On the lower bounded slope condition. (English) Zbl 1132.49031

Bousquet, Pierre

J. Convex Anal. 14, No. 1, 119-136 (2007).

Summary: Let \(\Omega\) be a bounded open convex set in \({\mathbb R}^n\) and let \(\varphi:\Gamma:=\partial\Omega\to {\mathbb R}\) be a function defined on its boundary. The lower bounded slope condition (on \(\varphi\)) is a hypothesis recently introduced by F. Clarke [Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 4, No. 3, 511–530 (2005; Zbl 1127.49001)], who has shown its relevance to regularity theory in the calculus of variations. It corresponds to a weaker version of the traditional bounded slope condition, which also appears in the theory of elliptic differential equations. In this paper, we study the regularity properties of these functions and give intrinsic characterizations of them. Semiconvexity turns out to be a central tool in the proofs.

MSC:

49N60

Regularity of solutions in optimal control

Keywords:

regularity theory in the calculus of variations; semiconvexity; Hilbert-Haar theory

Citations:

Zbl 1127.49001