Minimizing $L^{\infty }$-norm functional on divergence-free fields

Abstract

In this paper, we study the minimization problem on the $L^{\infty }$-norm functional over the divergence-free fields with given boundary normal component. We focus on the computation of the minimum value and the classification of certain special minimizers including the so-called absolute minimizers. In particular, several alternative approaches for computing the minimum value are given using $L^q$-approximations and the sets of finite perimeter. For problems in two dimensions, we establish the existence of absolute minimizers using a similar technique for the absolute minimizers of $L^{\infty }$-functionals of gradient fields. In some special cases, precise characterizations of all minimizers and the absolute minimizers are also given based on equivalent descriptions of the absolutely minimizing Lipschitz extensions of boundary functions.