PDE Geometric Analysis seminar

The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.

Seminar Schedule Spring 2016

date	speaker	title	host(s)
January 25	Tianling Jin (HKUST and Caltech)	Holder gradient estimates for parabolic homogeneous p-Laplacian equations	Zlatos
February 1	Russell Schwab (Michigan State University)	Neumann homogenization via integro-differential methods	Lin
February 8	Jingrui Cheng (UW Madison)	Semi-geostrophic system with variable Coriolis parameter	Tran & Kim
February 15	Paul Rabinowitz (UW Madison)	On A Double Well Potential System	Tran & Kim
February 22	Hong Zhang (Brown)	On an elliptic equation arising from composite material	Kim
February 29	Aaron Yip (Purdue university)	Discrete and Continuous Motion by Mean Curvature in Inhomogeneous Media	Tran
March 7	Hiroyoshi Mitake (Hiroshima university)	Selection problem for fully nonlinear equations	Tran
March 15	Nestor Guillen (UMass Amherst)	Min-max formulas for integro-differential equations and applications	Lin
March 21 (Spring Break)
March 28	Ryan Denlinger (Courant Institute)	The propagation of chaos for a rarefied gas of hard spheres in vacuum	Lee
April 4	No seminar
April 11	Misha Feldman (UW)	Shock reflection, free boundary problems and degenerate elliptic equations
April 14: 2:25 PM in VV 901-Joint with Probability Seminar	Jessica Lin (UW-Madison)	Optimal Quantitative Error Estimates in Stochastic Homogenization for Elliptic Equations in Nondivergence Form
April 18	Sergey Bolotin (UW)	Degenerate billiards in celestial mechanics
April 21-24, KI-Net conference: Boundary Value Problems and Multiscale Coupling Methods for Kinetic Equations	Link: http://www.ki-net.umd.edu/content/conf?event_id=493
April 25	Moon-Jin Kang (UT-Austin)		Kim
May 3 (Joint Analysis-PDE seminar )	Stanley Snelson (University of Chicago)		Seeger & Tran.
May 16-20, Conference in Harmonic Analysis in Honor of Michael Christ	Link: https://www.math.wisc.edu/ha_2016/

Abstracts

Tianling Jin

Holder gradient estimates for parabolic homogeneous p-Laplacian equations

We prove interior Holder estimates for the spatial gradient of viscosity solutions to the parabolic homogeneous p-Laplacian equation u_t=|\nabla u|^{2-p} div(|\nabla u|^{p-2}\nabla u), where 1<p<\infty. This equation arises from tug-of-war like stochastic games with white noise. It can also be considered as the parabolic p-Laplacian equation in non divergence form. This is joint work with Luis Silvestre.

Russell Schwab

Neumann homogenization via integro-differential methods

In this talk I will describe how one can use integro-differential methods to attack some Neumann homogenization problems-- that is, describing the effective behavior of solutions to equations with highly oscillatory Neumann data. I will focus on the case of linear periodic equations with a singular drift, which includes (with some regularity assumptions) divergence equations with non-co-normal oscillatory Neumann conditions. The analysis focuses on an induced integro-differential homogenization problem on the boundary of the domain. This is joint work with Nestor Guillen.

Jingrui Cheng

Semi-geostrophic system with variable Coriolis parameter.

The semi-geostrophic system (abbreviated as SG) is a model of large-scale atmospheric/ocean flows. Previous works about the SG system have been restricted to the case of constant Coriolis force, where we write the equation in "dual coordinates" and solve. This method does not apply for variable Coriolis parameter case. We develop a time-stepping procedure to overcome this difficulty and prove local existence and uniqueness of smooth solutions to SG system. This is joint work with Michael Cullen and Mikhail Feldman.

Paul Rabinowitz

On A Double Well Potential System

We will discuss an elliptic system of partial differential equations of the form \[ -\Delta u + V_u(x,u) = 0,\;\;x \in \Omega = \R \times \mathcal{D}\subset \R^n, \;\;\mathcal{D} \; bounded \subset \R^{n-1} \] \[ \frac{\partial u}{\partial \nu} = 0 \;\;on \;\;\partial \Omega, \] with $u \in \R^m$,\; $\Omega$ a cylindrical domain in $\R^n$, and $\nu$ the outward pointing normal to $\partial \Omega$. Here $V$ is a double well potential with $V(x, a^{\pm})=0$ and $V(x,u)>0$ otherwise. When $n=1, \Omega =\R^m$ and \eqref{*} is a Hamiltonian system of ordinary differential equations. When $m=1$, it is a single PDE that arises as an Allen-Cahn model for phase transitions. We will discuss the existence of solutions of \eqref{*} that are heteroclinic from $a^{-}$ to $a^{+}$ or homoclinic to $a^{-}$, i.e. solutions that are of phase transition type.

