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4 The mean field limit where Cd is a constant depending only on the dimension d . 44) for ’ n large enough. 1/ when Á ! x y/dxdy < C1. 1/: ´ ´ Also Vd n ! C1 1 Hn . 45) are supported Ä I. 1/ and letting Á ! 0 finally gives us the €-lim sup inequality, which concludes the proof. 19. x y/ dx dy < 1. This shows that the result still holds for all such interaction kernels. 20. To prove the €-liminf relation, we have only used that V is l. s. c. and bounded below. To prove the €-limsup relation, we have assumed that V is continuous for convenience.

2. The coincidence set for a one-dimensional obstacle problem. 49) has two options at each point: to touch the obstacle or not (and typically uses both possibilities). x/ q: e:g is closed and called the coincidence set or the contact set. ” The obstacle problem thus belongs to the class of so-called free-boundary problems, cf. [Fri]. Trying to compute the Euler–Lagrange associated to this problem by perturbing h by a small function, one is led to two possibilities depending on whether h D or h > .

10. 1. Since we assumed for simplicity that V is continuous and finite, it suffices to assume (A2) to have (A1) and (A3). With all the preceding, we may conclude with the following result, which goes back to [Cho]. 2 (Convergence of minimizers and minima of Hn ). Assume that V is continuous and satisfies (A2). x1 ; : : : ; xn /gn is a minimizer of Hn . Then, 1X ıxi ! x1 ; : : : ; xn / D I. 48) Proof. C1 n12 min Hn is bounded above (by I. 21. 5 Linking the equilibrium measure with the obstacle problem 29 P follows that, up to a subsequence, we have n1 niD1 ıxi !