Convex sets and the bounded slope condition

Philip Hartman

Vol. 25 (1968), No. 3, 511–522

DOI: 10.2140/pjm.1968.25.511

Abstract

Let Ω be a bounded open convex set in Rⁿ with boundary Γ. This paper concerns the class B(Γ) of functions ϕ(x), defined on Γ, satisfying a bounded slope condition and its closure B(Γ) in C⁰(Γ). The class B(Γ) is of interest because of its occurrence in the theory of nonlinear, nonuniformly elliptic, boundary value problems. It is shown that B(Γ) is the set of continuous functions on Γ which, on flat pieces of Γ, are restrictions of linear functions of x. Thus B(Γ) = C⁰(Γ) if and only if there are no line segments on Γ.

Mathematical Subject Classification

Primary: 46.25

Secondary: 35.00

Milestones

Received: 29 May 1967

Published: 1 June 1968