Uniform Poincar\'e inequality in o-minimal structures

TL;DR

This paper establishes a Poincaré inequality for functions in Sobolev spaces on definable domains within o-minimal structures, demonstrating the boundedness of the inequality constant across definable families with volume control.

Contribution

It introduces a trace definition on o-minimal definable domains and proves a uniform Poincaré inequality with bounded constants for definable domain families.

Findings

Poincaré inequality holds for Sobolev functions with boundary trace in o-minimal domains.

The inequality constant remains bounded for definable domain families with bounded volume.

The trace operator is well-defined on definable domains within o-minimal structures.

Abstract

We first define the trace on a domain $\Omega$ which is definable in an o-minimal structure. We then show that every function $u\in W^{1,p}(\Omega)$ vanishing on the boundary in the trace sense satisfies Poincar\'e inequality. We finally show, given a definable family of domains $(\Omega_t)_{t\in \mathbb{R}^k}$, that the constant of this inequality remains bounded, if so does the volume of $\Omega_t$.