W^{2,1} estimate for singular solutions to the Monge-Ampere equation
WW , ESTIMATE FOR SINGULAR SOLUTIONS TO THEMONGE-AMP`ERE EQUATION
CONNOR MOONEY
Abstract.
We prove an interior W , estimate for singular solutions to theMonge-Amp`ere equation, and construct an example to show our results areoptimal. Mathematics Subject Classification Introduction
Interior W ,p estimates for the Monge-Amp`ere equationdet D u = f in Ω , u | ∂ Ω = 0were first obtained by Caffarelli assuming that f has small oscillation dependingon p (see [C2]).In the case that we only have λ ≤ f ≤ Λ , De Philippis, Figalli and Savin recentlyobtained interior W , (cid:15) estimates for some (cid:15) depending only on n, Λ and Λ (see[DF],[DFS]). This result is optimal in light of counterexamples due to Wang ([W])obtained by seeking solutions with the homogeneity u ( x, y ) = 1 λ α u ( λx, λ α y ) . These can be viewed as estimates for strictly convex solutions to the Monge-Amp`ere equation. Indeed, at a point x where u is strictly convex we can find atangent plane that touches only at x and lift it a little to carve out a set where u has linear boundary data.In [M] we show that solutions to λ ≤ det D u ≤ Λ are strictly convex away froma singular set of Hausdorff n − W , regularity for singular solutions. We also construct for any (cid:15) asingular solution to det D u = 1 in B ⊂ R n ( n ≥
3) with a singular set ofHausdorff dimension at least n − − (cid:15) which is not in W , (cid:15) . However, as (cid:15) → Theorem 1.1.
Assume that λ ≤ det D u ≤ Λ in B ⊂ R n , (cid:107) u (cid:107) L ∞ ( B ) < K. Then for some (cid:15) ( n ) and C ( n, λ, Λ , K ) we have ∆ u ∈ L log (cid:15) L and (cid:90) B / ∆ u (log(1 + ∆ u )) (cid:15) dx ≤ C. a r X i v : . [ m a t h . A P ] D ec CONNOR MOONEY
We also construct an example with a singular set of Hausdorff dimension exactly n − L log M L for M large, showing that the maintheorem is in a sense optimal and that we cannot improve our estimate on theHausdorff dimension to n − − (cid:15) for any (cid:15) . Since solutions in two dimensions arestrictly convex, this result is interesting for n ≥ n − Acknowledgements
I would like to thank my thesis advisor Ovidiu Savin for his patient guidanceand encouragement. The author was partially supported by the NSF GraduateResearch Fellowship Program under grant number DGE 1144155.2.
Preliminaries
Let u : Ω ⊂ R n → R be a convex function. Then u has an associated Borelmeasure M u , called the Monge-Amp`ere measure, defined by
M u ( A ) = |∇ u ( A ) | where |∇ u ( A ) | represents the Lebesgue measure of the image of the subgradientsof u in A (see [Gut]). We say that u solves det D u = f in the Alexandrov sense if M u = f dx. We define a section of u by S h ( x ) = { y ∈ Ω : u ( y ) < u ( x ) + ∇ u ( x ) · ( y − x ) + h } for some subgradient ∇ u ( x ) at x . Finally, we define D n,λ, Λ ,K to be the collectionof convex functions satisfying λ ≤ det D u ≤ Λ in B ⊂ R n , (cid:107) u (cid:107) L ∞ ( B ) ≤ K in the Alexandrov sense and we say that a constant depending only on n, λ, Λ and K is a universal constant. In this section we recall some geometric observationsabout sections of solutions in D n,λ, Λ ,K . Lemma 2.1. (John’s Lemma). If S ⊂ R n is a bounded convex set with nonemptyinterior, and is the center of mass of S , then there exists an ellipsoid E and adimensional constant C ( n ) such that E ⊂ S ⊂ C ( n ) E. We call E the John ellipsoid of S . There is some linear transformation A such that A ( B ) = E , and we say that A normalizes S .In the following two lemmas we present an important observation on the vol-ume growth of sections that are not compactly contained and relate the volume ofcompactly contained sections to the Monge-Amp`ere mass of these sections. Shortproofs can be found in [M]. , ESTIMATE FOR SINGULAR SOLUTIONS 3
Lemma 2.2.
Assume that det D u ≥ λ in Ω ⊂ R n . Then if S h ( x ) is any sectionof u , we have | S h ( x ) | ≤ Ch n/ for some constant C depending only on λ and n . The proof is just a barrier by above in the John ellipsoid for S h ( x ). Lemma 2.3.
Let v be any convex function on Ω ⊂ R n with v | ∂ Ω = 0 . Then M v (Ω) | Ω | ≥ c ( n ) | min Ω v | n . The proof is by comparing to the Monge-Amp`ere mass of the function whose graphis the cone generated by the minimum point of v and ∂ Ω.Next, we recall the following geometric observation of Caffarelli for solutions tothe Monge-Amp`ere equation with bounded right hand side (see [C1]). It says thatcompactly contained sections S h ( x ) are balanced around x . Lemma 2.4.
