A nonlocal free boundary problem with Wasserstein distance

We study the probability measures ρ∈M(R2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho \in \mathcal M(\mathbb R^{2})$$\end{document} minimizing the functional J[ρ]=∬log1|x-y|dρ(x)dρ(y)+d2(ρ,ρ0),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} J[\rho ]=\iint \log \frac{1}{|x-y|}d\rho (x)d\rho (y)+d^2(\rho , \rho _0), \end{aligned}$$\end{document}where ρ0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho _0$$\end{document} is a given probability measure and d(ρ,ρ0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d(\rho , \rho _0)$$\end{document} is the 2-Wasserstein distance of ρ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho $$\end{document} and ρ0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho _0$$\end{document}. J[ρ]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J[\rho ]$$\end{document} appears in aggregation models when the movement of particles is advanced by the potential -log|x|∗ρ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-\log |x|*\rho $$\end{document}. We prove the existence of minimizers ρ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho $$\end{document} and show that the potential Uρ=-log|x|∗ρ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U^\rho =-\log |x|*\rho $$\end{document} solves a degenerate obstacle problem, the obstacle being the transport potential. Every minimizer ρ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho $$\end{document} is absolutely continuous with respect to the Lebesgue measure. The singular set of the free boundary of the obstacle problem is contained in a rectifiable set, and its Hausdorff dimension is <n-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$< n-1$$\end{document}.Moreover, Uρ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U^\rho $$\end{document} solves a nonlocal Monge–Ampère equation, which after linearization leads to the equation ρt=div(ρ∇Uρ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho _t={\text {div}}(\rho \nabla U^\rho )$$\end{document}. The methods we develop use Fourier transform techniques. They work equally well in high dimensions n≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge 2$$\end{document} for the energy J[ρ]=∬|x-y|2-ndρ(x)dρ(y)+d2(ρ,ρ0).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} J[\rho ]=\iint |x-y|^{2-n}d\rho (x)d\rho (y)+d^2(\rho , \rho _0). \end{aligned}$$\end{document}


Introduction
In this paper we are concerned with the minimization of the functional among all probability measures ρ with finite second momentum.Here d 2 (ρ, ρ 0 ) = inf γ 1 2 |x − y| 2 dγ (x, y) is the square of the Wasserstein distance between ρ and a given probability measure ρ 0 , and γ is a joint probability measure with marginals π x# γ = ρ, π y# γ = ρ 0 .The support of ρ is a priori unknown (or free) and our main goal is to analyze the regularity of the free boundary, i.e. the boundary of the set where ρ = 0.
An analogous problem arises in high dimensions if we replace the logarithmic kernel by K (x − y) = |x − y| 2−n , n ≥ 3. The methods we employ do not depend on the dimension.We focus on the logarithmic kernels since the potential U ρ = −ρ * log |x| may change sign and log-interaction phenomenon has a number of important applications [29,32] (in Sect. 2 we also give a connection with random matrices).
An interesting feature of the variational problem for J [ρ] is that it leads to an obstacle problem involving the potential of the optimal transport of ρ to ρ 0 .Let U ρ be the logarithmic (or the Newtonian potential if n ≥ 3) of the probability measure ρ and ψ the potential of the transport map, then formally we have U ρ = ψ in {ρ > 0} and U ρ ≥ ψ elsewhere. (1.2) Since U ρ = −2πρ then it follows that Thus combining (1.2) and (1.3) we have the obstacle problem In this formulation the position of the obstacle is a priori unknown as opposed to the classical case [7].Note that ψ is semiconvex function, hence from Aleksandrov's theorem it follows that D 2 ψ exists a.e.Consequently, the first equation in (1.4) is satisfied in a.e.sense provided that ρ is absolutely continuous with respect to the Lebesgue measure.
The partial mass transport and Monge-Ampère obstacle problems had been developed in the seminal work of Caffarelli and McCann [6], see also [16,19] and the references given there.
Several papers introduced variational problems for measures.In [26] McCann formulated a variational principle for the energy which allowed to prove existence and uniqueness for a family of attracting gas models, and generalized the Brunn-Minkowski inequality from sets to measures.Another interesting energy appears in the large deviation laws and log-gas interactions [29,32].This problem is also related to the classical obstacle problem [3].Thanks to the quadratic potential every measure minimizing F [•] is confined in some ball.Furthermore, one can prove transport inequalities and bounds for the Wasserstein distance in terms of F[ρ] [25].
There is a vast literature on interaction energies for probability measures governed by the Wasserstein metric [8,10,11,20].In particular, [13] contains an L ∞ estimate for the equilibrium measure and [14] a connection to obstacle problems.The energy J [ρ] appears in a number of physical considerations, for example in aggregation models where the movement of particles is advanced by the global potential U ρ .The corresponding gradient flow is ρ t = div(ρ∇U ρ ) [30], page 307.Another application appears in thermalization of granular media [12].
In [31] Savin considered the optimal transport of the probability measures in periodic setting for the energy The resulted obstacle problem takes the form where ψ is the transport potential of ρ → ρ 0 with given initial periodic probability measure ρ 0 with H 1 density.The aim of this paper is to study the free boundary of the obstacle problem (1.4) for the minimizers ρ of J [ρ].
In [4], the authors consider a gradient flow of the interaction energy and study the convergence of radially symmetric solutions under conditions imposed on K, namely radially symmetric solutions converge exponentially fast in some transport distance toward a spherical shell stationary state.
In [5], they study the minimizers of E 0 in d ∞ topology.Their main result being a Hausdorff dimension estimate of the support of the minimizer.
In [9], the minimization problem for E 0 under the constraint d ∞ (μ, μ 0 ) < is studied.As a result they obtain an obstacle type problem for the potential of transport.They also prove finite perimeter of suppρ and C 1,1 regularity of transport potential under some conditions on K.
In our paper the energy is different, it has the additional term.Moreover, the set of admissible measures is not constrained to a neighborhood of a given initial measure ρ 0 in d ∞ topology.Note also that above papers do not study the singular set of the free boundary, whereas we estimate the dimension of the singular set of suppρ 0 ∩ suppρ.This estimate is quite different as opposed to the classical obstacle problem.

