Locally C1,1 convex extensions of 1-jets

• Daniel Azagra ^[1]
  1. [1] Universidad Complutense de Madrid

     Universidad Complutense de Madrid

     Madrid, España

     
• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 38, Nº 1, 2022, págs. 131-174
• Idioma: inglés
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• 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.