Assume that λ ≤ det D u ≤ Λ in Ω ⊂ R n . Then there exist c, C ( n, λ, Λ) such that for all S h ( x ) ⊂⊂ Ω , there is an ellipsoid E centered at of volume h n/ with cE ⊂ S h ( x ) − x ⊂ CE.
Finally, we give the following engulfing and covering properties of compactlycontained sections (see [CG] and [DFS]). In the following αS h ( x ) will denote the α dilation of S h ( x ) around x . Lemma 2.5.
Assume that λ ≤ det D u ≤ Λ in Ω . Then there exists δ > universal such that: (1) If S h ( x ) ⊂⊂ Ω then S δh ( x ) ⊂ S h ( x ) . (2) Suppose that for some compact D ⊂ Ω , we can associate to each x ∈ D some S h ( x ) ⊂⊂ Ω . Then we can find a finite subcollection { S h i ( x i ) } Mi =1 such that S δh i ( x i ) are disjoint and D ⊂ ∪ Mi =1 S h i ( x i ) . Statement of Key Proposition and Proof of Theorem 1.1
In this section we state the key proposition and use it to prove our main theorem.In [M] we show that the Monge-Ampere mass of u + | x | in small balls aroundsingular points is large compared to the mass of ∆ u . The proposition is a moreprecise, quantitative version of this statement for long, thin sections. Let ¯ h ( x ) ≥ h such that S h ( x ) ⊂⊂ B . We say that S ¯ h ( x ) ( x ) is the maximalsection at x . If ¯ h ( x ) = 0 then x is a singular point. Proposition 3.1. If u ∈ D n,λ, Λ ,K , v = u + | x | , x ∈ B / and h > ¯ h ( x ) thenthere exist η ( n ) and c universal such that for some r with | log r | > c | log h | / , we have M v ( B r ( x )) > cr n − | log r | η . CONNOR MOONEY
Remark . Let Σ denote the singular set of u , where ¯ h = 0. It follows fromproposition 3.1 and a covering argument thatinf δ> (cid:40) ∞ (cid:88) i =1 r n − i | log r i | η : { B r i ( x i ) } ∞ i =1 cover Σ , r i < δ (cid:41) = 0for some small η ( n ), giving a quantitative version of the main theorem in [M] forsolutions to λ ≤ det D u ≤ Λ.We will give a proof of Proposition 3.1 in the next section by closely examiningthe geometric properties of maximal sections.The idea of the proof of Theorem 1.1 is to apply Proposition 3.1 in the thinmaximal sections, and then apply the W , (cid:15) estimate of [DFS] in the larger sectionsto show the following decay of the integral of ∆ u over its level sets:(1) (cid:90) { ∆ u>t } ∆ u dx ≤ C | log t | (cid:15) , for some (cid:15) ( n ). Assuming this is true, theorem 1.1 follows easily by Fubini: (cid:90) B / ∆ u (log(1 + ∆ u )) (cid:15)/ dx ≤ C (cid:90) B / ∆ u (cid:90) u t (log t ) − (cid:15)/ dt dx ≤ C + C (cid:90) ∞ t (log t ) − (cid:15)/ (cid:90) { ∆ u>t } ∆ u dx dt ≤ C + C (cid:90) ∞ t (log t ) (cid:15)/ dt ≤ C ( (cid:15) ) . To prove (1), We first recall the following theorem of De Philippis, Figalli andSavin:
Theorem 3.3.
Assume that λ ≤ det D u ≤ Λ in S H (0) , u | ∂S H (0) = 0 and B is the John ellipsoid for S H (0) . Then there exist C, (cid:15) depending only on λ, Λ and n such that (cid:90) S H/ (0) ∩{ ∆ u>t } ∆ u dx < Ct − (cid:15) . We will use the rescaled version of this theorem in the larger maximal sections.
Lemma 3.4. If u ∈ D n,λ, Λ ,K with x ∈ B / and S h ( x ) ⊂⊂ B , then for C universaland (cid:15) ( n, λ, Λ) we have (cid:90) S h/ ( x ) ∩{ ∆ u>t } ∆ u dx < Ch n/ − − (cid:15) t − (cid:15) . Proof.
By subtracting a linear function and translating assume that x = 0 and u | ∂S h (0) = 0. Let u ( x ) = (det A ) /n ˜ u ( A − x )where A normalizes S h ( x ) and ˜ u has height H . Then D u ( x ) = C | S h (0) | /n ( A − ) D ˜ u ( A − x )( A − ) T . , ESTIMATE FOR SINGULAR SOLUTIONS 5
Applying the estimate on | S h (0) | from Lemma 2.4 and letting d denote the lengthof the smallest axis for the John ellipsoid of S h (0), it follows that∆ u ( x ) ≤ C (cid:18) hd (cid:19) ∆˜ u ( A − x ) . Using change of variables and Theorem 3.3 we obtain that (cid:90) S h/ (0) ∩{ ∆ u>t } ∆ u dx ≤ C (det A ) (cid:18) hd (cid:19) (cid:90) S H/ (0) ∩{ ∆˜ u>c d h t } ∆˜ u ( y ) dy ≤ C (det A ) (cid:18) hd (cid:19) (cid:15) t − (cid:15) . Since det A = h n/ up to a universal constants and d > ch since u is locallyLipschitz, the conclusion follows. (cid:3) Let F γ = { x ∈ B / : γ ≤ ¯ h ( x ) < γ } . Lemma 3.5.