This is joint work with Jaeyoung Byeon (KAIST) and Piero Montecchiari (Ancona).

Hong Zhang

On an elliptic equation arising from composite material

I will present some recent results on second-order divergence type equations with piecewise constant coefficients. This problem arises in the study of composite materials with closely spaced interface boundaries, and the classical elliptic regularity theory are not applicable. In the 2D case, we show that any weak solution is piecewise smooth without the restriction of the underling domain where the equation is satisfied. This completely answers a question raised by Li and Vogelius (2000) in the 2D case. Joint work with Hongjie Dong.

Aaron Yip

Discrete and Continuous Motion by Mean Curvature in Inhomogeneous Media

The talk will describe some results on the behavior of solutions of motion by mean curvature in inhomogeneous media. Emphasis will be put on the pinning and de-pinning transition, continuum limit of discrete spin systems and the motion of interface between patterns.

Hiroyoshi Mitake

Selection problem for fully nonlinear equations

Recently, there was substantial progress on the selection problem on the ergodic problem for Hamilton-Jacobi equations, which was open during almost 30 years. In the talk, I will first show a result on the convex Hamilton-Jacobi equation, then tell important problems which still remain. Next, I will mainly focus on a recent joint work with H. Ishii (Waseda U.), and H. V. Tran (U. Wisconsin-Madison) which is about the selection problem for fully nonlinear, degenerate elliptic partial differential equations. I will present a new variational approach for this problem.

Nestor Guillen

Min-max formulas for integro-differential equations and applications

We show under minimal assumptions that a nonlinear operator satisfying what is known as a "global comparison principle" can be represented by a min-max formula in terms of very special linear operators (Levy operators, which involve drift-diffusion and integro-differential terms). Such type of formulas have been very useful in the theory of second order equations -for instance, by allowing the representation of solutions as value functions for differential games. Applications include results on the structure of Dirichlet-to-Neumann mappings for fully nonlinear second order elliptic equations.

Ryan Denlinger

The propagation of chaos for a rarefied gas of hard spheres in vacuum

We are interested in the rigorous mathematical justification of Boltzmann's equation starting from the deterministic evolution of many-particle systems. O. E. Lanford was able to derive Boltzmann's equation for hard spheres, in the Boltzmann-Grad scaling, on a short time interval. Improvements to the time in Lanford's theorem have so far either relied on a small data hypothesis, or have been restricted to linear regimes. We revisit the small data regime, i.e. a sufficiently dilute gas of hard spheres dispersing into vacuum; this is a regime where strong bounds are available globally in time. Subject to the existence of such bounds, we give a rigorous proof for the propagation of Boltzmann's ``one-sided molecular chaos.

Misha Feldman

Shock reflection, free boundary problems and degenerate elliptic equations.

Abstract: We will discuss shock reflection problem for compressible gas dynamics, and von Neumann conjectures on transition between regular and Mach reflections. We will discuss existence of solutions of regular reflection structure for potential flow equation, and also regularity of solutions, and properties of the shock curve (free boundary). Our approach is to reduce the shock reflection problem to a free boundary problem for a nonlinear equation of mixed elliptic-hyperbolic type. Open problems will also be discussed, including uniqueness. The talk is based on the joint works with Gui-Qiang Chen, Myoungjean Bae and Wei Xiang.

Jessica Lin

Optimal Quantitative Error Estimates in Stochastic Homogenization for Elliptic Equations in Nondivergence Form

Abstract: I will present optimal quantitative error estimates in the stochastic homogenization for uniformly elliptic equations in nondivergence form. From the point of view of probability theory, stochastic homogenization is equivalent to identifying a quenched invariance principle for random walks in a balanced random environment. Under strong independence assumptions on the environment, the main argument relies on establishing an exponential version of the Efron-Stein inequality. As an artifact of the optimal error estimates, we obtain a regularity theory down to microscopic scale, which implies estimates on the local integrability of the invariant measure associated to the process. This talk is based on joint work with Scott Armstrong.