14 Introduction Had the author had more time and/or courage, he would have been really pleased to add two chapters on the semi-geostrophic equations on one hand, on electromagnetism on the other. The semi-geostrophic equations provide a beautiful example of a Hamiltonian system arising in fluid mechanics, in which the quadratic Wasserstein distance, amazingly, plays an explicit role. They were first studied by Cullen and collaborators [101, 100, 97], then by various mathematicians [36, 98, 99]. It is absolutely remarkable that Cullen’s group was led to the intuition of Brenier’s theorem on the sole basis of physical considerations like stability of some weather patterns. To get an idea of this field, the reader may consult Problem 9 in Chapter 10. As for electromagnetism, what we have in mind is the series of recent works by Brenier [55] about the interpolation of currents and the way to use the Monge-Amp`ere equation as a replacement for the Poisson equation arising from Coulomb interaction. On both topics current research is advancing rather fast. Warm-up exercises The following exercises are specifically intended for students who would not feel so comfortable with some of the basic notions used in this introduc- tion, and would like to practise just a little bit before more serious matters begin. In case the reader needs to review measure theory, he or she can consult for instance Rudin [217]. 1. About the definition of image measure. Let X be a probability space, equipped with a probability measure μ, and let Y be an abstract set. Given any map T : X Y , show that Y can be equipped with a structure of measure space, in such a way that T is measurable. Show that the formula ν[B] = μ[T −1(B)] uniquely defines a probability measure ν on Y . Prove that this measure is characterized by equation (9) holding true for all bounded measurable ψ. Show that this identity actually holds true for all ψ L1(dν). 2. About the change of variable formula. Construct examples in which T = ν, dμ(x) = f(x) dx, dν(y) = g(y) dy, but formula (15) does not apply, because (a) T is not smooth enough, or (b) T is not one-to-one. 3. About Choquet’s theorem. Let K be a compact convex set of a Banach space E, and let E(K) stand for the set of all extremal points of K. Let : K R be the restriction of a continuous linear functional on E. (i) Recall why admits a minimizer in K. The goal is now to show that at least one of these minimizers lies in E(K).
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