Let u ∈ D n,λ, Λ ,K . Then there is some C universal and (cid:15) ( n, λ, Λ) such that (cid:90) F γ ∩{ ∆ u>t } ∆ u dx < Cγ − (cid:15) t − (cid:15) Proof.
By Lemma 2.5 we can take a cover of F γ by sections { S ¯ h i ( x i ) / ( x i ) } M γ i =1 with x i ∈ F γ and S δ ¯ h i ( x i ) ( x i ) disjoint for some universal δ . Then (cid:90) F γ ∩{ ∆ u>t } ∆ u dx ≤ CM γ γ n/ − − (cid:15) t − (cid:15) by Lemma 3.4. We need to estimate the number of sections M γ in our Vitali coverof F γ .Take x ∈ F γ and consider S ¯ h ( x ) ( x ), which touches ∂B . By translation andsubtracting a linear function assume that x = 0 and u | ∂S δ h (0) (0) = 0. By rotatingand applying Lemma 2.4 assume that S δ ¯ h (0) (0) contains the line segment from − ce n to ce n , with c universal.Let w t be the restriction of u to { x n = t } and let S w t = S δ ¯ h (0) (0) ∩ { x n = t } be the slice of S δ ¯ h (0) (0) at x n = t . Since | S δ ¯ h (0) (0) | ≤ Cγ n/ and this section haslength 2 c in the e n direction, it follows from convexity that | S w t | H n − ≤ Cγ n/ . By convexity, u ( te n ) < − δ ¯ h (0) / − c/ ≤ t ≤ c/
2. Applying Lemma 2.3, weconclude that for t ∈ [ − c/ , c/ M w t ( S w t ) > cγ n/ − . Let r be the distance between ∂S δ ¯ h (0) (0) and ∂ (2 S δ ¯ h (0) (0)). Divide 2 S δ ¯ h (0) (0)into the slices S k = 2 S δ ¯ h (0) (0) ∩ { kr < x n < ( k + 1) r } for k = − c r to c r . Let v = u + | x | . Then ∇ v ( S k ) contains a ball of radius r/ ∇ v ( S w ( k +1 / r ) (see Figure 1), so M v ( S k ) ≥ crM v ( S w ( k +1 / r ) ≥ crγ n/ − . CONNOR MOONEY S k S kw ( k + / ) r ∇ v ( S kw ( k + / ) r )∇ vS δ ̄ h ( ) δ ̄ h ( ) B r / ( y ) ∇ v ( B r / ( y )) Figure 1. ∇ v ( S k ) contains an r/ ∇ v ( S w ( k +1 / r ), which projects in the x n direction to a set of H n − measure at least cγ n/ − .Summing from k = − c r to c r we obtain that |∇ v (2 S δ ¯ h (0) (0)) | ≥ cγ n/ − . Using that 2 S δ ¯ h i ( x i ) ⊂ S δ ¯ h i ( x i ) are disjoint and summing over i we obtain that M γ γ n/ − < C and the conclusion follows. (cid:3) Proof of Theorem 1.1 . We first consider the set where ¯ h ( x ) ≤ t / . At any pointin this set, by Proposition 3.1, we can find some r > | log r | > c | log t | / and M v ( B r ( x )) > cr n − (log t ) η/ . We conclude that (cid:90) B r ( x ) ∆ u dx ≤ Cr n − ≤ C (log t ) η/ M v ( B r ( x )) . Covering { ∆ u > t } ∩ { ¯ h ( x ) ≤ t / } with these balls and taking a Vitali subcover { B r i ( x i ) } , we obtain that (cid:90) { ∆ u>t }∩{ ¯ h ( x ) < t / } ∆ u dx ≤ C (log t ) η/ (cid:88) i M v ( B r i ( x i )) ≤ C (log t ) η/ , giving the desired bound over the “near-singular” points.We now study the integral of ∆ u over the remaining subset of { ∆ u > t } . Take k so that 2 k − ≤ t / < k . , ESTIMATE FOR SINGULAR SOLUTIONS 7
Applying Lemma 3.5 we obtain that (cid:90) { ∆ u>t }∩{ ¯ h ( x ) > t / } ∆ u dx ≤ k (cid:88) i =0 (cid:90) { ∆ u>t }∩ F − i ∆ u dx ≤ Ct − (cid:15) k (cid:88) i =1 (cid:15)i ≤ Ct − (cid:15)/ , giving the desired bound. (cid:3) Quantitative Behavior of Maximal Sections
In this section we closely examine the geometric properties of maximal sectionsof solutions in D n,λ, Λ ,K to prove Proposition 3.1.Let u ∈ D n,λ, Λ ,K and fix x ∈ B / . Then for any h > ¯ h ( x ), S h ( x ) is notcompactly contained in ∂B . If ¯ h ( x ) >
0, then by Lemma 2.4, S ¯ h ( x ) ( x ) contains anellipsoid E centered at x with a long axis of universal length 2 c .If ¯ h ( x ) = 0 and L is the tangent to u at x then it is a consequence of lemma 2.4(see [C1]) that { u = L } has no extremal points, and in particular for any h > S h ( x ) contains a line segment (independent of h ) exiting ∂B at both ends.By translating and subtracting a linear function assume that x = 0 and ∇ u (0) =0. By rotating assume that S h (0) contains the line segment from − ce n to ce n forall h > ¯ h (0). For the rest of the section denote ¯ h (0) by just ¯ h .Let w be the restriction of u to { x n = 0 } with sections S wh . Since | S h (0) | < Ch n/ for all h and S ¯ h (0) contains a line segment of universal length in the e n direction,we have | S wh (0) | H n − < Ch n/ for h ≥ ¯ h . In the following analysis we need to focus on those sections of w withthe same volume bound. The following property is sufficient: Property F : We say S wh ( y ) satisfies property F if w ( y ) + ∇ w ( y ) · ( − y ) + h ≥ ¯ h. (See Figure 2). Lemma 4.1. If S wh ( y ) satisfies property F then | S wh ( y ) | < Ch n/ . Proof.
