Regularity of the Monge-Ampère equation in Besov's space

0.17 MB·

by  Alexander V. Kolesnikov

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Regularity of the Monge–Amp`ere equation
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2 in Besov’s spaces

n
a Alexander V. Kolesnikov and Sergey Yu. Tikhonov
J

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1 Abstract. Let µ = e−V dx be a probability measure and T = ∇Φ be the
optimal transportation mapping pushing forward µ onto a log-concave com-
] pactlysupportedmeasureν=e−W dx. Inthispaper,weintroduceanewap-
P proachtotheregularityproblemforthecorrespondingMonge–Amp`ereequa-
A tion e−V = detD2Φ·e−W(∇Φ) in the Besov spaces Wγ,1. We prove that
loc
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t loc
a previouslyknownregularityresults.
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1. Introduction
7
5 We consider probability measures µ = e−V dx, ν = e−W dx on Rd and the
4
optimal transportation mapping T pushing forward µ onto ν and minimizing the
3
. Monge–Kantorovichfunctional
3
0 kx−T(x)k2 dµ.
2 Z
1
It is known (see, e.g., [Vi1], [BoKo2]) that T has the form T = ∇Φ, where Φ
:
v is a convex function. If Φ is smooth, it satisfies the following change of variables
Xi formula
r (1) e−V =e−W(∇Φ)detD2Φ.
a
This relation can be considered as a non-linear second order PDE with unknown
Φ, the so-called Monge–Amp`ere equation.
The regularity problem for the Monge–Amp`ere equation has a rather long
history. The pioneering results have been obtained by Alexandrov, Bakelman,
Pogorelov,Calabi,Yau. Theclassicaltheorycanbe foundin[Po],[Ba],and[GT].
See also an interesting survey [Kr] on nonlinear PDE’s.
2000 Mathematics Subject Classification. Primary35J60,35B65; Secondary46E35.
Key words and phrases. Monge–Kantorovich, optimal transport, Monge–Amp`ere equation,
regularity,Besov’sspaces.
This study was carried out within ”The National Research University Higher School of
Economics“ Academic Fund Program in 2012-2013, research grant No. 11-01-0175. The first
authorwassupportedbyRFBRprojects10-01-00518,11-01-90421-Ukr-f-a,andtheprogramSFB
701 at the University of Bielefeld. The second author was partially supported by MTM 2011-
27637,2009SGR1303,RFFI12-01-00169, andNSH-979.2012.1.
1
Despite thelonghistory,thesharpestHo¨lderregularityresultsofclassicaltype
have been obtained only in the 90’th by L. Caffarelli [Ca1] (see also [CC, Gu,
Vi1]). In particular, Caffarelli proved that D2Φ is Ho¨lder if V and W are Ho¨lder
onbounded sets A and B, where A and B are supports of µ and ν respectively. In
addition, B is supposed to be convex. It seems that the latter assumption cannot
be dropped as demonstrated by famous counterexamples. A nice exposition with
new simplified proofs and historical overview can be found in [TW].
Another result from [Ca1] establishes sufficient conditions for Φ to belong to
the second Sobolev class W2,p with p > 1. More precisely, Caffarelli considered
loc
solution of the Monge–Amp´ere equation
detD2Φ=f
on a convex set Ω with Φ| = 0. Assume that Ω is normalized: B ⊂ Ω ⊂ B
∂Ω 1 d
(anarbitraryconvex setΩ can be normalizedusing an affine transformation). It is
shownin[Ca1]thatforeveryp>0thereexists ε(p)>0suchthatif|f−1|<ε(p)
then kΦk ≤C(ε). Wang [Wa] provedthat for a fixed ε in |f−1|<ε the
W2,p(B1/2)
value of p in the inclusion Φ∈W2,p cannot be chosen arbitrary large.
loc
