Non-trivial Poincare Constant in any Subset of $\mathbb{R}^n$

Abstract

The Poincare Inequality is an extremely useful tool in the analysis of PDEs. A significant amount of literature has dealt with finding the optimal constant $C(p,\Omega)$, depending only the domain $\Omega$, and the $L^p$ norm in question. For convex subsets $\Omega\subset \mathbb{R}^n$, an optimal constant is known. There has also been some work done for $\textit{specific}$ non-convex subsets. For a general open, bounded subset, finding the optimal constant is a non-trivial task. This paper will outline a method for finding a non-trivial estimate for $C$ when $\Omega\subset \mathbb{R}^n$ is non-convex, by first observing $\Omega\subset \mathbb{R}^n$ has finite Lebesgue measure, and using local estimates on subsets of $\Omega$ to obtain a global estimate on $\Omega$.