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 =Abstracts=   =Abstracts= 
   
−  ===Tianling Jin===  +  === === 
−   
−  Holder gradient estimates for parabolic homogeneous pLaplacian equations
 
−   
−  We prove interior Holder estimates for the spatial gradient of viscosity solutions to the parabolic homogeneous pLaplacian equation
 
−  u_t=\nabla u^{2p} div(\nabla u^{p2}\nabla u),
 
−  where 1<p<\infty. This equation arises from tugofwar like stochastic games with white noise. It can also be considered as the parabolic pLaplacian equation in non divergence form. This is joint work with Luis Silvestre.
 
−   
−  ===Russell Schwab===
 
−   
−  Neumann homogenization via integrodifferential methods
 
−   
−  In this talk I will describe how one can use integrodifferential 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 nonconormal oscillatory Neumann conditions. The analysis focuses on an induced integrodifferential homogenization problem on the boundary of the domain. This is joint work with Nestor Guillen.
 
−   
−  ===Jingrui Cheng===
 
−   
−  Semigeostrophic system with variable Coriolis parameter.
 
− 
 
−  The semigeostrophic system (abbreviated as SG) is a model of largescale 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 timestepping 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^{n1}
 
−  \]
 
−  \[
 
−  \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 AllenCahn 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 secondorder 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 depinning 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 HamiltonJacobi equations, which was open during almost 30 years. In the talk, I will first show a result on the convex HamiltonJacobi 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. WisconsinMadison) 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===
 
−   
−  Minmax formulas for integrodifferential 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 minmax formula in terms of very special linear operators (Levy operators, which involve driftdiffusion and integrodifferential 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 DirichlettoNeumann 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
 
−  manyparticle systems. O. E. Lanford was able to derive Boltzmann's
 
−  equation for hard spheres, in the BoltzmannGrad 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 ``onesided'' 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 elliptichyperbolic type. Open problems
 
−  will also be discussed, including uniqueness.
 
−  The talk is based on the joint works with GuiQiang 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
 
−  EfronStein 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.
 
−   
−  ===Sergey Bolotin===
 
−   
−  Degenerate billiards
 
−   
−  In an ordinary billiard trajectories of a Hamiltonian system
 
−  are elastically reflected when colliding with a hypersurface (scatterer).
 
−  If the scatterer is a submanifold of codimension more than one, then collisions are rare.
 
−  Trajectories with infinite number of collisions form a lower dimensional dynamical system.
 
−  Degenerate billiards appear as limits of ordinary billiards and in celestial mechanics.
 
−   
−  ===MoonJin Kang===
 
−   
−  On contraction of large perturbation of shock waves, and inviscid limit problems
 
−   
−  This talk will start with the relative entropy method to handle the contraction of possibly large perturbations around viscous shock waves of conservation laws. In the case of viscous scalar conservation law in one space dimension, we obtain $L^2$contraction for any large perturbations of shocks up to a Lipschitz shift depending on time. Such a timedependent Lipschitz shift should be constructed from dynamics of the perturbation. In the case of multidimensional scalar conservation law, the perturbations of planar shocks are $L^2$contractive up to a more complicated shift depending on both time and space variable, which solves a parabolic equation with inhomogeneous coefficient and force terms reflecting the perturbation. As a consequence, the $L^2$contraction property implies the inviscid limit towards inviscid shock waves. At the end, we handle the contraction properties of admissible discontinuities of the hyperbolic system of conservation laws equipped with a strictly convex entropy.
 