Main results
The energy J [ρ] has nonlocal character due to the presence of the logarithmic kernel.However, thanks to the Wasserstein distance ρ is forced to have compact support provided that suppρ 0 is compact.Observe that if ρ has atoms then J [ρ] = ∞, since the logarithmic term is unbounded, see also the discussion on page 133 [2] for the pure optimal transport case for the energy without the logarithmic term.
Theorem A If ρ 0 has compact support then there is a probability measure ρ minimizing J such that suppρ is compact.Moreover, ρ cannot have atoms and hence there is a measure preserving transport map y = T (x) such that ρ 0 is the push forward of ρ.
The second part of the theorem follows from the standard theory of optimal transport [1], [2].The chief difficulty in proving the first part is to show that there is a minimizing sequence of probability measure with uniformly bounded supports.In order to establish this we use Carleson's estimate from below for the nonlocal term and a localization argument for the Fourier transforms of these measures.For other applications of Fourier transforms see [23] and reverences therein.
Next we want to analyze the character of equilibrium measures.We show that ρ ∈ L ∞ .To see this we compute and explore the first variation of J .The weak form of the Euler-Lagrange equation implies that ρ, the Fourier transform of ρ, is in L 2 .
Theorem B Let ρ, ρ 0 be as in Theorem A. Then ρ ∈ L 2 (R 2 ) and dρ = f dx on suppρ where f ∈ L ∞ (R 2 ).In particular, the transport map y = T (x) (as in Theorem A) is given by where U ρ = ρ * K is the potential of ρ and ∇U ρ is log-Lipschitz continuous.
The log-Lipschitz continuity of ∇U ρ follows from Judovič's theorem [21].In fact from the Calderòn-Zygmund estimates it follows that D 2 U ρ ∈ L p loc for every p > 1.The local mass balance condition for the optimal transport leads to a nonlocal Monge-Ampère equation . (1.6) (1.6) implies that suppρ ⊂ suppρ 0 .If we linearize (1.6) using a time discretization scheme, the resulted equation is ρ t = div(ρ∇U ρ ).
The analysis of the structure of singular set in the obstacle problems is the central problem of the regularity theory.Let MD(suppρ ∩ B r (x)) be the infimum of distances between pairs of parallel planes such that suppρ ∩ B r (x) is contained in the strip determined by them [7].Let From here we can deduce the equation (1.8) Consequently, the standard regularity theory for the Monge-Ampère equation (see [33]) implies that we can get higher regularity for ρ if ρ 0 is sufficiently smooth.
Theorem C Let ω(R) be the modulus of continuity of the slab height (see a collection of disjoint balls included in B R with x i ∈ S, where S is the singular set.Then for every β > n − 1 we have The paper is organized as follows: In Sect. 2 we recall some facts on the Wasserstein distance and Fourier transformation of measures.One of the key facts that we use is that the logarithmic term can be written as a weighted L 2 norm of the Fourier transformation of ρ.
Section 3 contains the proof of Theorem A. The chief difficulty in the proof is to control the supports of the sequence of minimizing measures.Section 4 contains some basic discussion of cyclic monotonicity and maximal Kantorovich potential.Then we derive the Euler-Lagrange equation.From here we infer that ρ has L ∞ density with respect to the Lebesgue measure.Theorem B follows from Theorem 4.4 and Corollary 4.6.
In Sect. 5 we study the regularity of free boundary and prove Theorem C. The last two sections contain some final remarks and possible applications.First, in Sect.6 we discuss the relation of J [ρ] with the large deviations laws for the random matrices with interaction and provide a simple model with energy J .Finally, Sect.7 is devoted to the nonlocal Monge-Ampère equation and its linearization ∂ t ρ = div(ρ∇U ρ ).

