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.
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 Hamilton--Jacobi 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) | TBA | |
| October 5 | Hyung-Ju 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 Gomez-Serrano (Princeton) | TBA | Zlatos |
| November 16 | Yifeng Yu (UC Irvine) | TBA | Tran |
| November 23 | Nam Le (Indiana) | TBA | Tran |
| November 30 | Lu Wang (Madison) | TBA | |
| December 7 | Qin Li (Madison) | TBA | |
| December 14 | reserved | Zlatos |
Abstract
Hung Tran
Some inverse problems in periodic homogenization of Hamilton--Jacobi 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 $B(2-2/N,N)$ 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 $B(2-2/N,N)$ 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 $B(2-2/N,p)$ 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.