The Poincare Inequality on Smooth and Bounded Domains in R^d

Abstract

In this thesis we study for which domain types the Poincare inequality holds for all functions having continuous first derivative. We first consider the classical Poincare inequality, which we prove holds for a very large class of open sets in R^d. We then constructively prove that bounded, open, and connected domains in R^d, which also possess a smooth C¹-boundary, must satisfy the Poincare-Wirtinger inequality. We do this in six successive steps.
First, we show that an arbitrary open rectangle in R^d must satisfy the inequality.Second, we prove that a C¹-diffeomorphism with a sufficient condition, between a set which satisfies the inequality and an open, bounded and connected set implies the open, bounded and connected set also satisfies the Poincare-Wirtinger inequality. Third, we show that there exists such a C¹-diffeomorphism between a domain in the class of open rectangles with one face distorted by a C¹-function and another domain in the class of arbitrary open rectangles in R^d. Fourth, we show the class of all open rectangles with one face distorted by a C¹-function satisfies the Poincare-Wirtinger inequality. Fifth, we show the union of non-disjoint open sets which satisfy the inequality in turn also satisfies the Poincare-Wirtinger inequality. Lastly, we cover the open, bounded and connected domain with a C¹-boundary by a collection of rectangles from the classes of open rectangles with one face distorted by a C¹-function and arbitrary open rectangles to show that the domain satisfies thePoincare-Wirtinger inequality.

Finally, we extend our function space to the first-order Sobolev space and show that we can directly extend our results to this function space.