Locally C^1,1 convex extensions of 1-jets

2019/05/06 by Azagra, Daniel
#Classical Analysis and ODEs (math.CA) #Differential Geometry (math.DG) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1905.02127

Abstract

Let E be an arbitrary subset of ℝⁿ, and f:E→ℝ, G:E→ℝⁿ be given functions. We provide necessary and sufficient conditions for the existence of a convex function F∈ C^1,1_\textrmloc(ℝⁿ) such that F=f and ∇ F=G on E. We give a useful explicit formula for such an extension F, and a variant of our main result for the class C^{1, ω}_\textrmloc, where ω is a modulus of continuity. We also present two applications of these results, concerning how to find C^1,1_\textrmloc convex hypersurfaces with prescribed tangent hyperplanes on a given subset of ℝⁿ, and some explicit formulas for (not necessarily convex) C^1,1_\textrmloc extensions of 1-jets.

Related

• Convex C¹ extensions of 1-jets from compact subsets of Hilbert\n spaces
• Explicit formulas for C^{1, 1} and C^{1, ω}_\textrmconv extensions of 1-jets in Hilbert and superreflexive spaces
• Whitney Extension Theorems for convex functions of the classes C¹ and C^1,ω
• Differentiable convex extensions with sharp Lipschitz constants
• C^1,ω extension formulas for 1-jets on Hilbert spaces