{"product_id":"optimal-transportation-networks-models-and-theory-von-marc-bernot-vicent-caselles-jean-michel-morel","title":"Optimal Transportation Networks","description":"\u003cp\u003eThe transportation problem can be formalized as the problem of ?nding + ? the optimal paths to transport a measure ? onto a measure ? with the same mass. In contrast with the Monge-Kantorovich formalization, recent approaches model the branched structure of such supply networks by an energy functional whose essential feature is to favor wide roads. Given a ?ow ? in a tube or a road or a wire, the transportation cost per unit length ? is supposed to be proportional to ? with 0\u0026lt; 1. For the Monge- Kantorovich energy,? = 1 so that it is equivalent to have two roads with ?ow 1\/2oralargeronewith?ow1. Ifinstead0\u0026lt; 1,aroadwith?ow? +? is 1 2 ? ? ? preferable to two individual roads ? and ? because (? +? ) 1? N N such structures can irrigate a whole bounded open set ofR . The aim of this set of lectures is to give a mathematical proof of s- eral existence, structure and regularity properties empirically observed in transportation networks. This will be done in a simple mathematical fra- work (measures on the set of paths) unifying several di?erent approaches and results due to Brancolini, Buttazzo, Devillanova, Maddalena, Pratelli, Santambrogio, Solimini, Stepanov, Xia and the authors.\u003c\/p\u003e\u003cdiv class=\"aw-variant-hidden-subtitle-div\" id=\"aw-variant-subtitle-9783540693147\"\u003e\u003ch3\u003eModels and Theory\u003c\/h3\u003e\u003c\/div\u003e","brand":"Libri","offers":[{"title":"Softcover - 9783540693147","offer_id":39436517113949,"sku":"9783540693147","price":58.84,"currency_code":"EUR","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0940\/0622\/files\/ced4c395-0f5f-496b-a674-c2e48845083d.jpg?v=1772175823","url":"https:\/\/shop.autorenwelt.de\/products\/optimal-transportation-networks-models-and-theory-von-marc-bernot-vicent-caselles-jean-michel-morel","provider":"Autorenwelt Shop","version":"1.0","type":"link"}