# Difference between revisions of "PDE Geometric Analysis seminar"

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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. | 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. | ||

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+ | ===Eric Baer=== | ||

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+ | Optimal function spaces for continuity of the Hessian determinant as a distribution. | ||

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+ | Abstract: | ||

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+ | 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. |

## Revision as of 17:39, 14 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 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.*