Date: Wednesday, March 18, 2020
Time: 3:30pm
Location: Chandler-Ullmann Hall, Room 218
Speaker: Mikil Foss
University of Nebraska—Lincoln
Title: Local and Nonlocal Poincaré
Inequalities
Abstract: In 1890, as part of a study of the solutions for the heat equation, Henri Poincaré established an inequality providing a bound on a certain measure for the ``size'' of a function in terms of the ``size'' of its derivative. The inequality is a key component in proofs for existence of solutions to PDEs and variational problems. In the differential framework, the Poincaré inequality is also connected to the isoperimetric inequality (in 2D), the eigenvalues for differential operators (Sturm-Liouville problems), and provides a tool for analyzing the stability of numerical schemes.
In many situations, it is important to capture information across multiple scales and accommodate functions that lack differentiability. This leads to a nonlocal framework with integral operators. As in the classical setting, analogues of the Poincaré inequality play a critical role in the analysis of nonlocal problems. I will present some recently obtained nonlocal Poincaré-like inequalities for integral operators with a convolution-like structure.
There will be refreshments in Chandler-Ullmann Hall, Room 212 (Assmus Conference Room)
at 2:45pm preceding the talk.