# American Institute of Mathematical Sciences

April  2011, 30(1): 365-374. doi: 10.3934/dcds.2011.30.365

## Limit theorems for optimal mass transportation and applications to networks

 1 Department of Mathematics, Technion, Haifa 32000, Israel

Received  November 2009 Revised  August 2010 Published  February 2011

It is shown that optimal network plans can be obtained as a limit of point allocations. These problems are obtained by minimizing the mass transportation on the set of atomic measures of a prescribed number of atoms.
Citation: Gershon Wolansky. Limit theorems for optimal mass transportation and applications to networks. Discrete and Continuous Dynamical Systems, 2011, 30 (1) : 365-374. doi: 10.3934/dcds.2011.30.365
##### References:
 [1] G. Buttazzo and E. Stepanov, "Optimal Urban Networks via Mass Transportation," Lec. Notes in Math., 1961, Springer-Verlag, 2009. [2] E. N. Gilbert, Minimum cost communication networks, Bell Sys. Tech. J., 46 (1967), 2209-2227. [3] E. N. Gilbert and H. O. Pollak, Steiner minimal trees, SIAM J. Appl. Math., 16 (1968), 1-29. doi: 10.1137/0116001. [4] F. K. Hwang, D. S. Richards and P. Winter, "The Steiner Tree Problem," Elsevier, North-Holland, 1992. [5] M. Bernot, V. Caselles and J. M. Morel, "Optimal Transportation Networks," Lec. Notes in Math., 1955, Springer-Verlag, 2009. [6] F. Maddalena, S. Solimini and J. J. Morel, A variational model of irrigation pattern, Interfaces and Free Boundaries, 5 (2003), 391-416. doi: 10.4171/IFB/85. [7] E. Paolini and E. Stepanov, Optimal transportation networks as flat chains, Interfaces Free Bound., 8 (2006), 393-436. doi: 10.4171/IFB/149. [8] E. Paolini and E. Stepanov, Connecting measures by means of branched transportation networks at finite cost, J. Math. Sci. (N. Y.), 157 (2009), 858-873. doi: 10.1007/s10958-009-9362-x. [9] F. Santambrogio, Optimal channel networks, landscape function and branched transport, Interfaces and Free Boundaries, 9 (2007), 149-169. [10] Q. Xia, Optimal paths related to transport problems, Communications in Contemporary Math, 5 (2003), 251-279, doi: 10.1142/S021919970300094X. [11] G. Wolansky, Limit theorems for optimal mass transportation, to appear in Cal. Var PDE., \arXiv{0903.0145}., ().

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##### References:
 [1] G. Buttazzo and E. Stepanov, "Optimal Urban Networks via Mass Transportation," Lec. Notes in Math., 1961, Springer-Verlag, 2009. [2] E. N. Gilbert, Minimum cost communication networks, Bell Sys. Tech. J., 46 (1967), 2209-2227. [3] E. N. Gilbert and H. O. Pollak, Steiner minimal trees, SIAM J. Appl. Math., 16 (1968), 1-29. doi: 10.1137/0116001. [4] F. K. Hwang, D. S. Richards and P. Winter, "The Steiner Tree Problem," Elsevier, North-Holland, 1992. [5] M. Bernot, V. Caselles and J. M. Morel, "Optimal Transportation Networks," Lec. Notes in Math., 1955, Springer-Verlag, 2009. [6] F. Maddalena, S. Solimini and J. J. Morel, A variational model of irrigation pattern, Interfaces and Free Boundaries, 5 (2003), 391-416. doi: 10.4171/IFB/85. [7] E. Paolini and E. Stepanov, Optimal transportation networks as flat chains, Interfaces Free Bound., 8 (2006), 393-436. doi: 10.4171/IFB/149. [8] E. Paolini and E. Stepanov, Connecting measures by means of branched transportation networks at finite cost, J. Math. Sci. (N. Y.), 157 (2009), 858-873. doi: 10.1007/s10958-009-9362-x. [9] F. Santambrogio, Optimal channel networks, landscape function and branched transport, Interfaces and Free Boundaries, 9 (2007), 149-169. [10] Q. Xia, Optimal paths related to transport problems, Communications in Contemporary Math, 5 (2003), 251-279, doi: 10.1142/S021919970300094X. [11] G. Wolansky, Limit theorems for optimal mass transportation, to appear in Cal. Var PDE., \arXiv{0903.0145}., ().
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