Some quantitative Sobolev estimates for planar infinity harmonic functions

Speaker: 

Yi Zhang

Institution: 

Mathematical Institute of the University of Bonn

Time: 

Tuesday, January 16, 2018 - 3:00pm

Host: 

Location: 

RH306

Given a planar infinity harmonic function u, for each
$\alpha>0$ we show a quantitative $W^{1,\,2}_{\loc}$-estimate of
$|Du|^{\alpha}$, which is sharp when $\alpha\to 0$.  As a consequence we
obtain an $L^p$-Liouville property for infinity harmonic functions in
the whole plane
 

Mean Value Theorems for Riemannian Manifolds via the Obstacle Problem

Speaker: 

Jeremy LeCrone

Institution: 

University of Richmond

Time: 

Tuesday, February 13, 2018 - 3:00pm to 4:00pm

Host: 

Location: 

RH 306

In this talk, I will discuss recent results produced with co-authors Ivan Blank (KSU) and Brian Benson (UCR) regarding a formulation of the Mean Value Theorem for the Laplace-Beltrami operator on smooth Riemannian manifolds. We define the sets upon which mean values of (sub)-harmonic functions are computed via a particular obstacle problem in geodesic balls. I will thus begin by discussing the classical obstacle problem and then an intrinsic formulation on manifolds developed in our recent paper. After demonstrating how the theory of obstacle problems is leveraged to produce our Mean Value Theorem, I will discuss local and global theory for our family of mean value sets and potential connections between the properties of these sets and the geometry of the underlying manifold.

Effective dynamics of the nonlinear Schroedinger equation on large domains

Speaker: 

Zaher Hani

Institution: 

Georgia institute of Technology

Time: 

Tuesday, October 10, 2017 - 3:00pm

Location: 

RH 306

We consider the nonlinear Schroedinger equation posed on a large box of characteristic size $L$, and ask about its effective dynamics for very long time scales. After pointing out some “more or less” trivial time scales along which the effective dynamics can be easily described, we start inspecting some much longer time scales where we notice some non-trivial dynamical behaviors. Particularly, the end goal of such an analysis is to reach the so-called “kinetic time scale”, at which it is conjectured that the effective dynamics is governed by a kinetic equation called the “wave kinetic equation”. This is the subject of wave turbulence theory. We will discuss some recent advances towards this end goal. This is joint work with Tristan Buckmaster, Pierre Germain, and Jalal Shatah. 
 

Anomalous diffusion in passive scalar transport

Speaker: 

Gautam Iyer

Institution: 

Carnegie Mellon University

Time: 

Thursday, January 11, 2018 - 4:00pm to 5:00pm

Host: 

Location: 

RH306

Consider a diffusive passive scalar advected by a two
dimensional incompressible flow. If the flow is cellular (i.e.\ has a
periodic Hamiltonian with no unbounded trajectories), then classical
homogenization results show that the long time behaviour is an effective
Brownian motion. We show that on intermediate time scales, the effective
behaviour is instead a fractional kinetic process. At the PDE level this
means that while the long time scaling limit is the heat equation, the
intermediate time scaling limit is a time fractional heat equation. We
will also describe the expected intermediate behaviour in the presence
of open channels.
 

Derivation of multi-layered interface system and its application

Speaker: 

Hiroyoshi Mitake

Institution: 

Hiroshima University

Time: 

Thursday, September 21, 2017 - 3:00pm

Host: 

Location: 

410M

 In this talk, I will propose a multi-layered interface system which can be formally derived by the singular limit of the weakly coupled system of the Allen-Cahn equation.  By using the level set approach, this system can be written as a quasi-monotone degenerate parabolic system. We give results of the well-posedness of viscosity solutions, and study the singularity of each layers. 

This is a joint work with H. Ninomiya, K. Todoroki. 

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