Minimum of Quadratic Functionals of the Gradient on the Set of Convex Functions.Modelling, Analysis and Simulation (MAS)

摘要

We study the infimum of functionals of the form integral omega M Triangleu.Triangle u among all convex functions u epsilon H (sup 1) (sub 1) (omega) such that integral straight line Triangle u straight line (sup 2) equals 1. (Omega is a convex open subset of R (sup N) and M is a given symmetric N times N matrix.) We prove that this infimum is the smallest eigenvalue of M if omega is C (sup 1). Otherwise the picture is more complicated. We also study the case of an x-dependent matrix M.