The plane u ( y ) + ∇ u ( y ) · ( z − y ) + h is greater than ¯ h along z = te n foreither t > t <
0. Since u < ¯ h on the segment from − ce n to ce n , it follows that S h ( y ) contains the line segment from 0 to ce n or − ce n . Since | S h ( y ) | < Ch n/ theconclusion follows. (cid:3) The first key lemma says that w grows logarithmically faster than quadratic inat least two directions at a level comparable to ¯ h . Let d y ( h ) ≥ d y ( h ) ≥ ... ≥ d yn − ( h )denote the axis lengths of the John ellisoid for S wh ( y ). CONNOR MOONEY ̄ h h y w . Figure 2. S wh ( y ) satisfies property F if the tangent plane at y ,lifted by h , lies above ¯ h at 0. Lemma 4.2.
For any h > ¯ h there exist (cid:15) ( n ) , C universal, h < e −| log h | / and y such that S wh ( y ) satisfies property F and d yn − ( h ) < C h / | log h | − (cid:15) . The next lemma says that if w grows logarithmically faster than quadratic inat least two directions up to height h then the Monge-Amp`ere mass of u + | x | is logarithmically larger than the mass of ∆ u in a ball with radius comparable to h / . Lemma 4.3.
Fix (cid:15) > and assume that for some h > , S wh ( y ) satisfies property F . Then there exist η , η ( n, (cid:15) ) and C depending on universal constants and (cid:15) suchthat if d yn − ( h ) < h / | log h | − (cid:15) then for some r < Ch / | log h | − η we have M (cid:18) u + 12 | x | (cid:19) ( B r (0)) > C − r n − | log r | η . These lemmas combine to give the key proposition:
Proof of Proposision 3.1:
By Lemma 4.2, there is some S h ( y ) satisfying prop-erty F with d yn − ( h ) < C h / | log h | − (cid:15) , with (cid:15) ( n ), C universal and h < e −| log( δ +¯ h ( x )) | / for any δ . The conclusion followsfrom Lemma 4.3. (cid:3) We now turn to the proofs of Lemmas 4.2 and 4.3. , ESTIMATE FOR SINGULAR SOLUTIONS 9 x x b ( h / ) P P S hw ( ) S h / w ( ) . Figure 3.
Proof of Lemma 4.2 . Assume by way of contradiction that for all h < h and S wh ( y ) satisfying property F we have d yn − ( h ) > C h / | log h | − (cid:15) , for h depending on ¯ h and C , (cid:15) we will choose later. We divide the proof into twosteps. Step 1:
Define the breadth b ( h ) as the minimum distance between two paralleltangent hyperplanes to ∂S wh (0). We show that for ¯ h | log ¯ h | < h < h we have b ( h/ > (cid:18)
12 + C | log h | (cid:19) b ( h )for some C large depending on C . Let x be the center of mass of S wh (0) androtate so that the John ellipsoid for S wh (0) is A ( B ) + x , where A = diag( d ( h ) , ..., d n − ( h )) . Let P , P be the tangent hyperplanes to ∂S wh/ (0) a distance b ( h/
2) apart. Let x , x be points where P and P become tangent to ∂S wh (0) when we slide themout. Assume that the distance between 0 and the plane tangent at x is larger thanthat between 0 and the plane tangent at x . (See Figure 3).Let ˜ x be the image of x under A − and let˜ w ( x ) = (det A ) − /n w ( Ax ) . Observe that ˜ w is the restriction of ˜ u ( x ) = (det A ) − /n u ( Ax (cid:48) , x n ) which solves λ ≤ det D u ≤ Λ, so that sections S ˜ wh of ˜ w satisfying property F with ¯ h replacedby (det A ) − /n ¯ h have volume bounded above by Ch n/ . Furthermore, since thedistance between 0 and the plane tangent at x was larger and the images of thetangent planes under A − are separated by distance at least 2, we have | ˜ x | ≥ ̃ w ̃ x ̃ h ̃ y ( det A ) − / n ̄ hH . .. Figure 4.
