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	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	Yifeng Yu (UC Irvine)	TBA	Tran
November 16	Lu Wang (Madison)	TBA
November 23	Nam Le (Indiana)	TBA	Tran
November 30
December 7
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.