Resumen de Locally C1,1 convex extensions of 1-jets
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.
