Defining Functions for Unbounded $C^m$ Domains

TL;DR

This paper introduces the concept of uniformly $C^m$ defining functions for unbounded domains in Euclidean space, characterizes them via the signed distance function, and extends known results from bounded to unbounded cases.

Contribution

It defines uniformly $C^m$ defining functions for unbounded domains and characterizes them using the signed distance function, expanding the theory beyond bounded domains.

Findings

Characterization of uniformly $C^m$ defining functions via signed distance functions

Construction of examples of unbounded domains with these functions

Extension of bounded domain results to unbounded 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.