Singular Abreu equations and minimizers of convex functionals with a convexity constraint

Speaker: 

Nam Le

Institution: 

Indiana University Bloomington

Time: 

Friday, March 15, 2019 - 3:00pm to 4:00pm

Host: 

Location: 

RH 440R

Abreu type equations are fully nonlinear, fourth order, geometric PDEs which can be viewed as systems of two second order PDEs, one is a Monge-Ampere equation and the other one is a linearized Monge-Ampere equation. They first arise in the constant scalar curvature problem in differential geometry and in the affine maximal surface equation in affine geometry. This talk discusses the solvability and convergence properties of second boundary value problems of singular, fourth order equations of Abreu type arising from approximation of convex functionals whose Lagrangians depend on the gradient variable, subject to a convexity constraint. These functionals arise in different scientific disciplines such as Newton's problem of minimal resistance in physics and the Rochet-Choné model of monopolist's problem in economics. This talk explains how minimizers of the 2D Rochet-Choné model can be approximated by solutions of singular Abreu equations.

Mean field games on graphs

Speaker: 

Chenchen Mu

Institution: 

UCLA

Time: 

Friday, February 22, 2019 - 3:00pm to 4:00pm

Host: 

Location: 

RH 440R

Mean field game theory is the study of the limit of Nash
equilibria of differential games when the number of players tends to infinity. It
was introduced by J.-M. Lasry and P.-L. Lions, and independently by P.
Caines, M. Huang and R. Malhame. A fundamental object in the theory is the
master equation, which fully characterizes the limit equilibrium. In this
talk, we will introduce Mean field game and master equations on graphs. We will
construct solutions to both equations and link them to the solution to
a Hamilton-Jacobi equation on graph.

TBA

Speaker: 

Le Hai Khoi

Institution: 

Nanyang Technological University, Singapore

Time: 

Friday, February 15, 2019 - 3:00pm to 3:50pm

Host: 

Location: 

RH440R

This is a joint Nonlinear PDEs seminar with Analysis seminar

Bergman-Einstein metrics on Strongly pseudoconvex domains in a complex space.

Speaker: 

Xiaojun Huang

Institution: 

Rutgers University

Time: 

Friday, November 16, 2018 - 3:00pm to 3:50pm

Host: 

Location: 

RH440R

I will give a proof of S.Y.  Cheng's conjecture  that a bounded strongly pseudoconvex domain in C^n has  its Bergman metric being Einstein if and only if it is holomorphically equivalent to the ball.

Strong nonlinear instability and growth of Sobolev norms near quasiperiodic finite-gap tori of the 2D cubic NLS

Speaker: 

Zaher Hani

Institution: 

University of Michigan

Time: 

Tuesday, November 27, 2018 - 3:00pm

Location: 

RH 306

 We consider a family of quasiperiodic solutions of the nonlinear Schrodinger equation on the 2-torus, namely the family of finite-gap solutions (tori). These solutions are inherited by the 2D equation from its completely integrable 1D counterpart (NLS on the circle) by considering solutions that only depend on one variable. Despite being linearly stable, we prove that these tori (under some genericness conditions) are nonlinearly unstable in the following strong sense: there exists solutions that start very close to those tori in certain Sobolev spaces, but eventually become larger than any given factor at later times. This is the first instance where (unstable) long-time nonlinear dynamics near (linearly stable) quasiperiodic tori is studied and constructed. (joint work with M. Guardia (UPC, Barcelona), E. Haus (University of Naples), M. Procesi (Roma Tre), and A. Maspero (SISSA))

Spectral transitions for Schr\"odinger operators with decaying potentials and Laplacians on asymptotically flat (hyperbolic) manifolds

Speaker: 

Wencai Liu

Institution: 

UCI

Time: 

Friday, October 26, 2018 - 3:00pm

Location: 

RH 440R

We apply piecewise constructions and gluing  techniques  to  construct

asymptotically flat (hyperbolic) manifolds such that associated

Laplacians have dense embedded eigenvalues or singular continuous

spectra.  The method also allows us to provide various examples of

operators with embedded singular spectra. With additional perturbation theory,    several sharp

spectral transitions (even criteria) for singular spectra are obtained.

In this talk, I will focus on two models-Laplacians on manifolds and Stark operators.

Instability of the Couette flow in low regularity spaces

Speaker: 

Yu Deng

Institution: 

USC

Time: 

Friday, November 2, 2018 - 3:00pm

Host: 

Location: 

RH 440R

 In an exciting paper, J. Bedrossian and N. Masmoudi established the stability of the 2D Couette flow in Gevrey spaces of index greater than 1/2. I will talk about recent joint work with N. Masmoudi, which proves, in the opposite direction, the instability of the Couette flow in Gevrey spaces of index smaller than 1/2. This confirms, to a large extent, that the transient growth predicted heuristically in earlier works does exist and has the expected strength. The proof is based on the fremawork of the stability result, with a few crucial new observations. I will also discuss related works regarding Landau damping, and possible extensions to infinite time.

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.

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