Defining functions for unbounded $C^m$ domains

Abstract

Abstract. For a domain Ω ⊂ 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.

A remark on boundary estimates on unbounded Z(q) domains in

“…In [HR13], we show that we may assume |∇ρ| = 1 on U without loss of generality. In fact, the existence of any uniformly C m defining function implies that the signed distance function is uniformly C m .…”

Bounded Plurisubharmonic Exhaustion Functions for Lipschitz Pseudoconvex Domains in $${\mathbb {C}}{\mathbb {P}}^n$$ C P n

“…Since Ω is assumed to have nonempty interior, we will always have sup CP n δ < π 2 . Basic properties of the distance function in R n were developed in Section 4 of [10] (see also [24], [19], [17], and [14]). Following Federer, for p ∈ ∂Ω, we define Tan(∂Ω, p) to be the set of all u ∈ Tan(CP n , p) such that either u = 0 or for every π 2 > ε > 0 there exists q ∈ ∂Ω…”

Closed range of∂¯in L2-Sobolev spaces on unbounded domains inCnHarrington1, Raich2 2018Journal of Mathematical Analysis and ApplicationsSelf Cite7010View full textAdd to dashboardCite

“…With the tools of [HR15], [HR13], [HR14], and the L 2 theory established in [HRb], we are now able to prove closed range of the Cauchy-Riemann operator on appropriately defined Sobolev spaces for a large class of unbounded domains. We review our key definitions in Section 2.…”

“…There is no difference between uniform C m and C m on domains with compact boundary. On unbounded domains, however, we provided counterexamples, a large class of examples, and a complete characterization in terms of the signed distance function in [HR13].…”