On a variational problem related to a model of black and white printing

Abstract

A variational problem, which might be considered as a case of shape optimisation, is studied characterising the existence of rrrinimisers. The problem can be understood as a model for black and white printing on digitalised printers. A direct construction of a minimising sequence is presented, based on measure theoretic tools. Another possible construction, using the relationship between the functional in questions and weak $ast$ topology of $L^{infty}$, is given as well. The problem has the unique solution in a relaxed sense: every minimising sequence determines the unique Young measure.