# Difference between revisions of "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 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) FLUIDS WITH FREE-SURFACE AND VANISHING VISCOSITY LIMIT


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 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 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.