Duality Methods for Linear Variational Problems in $L^\infty $

Abstract

Minimization problems of the form $\{ \| Lu \|_{L^\infty } :u \in U\} $ are considered where L is a linear operator and U is a convex subset of some Hilbert space determined by a finite number of linear functional constraints. The usual variational techniques which provide an Euler equation for such problems in U for $p < \infty$ are not applicable in $L^\infty $; the solution is obtained and necessary conditions that it must satisfy are found by dualizing the problem to an appropriate extremal problem in a subspace of $L^1 $. Several applications are given where L is a linear differential operator acting on a Sobolev space.