Notation
We will denote by M(R n ) the set of probability measures on R n , and let μ # f be the push (1.9)

Set-up
Let f : R n → R n be a map, for a Borel set E ⊂ R n the push forward is defined by For probability measures ρ, ρ 0 ∈ M(R n ) we define their Wasserstein distance as follows where γ 's are transport plans such that γ #π x = ρ, γ #π y = ρ 0 .We recall the following properties of the Wasserstein distance: See [34] for more details.
We also need the following definition of Wasserstein class: ) be a Polish space (i.e.complete separable metric space equipped with its Borel σ -algebra).The Wasserstein space of order 2 is defined as where x 0 ∈ is arbitrary.This space does not depend on the choice of x 0 .Thus d defines a finite distance on P 2 .

Remark 2.2
If is compact then so is P 2 .If is only locally compact then P 2 ( ) is not locally compact, see [34].This introduces several difficulties in the proof of the existence of a minimizer.

Remark 2.3
Recall that the Fourier transformation of the truncated kernel where c 1 > 0 is a universal constant, B is the Bessel function of the first kind such that ) has compact support then from the weak Parseval identity we have that where K (x − y) = log 1 |x−y| and μ, K are the Fourier transforms of μ, K respectively, see [22] for the proof.This observation shows that the energy J is nonnegative for compactly supported μ ∈ M(R 2 ).
We say that μ ∈ M(R n ) has finite energy if has Hilbert structure, [24] page 82, and is a norm.It is remarkable that the standard mollifications μ k of μ converge to μ strongly, i.e. lim k→∞ μ − μ k = 0, see [24] Lemma 1.2 page 83.

