Locally C1,1 convex extensions of 1-jets

  1. Daniel Azagra 1
  1. 1 Universidad Complutense de Madrid
    info

    Universidad Complutense de Madrid

    Madrid, España

    ROR 02p0gd045

Revista:
Revista matemática iberoamericana

ISSN: 0213-2230

Año de publicación: 2022

Volumen: 38

Número: 1

Páginas: 131-174

Tipo: Artículo

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Otras publicaciones en: Revista matemática iberoamericana

Resumen

Let E be an arbitrary subset of Rn, and let f:E→R, G:E→Rn be given functions. We provide necessary and sufficient conditions for the existence of a convex function F∈C1,1loc(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 C1,ωloc, where ω is a modulus of continuity. We also present two applications of these results, concerning how to find C1,1loc convex hypersurfaces with prescribed tangent hyperplanes on a given subset of Rn, and some explicit formulas for (not necessarily convex) C1,1loc extensions of 1-jets.

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