Optimal region for the transport problem to the boundary

We consider a region omega subset of 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 sigma 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.(c) 2023 Elsevier Inc. All rights reserved.

Automated summary

In this paper, we consider a mass transportation problem in a two-dimensional region and aim to minimize the transportation cost by optimizing the free transport region. We study the regularity of the transport density on the boundary and prove the existence of an optimal set for shape optimization. Furthermore, we establish the regularity of the optimal set when the penalization term is given by the perimeter of the set.