Thursday 17 Oct 2019: Computing Poincare constants
Lyonell Boulton - Heriot Watt University
Peter Lanyon Lecture 4 15:00-16:00
The Poincare Inequality states that the amplitude of a regular function which vanishes on the boundary of a planar region is controlled in mean square by a constant times the gradient of that function. The smallest possible value of this constant allowing the inequality for all possible functions is non-zero and is called the Poincare constant of the region.
Except for rectangles, circles and a few triangles, the exact value of the Poincare constant is not given by a simple analytic expression. In this talk we will examine how to find approximate values of this constant for polygons, from the simplest to the very complicated. The first part of the talk will be a survey highlighting the importance of the Poincare constant in mathematics and its applications. We will then learn about numerical procedures for estimating accurately the precise value of this constant. In the final part of the presentation we will consider the challenging problem of finding accurate estimates for very complicated regions, such as fractals. A significant portion will hopefully be accessible to undergraduate Mathematics students.
During the talk I will report on results in collaboration with Banjai [To Appear in J Fractal Geom (2020)]. Further references can be found in the survey paper [App Num Math 99 (2016) 1-23].