Uniform Poincaré inequality in o-minimal structures

Abstract

We first define the trace on a domain $\hat{I}\text{c◯}$ which is definable in an o-minimal structure. We then show that every function $u\in W^{1,p}(\hat{I}\text{c◯})$ vanishing on the boundary in the trace sense satisfies Poincaré inequality. We finally show, given a definable family of domains $(\hat{I}{\text{c◯}}_t{)}_{t\in {\mathbb{R}}^k}$, that the constant of this inequality remains bounded, if so does the volume of $\hat{I}{\text{c◯}}_t$.