Geometry behind the Poincare inequalities

Nages Shanmugalingam (University of Cincinnati)

Thursday, October 29, 2015 4:00 pm
MSB 109A

Classical work on Euclidean potential theory and PDE extensively used the control of the variance of a function on a ball in terms of the average energy of the function on that ball. Such control, called Poincare inequality, is available in Euclidean spaces as well as Euclidean Lipschitz domains, but fail for more general Euclidean domains. Much of the recent development of analysis in metric measure spaces is done for metric measure spaces whose measure is doubling and supports a Poincare inequality. The work of Heinonen and Koskela indicates that there is geometric information of the metric measure space encoded in the Poincare inequality supported by that space. This talk will give a survey of results expanding on this connection between geometry and support of Poincare inequalities.