Wednesday, October 14, 2015 - 11:00 , Location: Skiles 05 , Blaz Mramor , Univ. Freiburg , Organizer: Rafael de la Llave
The Allen-Cahn equation is a second order semilinear elliptic PDE that arises in mathematical models describing phase transitions between two constant states. The variational structure of this equation allows us to study energy-minimal phase transitions, which correspond to uniformly bounded non-constant globally minimal solutions. The set of such solutions depends heavily on the geometry of the underlying space. In this talk we shall focus on the case where the underlying space is a Cayley graph of a group with the word metric. More precisely, we assume that the group is hyperbolic and show that there exists a minimal solution with any “nice enough” asymptotic behaviour prescribed by the two constant states. The set in the Cayley graph where the phase transition for such a solution takes place corresponds to a solution of an asymptotic Plateau problem.
Semiconcavity of viscosity solutions for a class of degenerate elliptic integro-differential equations in R^nMonday, September 28, 2015 - 11:00 , Location: Skiles 005 , Chenchen Mou , Georgia Institute of Technology , Organizer:
In this talk, we will consider semiconcavity of viscosity solutions for a class of degenerate elliptic integro-differential equations in R^n. This class of equations includes Bellman equations containing operators of Levy-Ito type. Holder and Lipschitz continuity of viscosity solutions for a more general class of degenerate elliptic integro-differential equations are also provided.
Monday, September 21, 2015 - 11:00 , Location: Skiles 005 , Marian Gidea , Yeshiva University , Organizer: Rafael de la Llave
We prove the existence of diffusion orbits drifting along heteroclinic chains of normally hyperbolic 3-dimensional cylinders, under suitable assumptions on the dynamics on the cylinders and on their homoclinic/heteroclinic connections. These assumptions are satisfied in the a priori stable case of the Arnold diffusion problem. We provide a geometric argument that extends Birkhoff's procedure for constructing connecting orbits inside a zone of instability for a twist map on the annuls. This is joint work with J.-P. Marco.
Monday, September 14, 2015 - 11:00 , Location: Skiles 005 , Willam T. Gignac , Georgia Tech (Math) , Organizer: Rafael de la Llave
Let f be a rational self-map of the complex projective plane. A central problem when analyzing the dynamics of f is to understand the sequence of degrees deg(f^n) of the iterates of f. Knowing the growth rate and structure of this sequence in many cases enables one to construct invariant currents/measures for dynamical system as well as bound its topological entropy. Unfortunately, the structure of this sequence remains mysterious for general rational maps. Over the last ten years, however, an approach to the problem through studying dynamics on spaces of valuations has proved fruitful. In this talk, I aim to discuss the link between dynamics on valuation spaces and problems of degree/order growth in complex dynamics, and discuss some of the positive results that have come from its exploration.
Friday, September 11, 2015 - 15:00 , Location: Skiles 006 , Yannick Sire , John Hopkins University , Organizer: Rafael de la Llave
I will describe several recent results with N. Nadirashvili where we construct extremal metrics for eigenvalues on riemannian surfaces. This involves the study of a Schrodinger operator. As an application, one gets isoperimetric inequalities on the 2-sphere for the third eigenvalue of the Laplace Beltrami operator.
Construction of quasi-periodic solutions of State-dependent delay differential equations by the parameterization methodWednesday, September 9, 2015 - 11:00 , Location: Skiles 006 , Xiaolong He , Georgia Tech (Math)/Hunan University , Organizer: Rafael de la Llave
We investigate the existence of quasi-periodic solutions for state-dependent delay differential equationsusing the parameterization method, which is different from the usual way-working on the solution manifold. Under the assumption of finite-time differentiability of functions and exponential dichotomy, the existence and smoothness of quasi-periodic solutions are investigated by using contraction arguments We also develop a KAM theory to seek analytic quasi-periodic solutions. In contrast with the finite differentonable theory, this requires adjusting parameters. We prove that the set of parameters which guarantee the existence of analytic quasi-periodic solutions is of positive measure. All of these results are given in an a-posterior form. Namely, given a approximate solution satisfying some non-degeneracy conditions, there is a true solution nearby.
Monday, August 31, 2015 - 11:00 , Location: Skiles 005 , Jiayin Jin , Georgia Inst. of Technology , Organizer: Rafael de la Llave
We construct invariant manifolds of interior multi-spike states for the nonlinear Cahn-Hilliard equation and then investigate the dynamics on it. An equation for the motion of the spikes is derived. It turns out that the dynamics of interior spikes has a global character and each spike interacts with all the others and with the boundary. Moreover, we show that the speed of the interior spikes is super slow, which indicates the long time existence of dynamical multi-spike solutions in both positive and negative time. This result is obtained through the application of a companion abstract result concerning the existence of truly invariant manifolds with boundary when one has only approximately invariant manifolds.
Friday, August 21, 2015 - 15:00 , Location: Skiles 005 , Cinzia Elia , Università degli Studi di Bari , Organizer: Rafael de la Llave
In this talk we examine the typical behavior of a trajectory of a piecewise smooth system in the neighborhood of a co-dimension 2 discontinuity manifold $\Sigma$. It is well known that (in the class of Filippov vector fields, and under commonly occurring conditions) one may anticipate sliding motion on $\Sigma$. However, this motion itself is not in general uniquely defined, and recent contributions in the literature have been trying to resolve this ambiguity either by justifying a particular selection of a Filippov vector field or by substituting the original discontinuous problem with a regularized one. However, in this talk, our concern is different: we look at what we should expect of a typical solution of the given discontinuous system in a neighborhood of $\Sigma$. Our ultimate goal is to detect properties that are satisfied by a sufficiently wide class of discontinuous systems and that (we believe) should be preserved by any technique employed to define a sliding solution on $\Sigma$.
Monday, August 17, 2015 - 23:00 , Location: Skiles 005 , Shangjiang Guo , College of Mathematics and Econometrics, Hunan University , Organizer: Rafael de la Llave
In this talk, the existence, stability, and multiplicity of spatially nonhomogeneous steady-state solution and periodic solutions for a reaction–diffusion model with nonlocal delay effect and Dirichlet boundary condition are investigated by using Lyapunov–Schmidt reduction. Moreover, we illustrate our general results by applications to models with a single delay and one-dimensional spatial domain.
Monday, August 10, 2015 - 11:00 , Location: Skiles 005 , Shangjiang Guo , College of Mathematics and Econometrics, Hunan University , Organizer: Rafael de la Llave
We study the existence and branching patterns of wave trains in a two-dimensional lattice with linear and nonlinear coupling between nearest particles and a nonlinear substrate potential. The wave train equation of the corresponding discrete nonlinear equation is formulated as an advanced-delay differential equation which is reduced by a Lyapunov-Schmidt reduction to a finite-dimensional bifurcation equation with certain symmetries and an inherited Hamiltonian structure. By means of invariant theory and singularity theory, we obtain the small amplitude solutions in the Hamiltonian system near equilibria in non-resonance and $p:q$ resonance, respectively. We show the impact of the direction $\theta$ of propagation and obtain the existence and branching patterns of wave trains in a one-dimensional lattice by investigating the existence of travelling waves of the original two-dimensional lattice in the direction $\theta$ of propagation satisfying $\tan\theta$ is rational