Difference between revisions of "PDE Geometric Analysis seminar"
(→Seminar Schedule Fall 2015) 
(→Seminar Schedule Fall 2015) 

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 Donghyun Lee (Madison)   Donghyun Lee (Madison)  
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+  ]]  
    
   
Revision as of 17:20, 23 September 2015
The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm  4:30pm, unless indicated otherwise.
Contents
Previous PDE/GA seminars
Tentative schedule for Spring 2016
Seminar Schedule Fall 2015
date  speaker  title  host(s) 

September 7 (Labor Day)  
September 14 (special room: B115)  Hung Tran (Madison)  Some inverse problems in periodic homogenization of HamiltonJacobi equations  
September 21 (special room: B115)  Eric Baer (Madison)  Optimal function spaces for continuity of the Hessian determinant as a distribution  
September 28  Donghyun Lee (Madison)  FLUIDS WITH FREESURFACE AND VANISHING VISCOSITY LIMIT


October 5  HyungJu Hwang (Postech & Brown Univ)  TBA  Kim 
October 12  Binh Tran (Madison)  TBA  
October 19  Bob Jensen (Loyola University Chicago)  TBA  Tran 
October 26  Luis Silvestre (Chicago)  TBA  Kim 
November 2  Connor Mooney (UT Austin)  TBA  Lin 
November 9  Javier GomezSerrano (Princeton)  TBA  Zlatos 
November 16  Yifeng Yu (UC Irvine)  TBA  Tran 
November 23  Nam Le (Indiana)  TBA  Tran 
November 30  Qin Li (Madison)  TBA  
December 7  Lu Wang (Madison)  TBA  
December 14  Christophe Lacave (Paris 7)  TBA  Zlatos 
Abstract
Hung Tran
Some inverse problems in periodic homogenization of HamiltonJacobi equations.
Abstract: We look at the effective Hamiltonian $\overline{H}$ associated with the Hamiltonian $H(p,x)=H(p)+V(x)$ in the periodic homogenization theory. Our central goal is to understand the relation between $V$ and $\overline{H}$. We formulate some inverse problems concerning this relation. Such type of inverse problems are in general very challenging. I will discuss some interesting cases in both convex and nonconvex settings. Joint work with Songting Luo and Yifeng Yu.
Eric Baer
Optimal function spaces for continuity of the Hessian determinant as a distribution.
Abstract: In this talk we describe a new class of optimal continuity results for the action of the Hessian determinant on spaces of Besov type into the space of distributions on $\mathbb{R}^N$, obtained in collaboration with D. Jerison. Inspired by recent work of Brezis and Nguyen on the distributional Jacobian determinant, we show that the action is continuous on the Besov space $B(22/N,N)$ of fractional order, and that all continuity results in this scale of Besov spaces are consequences of this result. A key ingredient in the argument is the characterization of $B(22/N,N)$ as the space of traces of functions in the Sobolev space $W^{2,N}(\mathbb{R}^{N+2})$ on the subspace $\mathbb{R}^N$ (of codimension 2). The most elaborate part of the analysis is the construction of a counterexample to continuity in $B(22/N,p)$ with $p>N$. Tools involved in this step include the choice of suitable ``atoms" having a tensor product structure and Hessian determinant of uniform sign, formation of lacunary series of rescaled atoms, and delicate estimates of terms in the resulting multilinear expressions.