Geodesic
An example of constructing the Bellman function for extremal problems in BMO
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 42, Tome 424 (2014), pp. 33-125 Cet article a éte moissonné depuis la source Math-Net.Ru

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An example of solution of a boundary value problem for a homogeneous Monge–Ampère equation is given, which produces a Bellman function for an extremal problem on the space BMO. The paper contains a step-by-step instruction for calculation of this function. The cases of a rather complicated foliation are considered. This illustrates the technique elaborated in a paper by Ivanishvili, Stoyanov, Vasyunin, and Zatitsky. 
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V. I. Vasyunin. An example of constructing the Bellman function for extremal problems in BMO. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 42, Tome 424 (2014), pp. 33-125. http://geodesic.mathdoc.fr/item/ZNSL_2014_424_a1/

[1] P. Ivanishvili, N. Osipov, D. Stolyarov, V. Vasyunin, P. Zatitskiy, “On Bellman function for extremal problems in BMO”, Comptes Rendus Mathematique, 350 (2012), 561–564 | DOI | MR | Zbl

[2] P. Ivanishvili, N. Osipov, D. Stolyarov, V. Vasyunin, P. Zatitskiy, “Bellman function for extremal problems in BMO”, Transactions of the Amer. Math. Soc. (to appear)

[3] V. I. Vasyunin, P. B. Zatitskii, P. Ivanishvili, N. N. Osipov, D. M. Stolyarov, Funktsiya Bellmana v prostranstve BMO, Preprint No 19, POMI, 2011, 101 pp. http://www.pdmi.ras.ru/preprint/2011/rus-2011.html

[4] V. I. Vasyunin, P. B. Zatitskii, P. Ivanishvili, N. N. Osipov, D. M. Stolyarov, Dobavlenie k preprintu POMI 19/2011 “Funktsiya Bellmana v prostranstve BMO”, Preprint No 10, POMI, 2012, 15 pp. http://www.pdmi.ras.ru/preprint/2012/rus-2012.html

[5] P. Ivanishvili, N. Osipov, D. Stolyarov, V. Vasyunin, P. Zatitskiy, Bellman function for extremal problems in BMO, 2012, 91 pp., arXiv: 1205.7018v3 | MR

[6] P. Ivanishvili, D. Stolyarov, V. Vasyunin, P. Zatitskiy, Bellman function for extremal problems in BMO: evolution, (in prepfration)