Existence of minimizers
and J be given by (1.1), then ) and ε k are the corresponding numbers from (ii) then there is ε 0 > 0 such that ε k ≤ ε 0 uniformly in k, where ε 0 depends only on C and R 0 .
Proof We split the proof into three steps: Step 1: Second moment estimate: Let ε > 0 be fixed.By Theorem 1 [28] there is transference plan Moreover, the projections of γ ε are μ ε = 1 μ(B ε ) μvB ε and μ 0 .Hence Since γ ε has marginals μ ε , μ 0 then 1 2 |x − y| 2 γ ε ≥ d 2 (μ ε , μ 0 ).Consequently, this in combination with the last inequality yields where we denote provided that ε > R 0 .From Hölder's inequality we have that Step 2: A bound for the logarithmic term: Now we want to estimate the logarithmic term from below using the method from Chapter 1.1 [29].To do so we denote Q(x) = c 0 |x| 2 , w(x) = e −c 0 |x| 2 and introduce the logarithmic energy with quadratic potential , with this we have Observe that Therefore for every large constant T 0 > 0 there is ε such that if max{|x|, |y|} ≥ ε then K w (x, y) ≥ T 0 .This yields the following estimate for I w Thus after some simplification we get (3.5) Step 3: Energy comparison in B ε : Combining (3.5) with (3.1) we get The last three terms on the last line can be further estimated from below as follows In particular from here and (2.3) we see that J [μ] > −∞ and hence (i) follows.Now if we choose then from (3.6) it follows that This implies 2 , hence it is enough to take the minimization over M(B ε ).
It remains to check (iii).First we estimate From (3.7) it follows that T 0 can be chosen to be the same for every μ k , say T 0 > Ĉ, satisfying 0 ≤ J [μ k ] ≤ C and the proof is complete.Now we are ready to finish the proof of Theorem A.
Then there exists a minimizer ρ ∈ M(R 2 ) of J .Moreover, the support of ρ is bounded.
Proof First note that if we take the uniform measure μ of some ball B having positive distance from B R 0 then J [μ] < +∞.Hence by Proposition 3.1 (i) we have that J ) is a minimizing sequence then without loss of generality we can assume that J [μ k ] ≤ C for some C > 0 uniformly in k.Moreover, from Proposition 3.1 (ii) it follows that there are positive numbers ε k > 0 such that for the restriction measures μ k,ε k we have On the other hand it follows from (2.3) that thanks to (3.8).Consequently, applying Proposition 3.1 (iii), we can use the weak compactness of μ k,ε k in M(B ε 0 ) to get a weakly converging subsequence still denoted μ k,ε k to some ρ ∈ M(B ε 0 ).The logarithmic term is lower-semicontinuous [29], or [24] page 78, hence from the lower-semicontinuity of d (see property 4) in Sect.2) it follows that and the desired result follows.
Cancelling the square terms from (4.1) we get Let γ be a transference plane with marginals ρ, ρ 0 .It is well known that the support of γ is cyclically monotone, see [1] Theorem 2.2.
|x − y| 2 and introduce the function where the supremum is taken over all cycles of finite length.It is easy to check that ψ defined in (4.3) satisfies ψ (x) ≤ 0 and the normalization condition ψ (x 0 ) = 0.If γ (x, y) is a transference plan then it is contained in the c superdifferential of the c concave function ψ constructed above.ψ is called the maximal Kantorovich potential.Moreover, we have that if (x , y ) ∈ suppγ then for every See Theorem 2.3 [1] for proof.

Remark 4.2
Recall that by Corollary 2.2 [1] if (CC) graphs are ρ negligible then the transference plan γ is unique and the transport map T = ∇v for some convex potential v.
We want to show that in (4.4) we can take ψ = 2U ρ , and ρ is absolutely continuous with respect to the Lebesgue measure.
Note that [24] Lemma 1.2 page 83 as k → ∞.Since U ρ ∈ H 1 (see [22]) is superharmonic (hence bounded below in B, say by C B ) then from Fatou's lemma we get that where C depends only on the dimension.

