capde.cl/scientific-activities-2/
Rémy Rodiac. Université Catholique de Louvain, Bélgica Axially Symmetric Minimizers of The Neo-Hookean Energy in 3D Sala 2 Abstract:
The neo-Hookean energy is an energy broadly used by physicists and engineers to describe the behavior of elastic materials undergoing large deformations. However to prove the existence of minimizers of this energy is still an open problem. We consider this problem in an axisymmetric setting and show that if the domain do not contain the axis of symmetry then minimizers do exist. Ours axisymmetric minimizers are also solutions of the weak form of the Euler-Lagrange equations of elasticity. This is a joint work with Duvan Henao.
Erwin Topp Paredes. Universidad de Santiago de Chile Parabolic Equations With Caputo Time Derivative Sala 2, Facultad de Matemáticas, PUC Abstract: In this talk we report results presented in [Topp & Yangari 2017] about well-posedness of fully nonlinear Cauchy problems in which the time derivative is of Caputo type. We address this question in the framework of viscosity solutions, obtaining the existence via Perron’s method, and comparison for bounded sub and supersolutions by a suitable regularization through inf and sup convolution in time. As an application, we prove the steady-state large time behavior in the case of proper nonlinearities and provide a rate of convergence by using the Mittag-Leffler operator. http://capde.cl/scientific-activities-2/
María Medina. Pontificia Universidad Católica de Chile A Mixed Fractional Problem. Moving The Boundary Conditions Sala 2, Facultad de Matemáticas, PUC Abstract: A natural question when one considers the mixed eigenvalue problem for the Laplacian (zero DIrichlet condition in D and Neumann homogeneous in N where $\Omega$ is a Lipschitz bounded domain in $R^N$ and D, N are disjoint submanifolds of ∂Ω) is whether the configuration of the sets D and N determines the behavior of u. Is it similar to the solution of the Dirichlet problem when N is small? or does it behave like the Neumann eigenfunction when N is large? Several authors have shown results where different configurations of D and N provide very different behaviors of u (see for example [Colorado & Peral 2003, Denzler 1999]) depending on the size of the sets, but also on their location.
In this talk we will try to understand the analogous non local problem where N and D are now two open sets of $R^N\Omega$. As we will see, the fact that the boundary now happens to be the whole $R^N\Omega$ instead of ∂Ω completely changes the possible configurations of the sets (one can even have both sets of unbounded measure). The purpose of this talk will be to understand what “a small boundary set” means here, and to analyze how D and N can move to recover the classical results.
This is a joint work with T. Leonori, I. Peral, A. Primo and F. Soria, that can be found at https://arxiv.org/pdf/1702.07644.pdf. http://capde.cl/scientific-activities-2/
Luis Fernando López. Università Degli Studi Roma Tre The Mean Field Equation in High Dimensions Sala 5 Abstract: In this talk we present an n-dimensional version of the well known 2D mean field equation. We exhibit a special set of solutions to the model and prove a nondegeneracy property associated with the linearization of the operator. Such property is used to study the behavior of blowing-up families of solutions. This phenomenon is inspired in related models, where the lack of compactness of the Sobolev embedding is closely related to the concentration of solutions. This is a joint work with Pierpaolo Esposito.
Juan Diego Dávila. Universidad de Chile Finite Time Blow-Up for The Harmonic Map Flow in 2 Dimensions Sala 5 Abstract: We describe precisely the finite time blow up behavior of some solutions of the harmonic map flow in 2 dimensions with values into the sphere, in a nonradial situation. One important quantity is the rate of blow up, which was established rigorously only in the 1-corrotational symmetric class by Raphael and Schweyer. This is joint work with Manuel del Pino (Universidad de Chile) and Juncheng Wei (University of British Columbia).
Gianmarco Sperone. Dim U. Chile Further Remarks on The Luo-Hou's Ansatz for a Self-Similar **solution To The 3D Euler Equations Sala de seminarios del 5to. piso del Departamento de Ingeniería Matemática DIM, U. de Chile Abstract: It is shown that the self-similar ansatz proposed by T. Hou and G. Luo to describe a blow-up solution of the 3D axisymmetric Euler equations leads, without assuming any asymptotic condition on the self-similar profi les, to an over-determined system of partial differential equations that produces two families of solutions: a class of trivial solutions in which the vorticity field is identically zero, and a family of solutions that blow-up immediately, where the vorticity field is governed by a stationary regime. In any case, the analytical properties of these solutions are not consisent with the numerical observations reported by T. Hou and G. Luo. Therefore, this result is a refi nement of the previous work published by D. Chae and T.-P. Tsai on this matter, where the authors find the trivial class of solutions under a rather unjusti fied decay condition of the blow-up profiles.
Duvan Henao. P. Universidad Católica de Chile Existence Theorems for Geometrically Nonlinear Models of Nematic Elastomers Sala 5 de la Facultad de Matemáticas a las 17:00 Hrs.
Mariel Sáez. P. Universidad Católica de Chile Fractional Laplacians and Extension Problems: The Higher Rank Case (Joint With M.m. Gonzalez) Sala 5 de la Facultad de Matemáticas entre las 16:00 Hrs.
Gyula Csató. Universidad de Concepción About Hardy-Sobolev, Moser-Trudinger and Isoperimetric Inequalities With Densities Sala 5 de la Facultad de Matemáticas a las 17:00 Hrs.
María Medina. Pontificia Universidad Católica de Chile The Effect of The Hardy Potential in Some Calderón-Zygmund Properties for The Fractional Laplacian Sala 2 Facultad de Matemáticas, P.U.C. a las 17:00 Hrs.
Carlos Román. Université Pierre Et Marie Curie - Paris Vi, On The First Critical Magnetic Field in The Three-Dimensional Ginzburg-Landau Model of Superconductivity Sala de seminarios D.I.M. (5to piso), U. de Chile a las 17:00 Hrs.
Michel Chipot. Universität Zürich Nonhomogeneous Boundary Value Problems for The Stationary Navier-Stokes Equations in Two-Dimensional Domains With Semi-Infinite Outlets Sala de seminarios del Departamento de Ingeniería Matemática de la Universidad de Chile 5º piso, Beauchef 851, Edificio Norte a las 17:00 Hrs.