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Uniform Poincaré inequality in o-minimal structures

Anna Valette, Guillaume Valette

Abstract

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

Uniform Poincaré inequality in o-minimal structures

Abstract

We first define the trace on a domain which is definable in an o-minimal structure. We then show that every function vanishing on the boundary in the trace sense satisfies Poincaré inequality. We finally show, given a definable family of domains , that the constant of this inequality remains bounded, if so does the volume of .

Paper Structure

This paper contains 3 sections, 4 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Trace operator
  3. Uniform bound for the Poincaré constants of definable families

Key Result

Theorem 3.1

Let $(A_t)_{t\in \mathbb{R}^k}$ be a definable family of subsets of $\mathbb{R}^n$ such that $A_t$ has empty interior for each $t\in\mathbb{R}^k$. There exists a uniformly bi-Lipschitz definable family of homeomorphisms $h_t:\mathbb{R}^n\to\mathbb{R}^n$ such that the vector $e_n$ is regular for the

Theorems & Definitions (11)

  • Definition 2.1
  • proof
  • proof
  • Remark 2.4
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 1 more