Poincar\'e inequality on subanalytic sets

TL;DR

This paper proves a Poincaré inequality for functions in Sobolev spaces on subanalytic bounded open sets in Euclidean space, even with singular boundaries, establishing a key functional inequality in geometric measure theory.

Contribution

It establishes the Poincaré inequality on subanalytic sets with possibly singular boundaries, extending classical results to more general geometric contexts.

Findings

Poincaré inequality holds on subanalytic sets with singular boundaries.

The inequality is valid for all p in [1, ∞).

A uniform constant C depends only on the set and p.

Abstract

Let $\Omega$ be a subanalytic bounded open subset of $\mathbb{R}^n$, with possibly singular boundary. We show that given $p\in [1,\infty)$, there is a constant $C$ such that for any $u\in W^{1,p}(\Omega)$ we have $||u-u_{\Omega}||_{L^p} \le C||\nabla u||_{L^p},$ where we have set $u_{\Omega}:=\frac{1}{|\Omega|}\int_{\Omega} u.$