Theorem 4.4
Let ρ be a minimizer.Suppose the infimum in d(ρ, ρ 0 ) is realized for a transference plan γ and (x * , y * ) ∈ suppγ .Then ρ has L ∞ density with respect to the Lebesgue measure, and for every x 0 we have Moreover, ∇U ρ is log-Lipschitz continuous.
For the other marginal, we have Observe that by (4.8) and the definition of γ * ε we have provided that t is small enough.Consequently we can use ρ − tϕ * + tϕ 0 against ρ and get from the convexity of d 2 (see Sect. 2) the following estimate For the nonlocal term we have Then the energy comparison yields Dividing by t and sending t → 0, t > 0 we get that Since γ ε is the push forward of γ * ε under translation x → x − x * + x 0 then we have from (4.9) (4.10) Fig. 1 The geometric construction of joint measures γ ε and γ * ε via restriction and translation Taking x * − x 0 = ±he j , where e j is the unit direction of the jth coordinate axis, h > 0, and adding the resulted inequalities (4.10) we get Note that by Lemma 4.3 the left hand side of (4.12) is well defined.
) be the discrete Laplacian.Then from (4.12) with a sequence of cut offs ξ k ↑ 1 on B ε 0 , using the dominated convergence theorem, since U ρ ρ is a signed Radon measure (see Lemma 4.3 and (4.6)), and recalling that ρ has compact support, it follows that Since suppρ is compact we can assume that K vanishes outside of B r 0 and consider the truncated kernel K r 0 = 1 B r 0 K .From the weak Parseval identity we get that Letting h → 0 and applying Fatou's lemma we get Since the left hand side of the previous inequality does not depend on r 0 we can let r 0 → ∞ and applying Fatou's lemma again we see that Since Fourier transform is isometry on L 2 then ρ, the inverse Fourier transform of ρ, exists and ρ ∈ L 2 .But then (ρ − ρ) = 0, and it follows that ρ has L 2 density.The proof of the claim is complete.
Returning to the localized inequality (4.12) with (x * , y * ) ∈ suppγ we get Using the weak convergence of second order finite differences in L 2 we finally obtain from (4.13) and .
Consequently, the upper Lebesgue density of the measure ρ is bounded by some universal constant and hence dρ = f dx for some f ∈ L ∞ (R n ) [18].Therefore from Judovič's theorem [21] ∇U ρ is log-Lipschitz continuous.Moreover, by construction Hence from (4.9) and the mean value theorem we get that Corollary 4.6 Let ρ be a minimizer of J , then U ρ = ψ on suppρ.Furthermore, suppρ has nonempty interior.
Proof In view of (4.4) and (4.7) U ρ and ψ have the same c-subdifferential on suppρ then it follows that U ρ = ψ and at free boundary point x * = y * we have ∇U ρ (x * ) = 0.The last claim follows from the log-Lipschitz continuity of ∇U ρ .

Regularity of free boundary
Let x * ∈ suppρ, then from (4.7) we have for every x and since U ρ is superharmonic in R 2 it follows from Hopf's lemma, applied to a with diameter x * y * , that the normal derivative ∂ ν U ρ (x * ) < 0 where ν = x * −y * |x * −y * | .Hence at the remaining free boundary points we must have x * = y * and hence ∇ψ(x * ) = 0. Definition 5.1 Let T be the transport map.We say that x ∈ suppρ ∩ suppρ 0 is a singular free boundary point if x = T (x), ∇U ρ (x) = 0 and lim sup t↓0 The set of singular points is denoted by S.
Lemma 5.2 Let 0 be a singular free boundary point and ρ 0 ≥ s > 0 on suppρ 0 .Then for every small ε > 0 there is R * > 0 such that the set of singular points in B R , R < R * can be trapped between two parallel planes at distance √ 8n+1 Proof Let K be the convex hull of the singular set in B R .Then there is x 0 ∈ B R and an ellipsoid E (John's ellipsoid [17] page 139) so that Let r be the smallest axis of E. By mass balance condition ρ 0 (y)dy. ( By assumption 0 is a singular point, so we have lim sup t↓0 1 |B t | B t ρ(x)dx = 0. Thus for every ε > 0 small there is a 0 such that B a ρ(x)dx ≤ εa n whenever a < a 0 . (5.2) By assumption ρ 0 > s > 0 then ρ 0 (y) dy > sr n c n (5.3) while B a ρ (x) dx < εa n .Consequently, combining (5.1)-( 5.3) we get εa n > sr n c n or It follows that (for small R and ε) there is a point A ∈ B r 2n (x 0 )∩{ρ 0 > 0} and B ∈ {ρ > 0} so that |O B| ∼ a and T −1 (A) = B.
Let x s be a singular point.Notice that x s = T (x s ), i.e. the singular free boundary points are fixed points.From the monotonicity (4.2) be the midpoint of the segment AB, then Fig. 2 The construction used in the proof of Lemma 5.2.The red ball is B r 2n (x 0 ), and Hence we arrive at From simple geometric considerations we have that (see Figure 2) Therefore S ∩ B R is on one side of the hyperplane containing the intersection B R and the ball with diameter AB, see Fig. 2. Hence or, in view of (5.4), we get 4R 2 ≥ r 2n r sc n ε R and the proof is complete.
Lemma 5. 3 Let ω(R) be the height of the slab containing S ∩ B R (see (1.7)), B i = B r i (x i ) a collection of disjoint balls included in B R with x i ∈ S. Then for every β > n − 1 we have r Proof Rotate the coordinate system such that x n points in the direction of the normal of the parallel planes which are ω(R) apart and contain S ∩ B R .Let F 0 be the collection of the balls satisfying Rω(R) 2 Rω(R).Therefore there are at most such balls.Thus we have Hence let F 1 be the collection of the balls B i contained in ∪ j B 4Rω(R) (y j ) and satisfying 2 and the number such balls B i is at most Again, as above we can choose at most then repeating the argument above we have that 1) .Now we can finish the proof of Theorem C.
Proof Let x ∈ S be such that there exists a unique normal in measure theoretic sense, see Definition 5.6 [18].Notice that at the point x, where such normal exists the set has approximate tangent plane.Therefore the projections of B r (x) ∩ S onto two dimensional planes have diameter at least 2R.Thus we let M 0 be the subset of S such that for x ∈ M 0 there is sequence R k → 0 such that the projections of B R i (x) onto some two dimensional plane is of order R 1+σ .Now let B r i (x i ) be a Besikovitch type covering of B R ∩ M 0 .Let us cover B r i (x i ) ∩ M 0 with balls of radius r , then there are at most 4 and β := n − 1 + δ and set We want to show that for this choice of β we get α = n − 1 − σ for some σ > 0 depending on n and σ .Indeed, we have