Lifting the tangent plane at ˜ y by h ∗ = ˜ h +det( A ) − /n ¯ h we obtain a section of ˜ w satisfying property F .By convexity we can find ˜ y on the line segment connecting 0 to ˜ x such that ∇ ˜ w (˜ y ) · ˜ x | ˜ x | = H | ˜ x | , where H = det A − /n h is the height of ˜ w . Let ˜ h be the smallest t such that0 ∈ S ˜ wt (˜ y ). We aim to bound ˜ h below, which heuristically rules out cone-likebehavior in the ˜ x direction. Let h ∗ = ˜ h + (det A ) − /n ¯ h. We have chosen h ∗ so that S ˜ wh ∗ (˜ y ) and S wδ ( y ) = A ( S ˜ wh ∗ (˜ y )) satisfy property F , where δ = (det A ) /n h ∗ . (See Figure 4). It follows that | S ˜ wh ∗ (˜ y ) | < C ( h ∗ ) n/ . We now bound the volume of S ˜ wh ∗ (˜ y ) by below. Since 0 , ˜ x are in this section,it has diameter at least 1. Since ˜ w has height H it has interior Lipschitz constant CH , so the smallest axis of the John ellipsoid for S ˜ wh ∗ (˜ y ) has length at least c h ∗ H . Weturn to the remaining axes.Let E y be the John ellipsoid for S wδ ( y ). By contradiction hypothesis for any n − P passing through the center of E y , we can find a n − P (cid:48) contained in P such that P (cid:48) ∩ E y is an n − d y ,P (cid:48) ≥ ... ≥ d yn − ,P (cid:48) satisfying d yn − ,P (cid:48) > C δ / | log δ | − (cid:15) . Take P such that A − ( P ) is perpendicular to the segment connecting 0 and ˜ x . Byusing the hypothesis and that w is locally Lipschitz we have d n − ( h ) d n − ( h ) > cC h / | log h | − (cid:15) . , ESTIMATE FOR SINGULAR SOLUTIONS 11
Since d ( h ) ...d n − ( h ) < Ch n , this gives d ( h ) ...d n − ( h ) < CC h n − | log h | (cid:15) . It follows that A − changes the n − P (cid:48) ∩ E y by a factor ofat least c ( n ) d ( h ) ...d n − ( h ) ≥ cC h − n − | log h | − (cid:15) . Since det
A > ch n/ | log h | − C ( n ) (cid:15) (by the contradiction hypothesis) and δ = (det A ) /n h ∗ we conclude that | S ˜ wh ∗ (˜ y ) ∩ A − ( P (cid:48) ) | H n − > C ( δ / | log δ | − (cid:15) ) n − d ( h ) ...d n − ( h ) ≥ C ( h ∗ ) n − (det A ) n − n h − n − ( C | log h | + | log h ∗ | ) − C ( n ) (cid:15) for some large C depending on C . We also have H = h (det A ) − /n ≤ | log h | C ( n ) (cid:15) . Using that the remaining axes have lengths at least 1 and c h ∗ H we obtain | S ˜ wh ∗ (˜ y ) | > C ( h ∗ ) n − | log h | − C ( n ) (cid:15) ( C | log h | + | log h ∗ | ) − C ( n ) (cid:15) . Using that | S ˜ wh ∗ (˜ y ) | < C ( h ∗ ) n/ we get a lower bound on h ∗ : h ∗ > C | log h | − C ( n ) (cid:15) . (See Figure 5 for the simple case n = 3.)Recalling the definition of h ∗ and using again the lower bound on det A it followsthat ˜ h + C ¯ hh | log h | C ( n ) (cid:15) > C | log h | − C ( n ) (cid:15) . Taking (cid:15) to be small enough that C ( n ) (cid:15) = 1 / h | log ¯ h | < h we get˜ h > C | log h | − / . Finally, let (cid:0) + γ (cid:1) ˜ x be the point where ˜ w = H . It is clear from convexity (seeFigure 6) that 2 γH ≥ ˜ h. Recalling that
H < c | log h | C ( n ) (cid:15) < c | log h | / , we obtain γ ≥ C | log h | − . Let l , l be the distances from 0 to the translations of P and P which aretangent to ∂S wh (0) so that b ( h ) ≤ l + l . The previous analysis implies that P and P have distance at least (cid:0) + γ (cid:1) l and l from 0. Since l ≥ l it follows that b ( h/ ≥ (cid:18)
12 + γ (cid:19) l + 12 l ≥ (cid:18) γ (cid:19) ( l + l ) . Since γ ≥ C | log h | , step 1 is finished. Step 2:
We iterate step 1 to prove the lemma. First assume that ¯ h > h | log ¯ h | = 2 − k and h = 2 − k . Note that d n − ( h ) > c ( n ) b ( h ) and that S h * ̃ w ( ̃ y ) > ch * / H ̃ x > 10 .. Figure 5.
