Defining Functions for Unbounded $C^m$ Domains

Abstract

For a domain $\Omega\subset\mathbb R^n$, we introduce the concept of a uniformly $C^m$ defining function. We characterize uniformly $C^m$ defining functions in terms of the signed distance function for the boundary and provide a large class of examples of unbounded domains with uniformly $C^m$ defining functions. Some of our results extend results from the bounded case.