On the bounded slope condition

Philip Hartman

Vol. 18 (1966), No. 3, 495–511

DOI: 10.2140/pjm.1966.18.495

Abstract

Let Ω be a bounded open set in Rⁿ and let φ(x), x ∈ ∂Ω, satisfy a “bounded slope condition”. The latter reduces to the classical “3-point condition” if n = 2 and occurs in papers on partial differential equations. The properties of φ(x) are studied. It is shown, for example, that if ∂Ω ∈ C¹ or C^1,λ, 0 < λ ≦ 1, then φ(x) ∈ C¹ or C^1,λ. Hence, if ∂Ω ∈ C^1,1 is uniformly convex, then φ(x), x ∈ ∂Ω, satisfies a bounded slope condition if and only if φ(x) ∈ C^1,1. The proofs use generalized convex functions of Beckenbach and, if n > 2, the equivalence of the bounded slope condition and an (n+1)-point condition”.

Mathematical Subject Classification

Primary: 26.52

Milestones

Received: 4 June 1965

Published: 1 September 1966