For the case n = 3, the above figure implies that | S ˜ wh ∗ (˜ y ) | > ch ∗ /H . This, combined with the volume estimate | S ˜ wh ∗ (˜ y ) | < C ( h ∗ ) / and the upper bound on H from the con-tradiction hypothesis give a lower bound of c | log h | − C(cid:15) for h ∗ . γ | ̃ x |2 γ | ̃ x | ̃ w HH / ̃ x Figure 6.
By convexity 2 γ is at least ˜ h/H , giving a quantitativemodulus of continuity for ∇ w near 0 which we exploit in Step 2 toobtain a contradiction. , ESTIMATE FOR SINGULAR SOLUTIONS 13 d n − ( h ) > c − k since u is locally Lipschitz. Iterating step 1 for C large weobtain d n − (2 − k ) ≥ c (1 / C /k )(1 / C / ( k − ... (1 / C /k )2 − k ≥ c − k exp( C k (cid:88) i = k i ) ≥ − k kk , showing that d n − (¯ h | log ¯ h | ) ≥ c ¯ h | log ¯ h | (cid:0) | log ¯ h || log h | − (cid:1) . Finally, take | log h | = | log ¯ h | / . We conclude using convexity that d n − (¯ h ) > | log ¯ h | − d (¯ h | log ¯ h | ) > c ¯ h | log ¯ h | / . Since d (¯ h ) ...d n − (¯ h ) < C ¯ h n/ we thus have d n − (¯ h ) < C ¯ h / | log ¯ h | − (cid:15) ( n ) , giving the desired contradiction.In the case that ¯ h = 0, we may run the above iteration for any h > h = e −| log h | / to obtain the contradiction. (cid:3) Proof of Lemma 4.3 . First assume that d y ( h ) < h / | log h | − α for some α .Since | S wh ( y ) | < Ch n/ , Lemma 2.3 gives M w ( S wh ( y )) > ch n − . Take C ( n ) large enough that for r = C ( n ) h / | log h | − α , S wh ( y ) ⊂ B r/ (0) . Clearly, M (cid:18) | x | + w (cid:19) ( S wh ( y )) > M w ( S wh ( y )) . Furthermore, ∇ (cid:0) u + | x | (cid:1) ( B r (0)) contains a ball of radius r/ ∇ (cid:0) u + | x | (cid:1) ( S wh ( y )) (see Figure 7). We conclude that M (cid:18) u + 12 | x | (cid:19) ( B r (0)) > crM w ( S wh ( y )) ≥ crh n − ≥ cr n − | log h | ( n − α ≥ cr n − | log r | ( n − α . We proceed inductively. Assume that d yi ( h ) > h / | log h | − α i for i = 1 , ..., k − d yk ( h ) < h / | log h | − α k for some α , ..., α k to be chosen shortly. We aim to apply Lemma 2.3 to slices of thesection S wh ( y ) at 0, but we need the height of the plane w ( y )+ ∇ w ( y ) · ( x − y )+ h at 0to be at least h . We thus consider S w h ( y ) instead. Note that d yi (2 h ) > h / | log h | − α i for i ≤ k − d yk (2 h ) < h / | log h | − α k . B r / ( x ) S hw ( y ) ∇ ( u + | x | / )∇ ( u + | x | / )( S hw ( y ))∇ ( u + | x | / )( B r / ( x )) B r ( ) , r << h Figure 7. ∇ ( u + | x | / B r (0)) contains an r/ ∇ ( u + | x | / S wh ( y )), which projects in the x n directionto a set of H n − measure at least cr n − | log r | ( n − α .Rotate so that the axes align with those for the John ellipsoid of S w h ( y ). Takethe restriction of w to the subspace spanned by e k , ..., e n − , and call this restriction w k . Let S w k = S w h ( y ) ∩ { x = ... = x k − = 0 } , the slice of the section S w h ( y ) in this subspace. Then since d y (2 h ) ...d yn − (2 h ) ≤ Ch n , by hypothesis we have | S w k | H n − k ≤ Ch n +1 − k | log h | α + ... + α k − . Since S wh ( y ) contains 0 and S w k is the slice of S w h ( y ), we know that w k has heightat least h in S w k . Using this and Lemma 2.3, M w k ( S w k ) ≥ ch n − k − | log h | − ( α + ... + α k − ) . Finally, take C ( n ) large enough that for r = C ( n ) h / | log h | − α k we have S w k ⊂ B r/ (0) . By strict quadratic growth, ∇ (cid:0) u + | x | (cid:1) ( B r (0)) contains a ball of radius r/ ∇ ( u + | x | )( S w k ). It follows that M (cid:18) u + 12 | x | (cid:19) ( B r (0)) ≥ cM w k ( S w k ) r k ≥ ch n − k − | log h | − ( α + ... + α k − ) r k ≥ cr n − | log r | ( n − k − α k − ( α + ... + α k − ) . , ESTIMATE FOR SINGULAR SOLUTIONS 15
Choose β i so that ( n − k − β k − ( β + ... + β k − ) = 1 and let α i = cβ i , with c chosen so that α n − = (cid:15) . If d y ( h ) < h / | log h | − α , we are done by the first step,so assume not. Then apply the inductive step for i = 2 , ..., n − (cid:3) Example