Random matrices: an example
In this section we discuss a problem related to random matrices which leads to the obstacle problem (1.4).Let H be a Hermitian matrix, i.e.H † i j = H ji (or H † = H for short) where Hi j are the complex conjugates of the entries of N × N matrix H .One of the well known random matrix ensembles is the Gaussian ensemble.The probability density of the random variables in the Gauss ensemble is given by the formula where κ > 0 and is the trace of the squared matrix [27].The dispersion is the same for every H in the ensemble.
The corresponding statistical sum is Z N can be rewritten in an equivalent form  The solution of this equation (given in terms of Hilbert's transform) has the form and this is Wigner's famous semicircle law [35], see also [32].
For the problem with d Consequently, the prescribed Jacobian equation is Note that this is a nonlocal Monge-Ampère equation.By standard W 2, p estimates for the potential U ρ it follows that suppρ 0 \ suppρ has vanishing Lebesgue measure.Let h > 0 be small and consider the perturbed energy Therefore, sending h → 0 we obtain the equation ∂ t ρ = ∇ρ∇U ρ + U ρ ρ = div(ρ∇U ρ ).

Lemma 4 . 3
Let ρ be a minimizer, then U ρ ρ is a signed Radon measure.Proof Let ξ ∈ C ∞ 0 (B)be a cut-off function of some ball B. Let {ρ k } ∞ k=1 be a sequence of mollifications of ρ.Recall that by Remark 2.3 I [ρ k ] < ∞, and ρ − ρ k

x 2 ig i x 2 iRemark 6 . 1 2 R
dx i = e −W dx 1 . . .dx N ,whereW = − i = j log |x i − x j | + Nand we replaced κ = Ng for convenience.If we assume that the particles (in the equilibrium) have density ρ then from approximation of Riemann's sum we get thatW ∼ −N 2 log |x − y|ρ(x)ρ(y)dxdy + N 2 g ρ(x)|x| 2 .As N → ∞ the main contribution comes from the minimum of the functionalF[ρ] = log |x − y|ρ(x)ρ(y)dxdy + g ρ(x)|x| 2 with respect to the constraint R ρ = 1.If in W the quadratic term is replaced by − 1 2 |x i − y i | 2 γ (x i , y i ), g ∼ N , H 0 = diag(y 1 , . . ., y N ), then we get the model corresponding to the energy J .Let n = 1, then the first variation of F[ρ] gives −log |x − y|ρ(y) + x 2 g = λ, where λ is the Lagrange multiplier of the constraint R ρ = 1.Differentiating in x we get P.V.