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Optimal Riemannian distances preventing mass transfer. (English) Zbl 1077.49032

Summary: We consider an optimization problem related to mass transportation: given two probabilities \(f^+\) and \(f^-\) on an open subset \(\Omega\subset \mathbb R^N\), we let vary the cost of the transport among all distances associated with conformally flat Riemannian metrics on \(\Omega\) which satisfy an integral constraint (precisely, an upper bound on the \(L^1\)-norm of the Riemannian coefficient). Then, we search for an optimal distance which prevents as much as possible the transfer of \(f^+\) into \(f^-\): higher values of the Riemannian coefficient make the connection more difficult, but the problem is non-trivial due to the presence of the integral constraint. In particular, the existence of a solution is a priori guaranteed only on the relaxed class of costs, which are associated with possibly non-Riemannian Finsler metrics. Our main result shows that a solution does exist in the initial class of Riemannian distances.

MSC:

49Q20 Variational problems in a geometric measure-theoretic setting
49J45 Methods involving semicontinuity and convergence; relaxation
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[1] Acerbi E., J. Anal. Math. 43 pp 183–
[2] Benamou J.-D., Numer. Math. 84 (3) pp 375– (2000)
[3] Bouchitte G., J. Eur. Math. Soc. 3 (2) pp 139– (2001)
[4] Bouchitte, G., Buttazzo, G., Seppecher, P., Shape optimization solutions via Monge-Kantorovich equation, C. R. Acad. Sci. Paris Ser. I Math. 324(10) (1997), 1185-1191. · Zbl 0884.49023
[5] Braides A., Asympt. Anal. 31 (2) pp 177– (2002)
[6] Buttazzo, G., Semicontinuity, relaxation and integral representation in the calculus of variations, Pitman Res. Notes Math. Ser. 207, Longman Scientific, Harlow 1989. · Zbl 0669.49005
[7] Buttazzo G., Discr. Contin. Dynam. Syst. 7 (2) pp 247– (2001)
[8] Caarelli L. A., J. Amer. Math. Soc. 15 (1) pp 1– (2002)
[9] De Cecco G., Math. Z. 218 (2) pp 223– (1995)
[10] De Pascale L., Equ. 14 (3) pp 249– (2002)
[11] Evans, L. C., Partial di erential equations and Monge-Kantorovich mass transfer, Current developments in mathematics (Cambridge, MA 1997), Int. Press, Boston, MA (1999), 65-126.
[12] Evans L. C., Mem. Amer. Math. Soc. pp 137– (1999)
[13] Feldman M., Equ. 15 (1) pp 81–
[14] Gangbo W., Acta Math. 177 (2) pp 113– (1996)
[15] Kantorovich L. V., Dokl. Akad. Nauk. SSSR 37 pp 227– (1942)
[16] Kantorovich L. V., Uspekhi Mat. Nauk. 3 pp 225– (1948)
[17] Monge, G., Memoire sur la theorie des Deblais et des Remblais, Hist. de L’Acad. des Sciences de Paris (1781).
[18] Rachev, S. T., R schendorf, L., Mass transportation problems, Vol. I II, Probability and its Applications, Springer-Verlag, New York 1998.
[19] Trudinger N. S., Equ. 13 (1) pp 19– (2001)
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