In this section we construct a solution to det D u = 1 in R such that Σ hasHausdorff dimension exactly 2. A small modification gives the analagous examplein R n with a singular set of Hausdorff dimension n −
1. This shows that the estimateon the Hausdorff dimension of the singular set in [M] cannot be improved to n − − δ for any δ .We proceed in several steps:(1) The key step is to construct a subsolution w in R satisfying det D w ≥ { x = x = 0 } and grows logarithmically faster thanquadratic in the x direction, in particular like x | log x | .(2) Next, we construct S ⊂ [ − ,
1] of Hausdorff dimension 1 and a convexfunction v on [ − ,
1] such that v separates from its tangent line faster than r | log r | at each point in S .(3) Finally, we obtain our example by solving the Dirichlet problemdet D u = 1 in Ω = {| x (cid:48) | < } × ( − , , u | ∂ Ω = C ( v ( x ) + | x | )and comparing with w at points in S × { } × {± } .In the following analysis c, C will denote small and large constants respectively. Construction of w : We first seek a function with just faster than quadraticgrowth in one direction and sections S h (0) with volume smaller than h / . To thatend, let g ( x , x ) = x | log x | α + | x || log x | β for some α, β to be chosen shortly. It is tempting to guess w = g ( x , x )(1 + x ).However, the dominant terms in the determinant of the Hessian near the x axisare | log x | α | log x | β (cid:18) | log g | − x (cid:19) , where the first comes from the diagonal entries and the second from the mixedderivatives. Thus, this function is not convex. This motivates the following modi-fication: w ( x (cid:48) , x ) = g ( x (cid:48) ) (cid:18) x | log g ( x (cid:48) ) | (cid:19) . It is straightforward to check that the leading terms in the determinant of theHessian (taking x small) are x | log x | α | x (log x ) β +1 log g | + | log x | α | (log x ) β log g | , since now the mixed derivative terms have the same homogeneity in log( g ) as thediagonal terms. For | x (cid:48) | small, the first term is large in {| x | < | x | } , and by taking α = 2 + 2 β the second term is bounded below by a positive constant in {| x | ≥ | x | } . Thus, up to rescaling and multiplying by a constant we havedet D w ≥ {| x (cid:48) | < } × ( − , β = 1 and α = 4 for the restof the example. Construction of S : Start with the interval [ − / , / from the center. At the k th step, removeintervals a fraction k +5 of the length of the remaining 2 k intervals from their centers.Denote the centers of the removed intervals by { x i,k } k i =1 , and the intervals by I i,k .Finally, let S = [ − , − ∪ i,k I i,k . Let l k = | I i,k | . It is easy to check l k = 10 k + 5 2 − k (cid:18) − k + 4 (cid:19) ... (cid:18) − (cid:19) ≤ Ck − k . One checks similarly that the length of the remaining intervals after the k th stepis at least 2 − k k − . It follows that(2) inf (cid:40) ∞ (cid:88) i =1 r i | log( r i ) | : { B r i ( x i ) } cover S, r i < δ (cid:41) > c for all δ >
0. In particular, the Hausdorff dimension of S is exactly 1. Construction of v : Let f ( x ) = (cid:26) | x | | x | ≤ | x | − | x | > f together to produce the desired function: v ( x ) = ∞ (cid:88) k =1 k l k f ( l − k ( x − x i,k )) . We now check that v satisfies the desired properties:(1) v is convex, as the sum of convex functions. Furthermore, using that l k There is some y ∈ ( x + r/ , x + r ) ∩ S . Then by the constructionof S it is easy to see that there is some interval I i,k such that I i,k ⊂ ( x, x + r ).On this interval, v grows by k l k ≥ cl k | log( l k ) | ≥ cr | log( r ) | . Case 2: Otherwise, there is an interval I i,j of length exceeding r/ x + r/ , x + r ) ⊂ I i,j . Then at the left point of I i,j , the slope of v jumps by at least k l k . It follows that at x + r , v is at least crk l k ≥ cr | log( r ) | . Thus, v has the desired properties. Construction of u : We recall the following lemma on the solvability of theMonge-Amp`ere equation (see [Gut]). Lemma 5.1. If Ω is open and convex, µ is a finite Borel measure and ϕ is con-tinuous on ∂ Ω then there exists a unique convex solution u ∈ C ( ¯Ω) to the Dirichletproblem det D u = µ, u | ∂ Ω = ϕ. Let ϕ ( x , x , x ) = C ( v ( x ) + | x | ) for a constant C we will choose shortly, andobtain u by solving the Dirichlet problemdet D u = 1 in Ω = {| x (cid:48) | < } × [ − , , u | ∂ Ω = ϕ. Take x ∈ S × { } × {± } . By translating and subtracting a linear function assumethat x = 0 and 0 is a subgradient for ϕ at x . Taking C large we guarantee that ϕ ( x , x , ± > C ( x | log( x ) | + | x | ) > w ( x , x , ± x , x , and that that ϕ > w on the sides of Ω. Thus, u ≥ w in all of Ω. Since u = 0 at both (0 , , ± 1) and w (0 , , x ) = 0 for all | x | < 1, we have by convexitythat u = 0 along (0 , , x ).This shows that for these examplesΣ ⊂ S × { } × ( − , , which has Hausdorff dimension exactly 2. Remark . To get the analagous example in R n , take u ( x , x , x ) + x + ... + x n . Optimality of Theorem 1.1 In [M] we construct for any (cid:15) solutions to det D u = 1 in R n that are not in W , (cid:15) , but as (cid:15) → W , (cid:15) for any (cid:15) , and in fact the second derivatives are not in L log M L for M large.Let φ ( x ) = (1+ x )(log(1+ x )) M for some M large. Then φ is convex for x ≥ 0, sofor any nonnegative integrable function f and ball B r we have by Jensen’s inequalitythat (cid:90) B r φ ( r n f ( x )) dx ≥ cr n φ (cid:18)(cid:90) B r f ( x ) dx (cid:19) . Taking f ( x ) = r − n ∆ u ( x ) we obtain (cid:90) B r (1 + ∆ u )(log(1 + ∆ u )) M dx ≥ c (cid:18)(cid:90) B r ∆ u dx (cid:19) (cid:18) log (cid:18) r − n (cid:90) B r ∆ u dx (cid:19)(cid:19) M . Recall that at points x ∈ S × { } × ( − , n − the subsolutions w touch u by below,and that w grows like | x || log x | − at x . It follows thatsup ∂B r ( x ) ( u − u ( x )) ≥ cr | log r | − . Applying convexity we conclude that (cid:90) B r ( x ) (1 + ∆ u ) (log(1 + ∆ u )) M dx ≥ c (cid:32)(cid:90) ∂B r ( x ) u ν (cid:33) (cid:32) log (cid:32) r − n (cid:90) ∂B r ( x ) u ν (cid:33)(cid:33) M ≥ cr n − | log r | − (cid:0) log( cr − | log( r ) | − ) (cid:1) M ≥ cr n − | log r | M − . Cover Σ ∩ B / with balls of radius less than δ and take a Vitali subcover { B r i } Ni =1 .We then have (cid:90) B / (1 + ∆ u ) (log(1 + ∆ u )) M dx ≥ c N (cid:88) i =1 r n − i | log r i | M − , and for M large the right side goes to ∞ as δ → u are not in L log M L for M large, and in par-ticular u is not in W , (cid:15) for any (cid:15) . References [C1] L. Caffarelli, A localization property of viscosity solutions to the Monge-Amp`ere equationand their strict convexity , Ann. of Math. (1990), 129-134.[C2] L. Caffarelli, Interior W ,p estimates for solutions of Monge-Amp`ere equation , Ann. ofMath. (1990), 135-150.[CG] L. Caffarelli and C. Gutierrez, Properties of solutions of the linearized Monge-Amp`ereequation , Amer. J. Math (1997), 423-465.[DF] G. De Philippis and A. Figalli, W , regularity for solutions of the Monge-Amp`ere equation ,Invent. Math. (2013), 55-69.[DFS] G. De Philippis, A. Figalli and O. Savin, A note on interior W , (cid:15) estimates for theMonge-Amp`ere equation , Math. Ann. (2013), 11-22.[Gut] C. Gutierrez, “The Monge-Amp`ere Equation,” Progress in Nonlinear Differential Equationsand their Applications 44, Birkh¨auser Boston, Inc., Boston, MA, 2001. , ESTIMATE FOR SINGULAR SOLUTIONS 19 [M] C. Mooney, Partial regularity for singular solutions to the Monge-Amp`ere equation , Comm.Pure Appl. Math., to appear.[W] X.-J. Wang, Regularity for Monge-Amp`ere equation near the boundary , Analysis (1996)101-107. Department of Mathematics, Columbia University, New York, NY 10027 E-mail address ::