Asymptotic self-similarity of entire solutions for nonlinear elliptic equations

Speaker: 

Soo-Hyun Bae

Institution: 

Hanbat National University (Daejeon, Korea)

Time: 

Friday, October 12, 2018 - 3:00pm

Location: 

RH 440R

I consider solutions with asymptotic self-similarity. The behavior shows an invariance which comes naturally from nonlinearity. The basic model is Lane-Emden equation. Solution structures depend on the dimension as well as the exponent describing the nonlinearity. More generally, I will explain the corresponding result for quasilinear equations in the radial setting.

The incompressible limit of a tumor growth model

Speaker: 

Olga Turanova

Institution: 

UCLA

Time: 

Tuesday, April 24, 2018 - 3:00pm

Host: 

Location: 

RH306

This talk concerns a PDE system that models tumor growth. We show that a novel free boundary problem arises via the incompressible limit of this model. We take a viscosity solutions approach; however, since the system lacks maximum principle, there are interesting challenges to overcome. This is joint work with Inwon Kim.

The stability of full dimensional KAM tori for nonlinear Schrödinger equation

Speaker: 

Hongzi Cong

Institution: 

UCI and Dalian University of Technology (China)

Time: 

Thursday, January 25, 2018 - 3:00pm

Host: 

Location: 

510N

In this talk, we will show  that the full dimensional invariant tori obtained by Bourgain [J. Funct. Anal., 229 (2005), no. 1, 62–94] is stable in a very long time for 1D nonlinear Schrödinger equation with periodic boundary conditions.

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.

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