Partial regularity of Brenier solutions of the Monge-Ampère equation

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4 ALESSIO FIGALLI AND YOUNG-HEON KIM so that, since c(n)| det L| −1 ≤ |Z| ≤ C(n)| det L| −1 with C(n), c(n) > 0, we get inf Z ϕ n ≥ c(n, α, λ) |(αZ) ∩ Ω| |Z| > 0. (3.1) On the other hand, let CL,x denote the cone generated by (L(x), ϕL(x)) and (∂(L(Z)), ϕL| ∂(L(Z))). Then (UA) implies that is |ϕL(L(x))| n ≤ C(n) |∂ − CL,x({L(x)})| dist ( L(x), ∂(L(Z)) ) , ϕ(x) n ≤ C(n) |∂ − Cx({x})| |Z| dist ( L(x), ∂(L(Z)) ) ≤ C(n) |∂ − ϕ(Z)| |Z| dist ( L(x), ∂(L(Z)) ) ≤ C(n, λ) |Z ∩ Ω| |Z| dist ( L(x), ∂(L(Z)) ) , where the second inequality follows from the inclusion ∂ − Cx({x}) ⊂ ∂ − ϕ(Z), which can be easily proven by moving down any supporting plane of Cx at x and lifting it up until it touches the graph of ϕ inside Z. (3.2) Using (3.1) and (3.2), the regularity theory for Alexandrov solutions to (1.1) goes as follows: • Strict convexity. Let y ∈ ∂ − ϕ(Ω), and set Sy := {z : y ∈ ∂ − ϕ(z)} = ∂ − ϕ ∗ (y). (Here ϕ ∗ denotes the Legendre transform of ϕ.) We want to prove that Sy is a singleton, which is equivalent to the strict convexity of ϕ. Assume that is not the case. Since Sy is convex, there are two possibilities: (1) Sy contains an infinite line (Rv) + w (v, w ∈ R n , v = 0). (2) There exists x0 an exposed point for Sy, namely, there is an affine function A : R n → R such that A(x0) = 0 and A < 0 on Sy \ {x0}. Case (1) is excluded by the Monge-Ampère equation, since this would imply that ∂ − ϕ(R n ) is contained inside the (n − 1)-dimensional subspace v ⊥ , so that |∂ − ϕ|(R n ) = 0, a contradiction to (1.1). (See [6] or [9, Lemma 2.4] for more details.) For the remaining case (2), with no loss of generality we can assume y = 0, x0 = 0 and A(z) = zn = z · en. First of all, one shows that x0 ∈ Ω. In fact, if this was the case, we could apply the Alexadrov estimates (3.1) and (3.2) to the sections Zε := {ϕ(z) < ε(zn − 1)} ∋ 0. Since 0 is almost a minimum point for ϕ − ε(zn − 1) and his image after renormalization is very close to the boundary of the renormalized section, as shown in [3, Theorem 1] this leads to a contradiction to (3.1) and (3.2) as ε → 0. Analogously one proves that x0 ∈ ∂Ω. Indeed, one considers the sections Zε := {ϕ(z) < ε(zn − 2R0)}, where R0 is such that Ω ⊂ BR0 . This ensures that after renormalization the measure |∂ − ϕLε| has some positive mass inside Lε(Zε) (uniformly as ε → 0), and this allows to conclude as above [6, Lemma 3]. Finally, since |∂ − ϕ| = 0 outside Ω but |∂ − ϕ| has always some positive mass inside a section Zε as the ones above, it is easily seen that x0 cannot belong to R n \ Ω (see [6, Lemma 3]). This together with above implies that there is no exposed point, yielding a contradiction. This concludes the proof of the strict convexity of ϕ.
PARTIAL REGULARITY OF BRENIER SOLUTIONS OF THE MONGE-AMPÈRE EQUATION 5 • Differentiability and C 1,α -regularity. As shown in [3, Corollary 1], (3.1) and (3.2) allow also to show the differentiability of ϕ at any point of strict convexity. This implies that ϕ is strictly convex and C 1 inside Ω. Finally, using again the Alexandrov estimates, the C 1,α -regularity of ϕ follows by a compactness argument [5, Lemma 2 and Theorem 2]. 3.2. Partial regularity of Brenier solutions. In this subsection we show the partial regularity result for Brenier solutions described in the introduction. As we already observed at the beginning of this section, if ϕ is a Brenier solutions of (1.1) then |∂ − ϕ| ≥ λ 2 χΩ. In particular (3.1) still holds true. The problem in this case is that we cannot deduce (3.2), since now no upper bound on |∂ − ϕ| is available [6]. To bypass this problem, we will use the following two preliminary results: Lemma 3.1. Let C ⊂ BR be a convex set such that 0 ∈ C, and fix δ ∈ (0, R). Then |C ∩ Bδ| ≥ C(δ, R) |C|. Proof. Since |C| = |C ∩ Bδ| + |C \ Bδ|, it suffices to bound |C \ Bδ|. Set D := C ∩ ∂Bδ, E := R + D = {tx : t > 0, x ∈ D}, Eδ := E ∩ Bδ, and ER := E ∩ BR. Since 0 ∈ C, by the convexity of C it is easily seen that the following inclusions hold: Hence C \ Bδ ⊂ ER, Eδ ⊂ C ∩ Bδ. |C \ Bδ| ≤ |ER| ≤ C(δ, R) |Eδ| ≤ C(δ, R) |C ∩ Bδ|, as desired. Lemma 3.2. Let ϕ be a Brenier solution of (1.1), and let Z be a section of ϕ. Then |∂ − ϕ(Z) ∩ Λ| ≤ 1 |Z ∩ Ω|. λ2 Proof. Let ϕ ∗ denote the Legendre transform of ϕ. Since ∇ϕ#χΩ ≥ λ 2 χΛ, for a.e. y ∈ Λ there exists x ∈ Ω such that x ∈ ∂ − ϕ ∗ (y). Thanks to (2.1), this implies that ∂ − ϕ ∗ (y) ∩ Ω = ∅ for all y ∈ Λ, so by the boundedness of Ω we deduce that ϕ ∗ is Lipschitz, and hence differentiable a.e., inside Λ. This gives |∂ − ϕ(Z) ∩ Λ| = |∂ − ϕ(Z) ∩ Λ ∩ (dom∇ϕ ∗ )|. Now, thanks to the duality relation (2.2), if y ∈ ∂ − ϕ(x1) ∩ ∂ − ϕ(x2) and y ∈ dom∇ϕ ∗ then x1 = x2. This implies (∇ϕ) −1( ∂ − ϕ(Z) ∩ (dom∇ϕ ∗ ) ) ⊂ Z. Combining all together and using the definition of Brenier solution we finally obtain |∂ − ϕ(Z) ∩ Λ| = |∂ − ϕ(Z) ∩ Λ ∩ (dom∇ϕ ∗ )| ≤ 1 λ 2 −1 (∇ϕ) ( ∂ − ϕ(Z) ∩ (dom∇ϕ ∗ ) ) ∩ Ω 1 ≤ |Z ∩ Ω|. λ2 Thanks to the two results above, we can show the following key estimate, which will play the role of (3.2) in our situation:
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Partial regularity of Brenier solutions of the Monge-Ampère equation

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