In this paper we study general transportation problems in Rn, in which m different goods are moved simultaneously. The initial and final positions of the goods are prescribed by measures μ−, μ+ on Rn with values in Rm. When the measures are finite atomic, a discrete transportation network is a measure T on Rn with values in Rn×m represented by an oriented graph G in Rn whose edges carry multiplicities in Rm. The constraint is encoded in the relation div(T)=μ−−μ+. The cost of the discrete transportation T is obtained integrating on G a general function C:Rm→R of the multiplicity. When the initial data (μ−,μ+) are arbitrary (possibly diffuse) measures, the cost of a transportation network between them is computed by relaxation of the functional on graphs mentioned above. Our main result establishes the existence of cost-minimizing transportation networks for arbitrary data (μ−,μ+). Furthermore, under additional assumptions on the cost integrand C, we prove the existence of transportation networks with finite cost and the stability of the minimizers with respect to variations of the given data. Finally, we provide an explicit integral representation formula for the cost of rectifiable transportation networks, and we characterize the costs such that every transportation network with finite cost is rectifiable.
A multi-material transport problem with arbitrary marginals / A. Marchese, A. Massaccesi, S. Stuvard, R. Tione. - In: CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS. - ISSN 0944-2669. - 60:3(2021 Jun), pp. 88.1-88.49.
|Titolo:||A multi-material transport problem with arbitrary marginals|
STUVARD, SALVATORE (Corresponding)
|Parole Chiave:||Transportation networks; Branched multi-material transportation; Normal currents|
|Settore Scientifico Disciplinare:||Settore MAT/05 - Analisi Matematica|
|Data di pubblicazione:||giu-2021|
|Digital Object Identifier (DOI):||http://dx.doi.org/10.1007/s00526-021-01967-x|
|Appare nelle tipologie:||01 - Articolo su periodico|
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