Optimal region for the transport problem to the boundary

Abstract

We consider a region Ω⊂R2 where a mass f is transported to the boundary and the aim is to find an optimal free transport region E that minimizes the total cost outside E of this transport problem plus a penalization term on E. First, we study the regularity of the transport density σ in this transport problem to the boundary. Then, we show existence of an optimal set E for this shape optimization problem and, we prove regularity on this optimal set E in the case where the penalization term on E is given by the perimeter (or the fractional perimeter) of E.