This Sobolev regularity result has been extended and generalized in different
ways in the recent papers [Sa], [DPhF], [DPhFS], and [Sm]. See also [H] for
some results onthe meanoscillationof D2Φ. It was shownin[DPhFS] that every
Φ satisfying detD2Φ = f on a normalized convex set Ω with Φ| = 0 belongs to
∂Ω
W2,1+ε(Ω′), where Ω′ =nx:Φ(x)≤−12kΦkL∞o provided 0<λ<f <Λ.
The main purpose of this paper is to develop an alternative approach to the
regularity problem of the Monge–Amp`ere equation. We prove that Φ belongs to a
Besov’sspaceunderthe assumptionthate−V isBesovandW isconvex. Wegivea
short proof which does not use previously known regularity results. Our estimates
rely on a generalization of the so-called above-tangent formalism which has been
widely used in the applications of the optimal transport theory in probability and
PDE’s(see[AGS],[BoKo2],[Vi1],and[Vi2]). We alsoapplyaresultofMcCann
on the change of variables formula and some classical results on equivalence of
functional norms.
Some estimates of the type considered in this paper have been previously ob-
tained in [Ko1] in the case of the Sobolev spaces. Applications to the infinite-
dimensional analysis and convex geometry can be found in [BoKo1] and [Ko2]
respectively.
Hereafter B denotes the ball of radius r centered at 0. We use notation D2Φ
r
for the Hessian matrix of Φ and k·k for the standard operator norm. We will
assume that the measures µ and ν satisfy the following assumptions:
Assumption A: The potential V :Rd →R has the representation
V =V +V ,
0 1
where |V | is globally bounded, V admits local Sobolev derivatives, and
0 1
|∇V |∈L1(µ).
1
Assumption B: The support of ν is a compact convex set B ⊂B .
R
2
Definition. Besov’s space (Fractional Sobolev’s space). The space
Ws,p(Q), where Q is a cube in Rd, consists of functions with the finite norm
|u(x)−u(y)|p 1
kuk =kuk + dxdy p.
Ws,p(Q) Lp(Q) (cid:16)ZQZQ |x−y|d+sp (cid:17)
The space Ws,p(Rd) is the completion of C∞(Rd) with the norm
0 0
|u(x)−u(y)|p 1
kukW0s,p(Rd) =(cid:16)ZRdZRd |x−y|d+sp dxdy(cid:17)p.
Definition. Log-concave measure. A probability measure ν is called log-
concave if it satisfies the following inequality for all compact sets A,B:
ν(αA+(1−α)B)≥να(A)ν(B)1−α
and any 0 ≤ α ≤ 1. If ν has a density ν = e−W dx, then W must be convex (we
assume that W = +∞ outside of supp(ν)). This is a classical result of C. Borell
[Bo].
Remark 1. We note that the second derivative of the convex function Φ can
be understood in different ways. In the generalized (weak) sense this is a measure
with absolutely continuous part D2Φ dx and singular part D2Φ. Throughout the
a s
paper we use the following agreement: the statement “D2Φ belongs to a certain
Besov or Sobolev class” means that the measure D2Φ has no singular component
and the corresponding Sobolev derivative D2Φ=D2Φ belongs to this class.
a
Theorem 2. Let Assumptions A and B be fulfilled and, moreover,
(1) there exist p≥1 and 0<γ <1 such that for every r >0
k(δ V) k
y + Lp(µ)
dy <∞,
ZBr |y|d+p1+γ
where δ V =V(x+y)+V(x−y)−2V(x) and (δ V) is the non-negative
y y +
part of δ V;
y
(2) e−V is locally bounded from below;
(3) ν is a compactly supported log-concave measure.
Then
kD2Φk <∞
Wε,1(Q)
for every cube Q⊂Rd and 0<ε< γ.
2
Taking p=+∞ we obtain the following result.
Corollary 3. Let Assumptions A, B and conditions (2)-(3) be fulfilled. As-
sume, in addition, that (δ V) ≤ω(|y|) for some ω :R+ →R+ and
y +
r ω(s)
ds<∞
Z s1+γ
0
for some γ >0 and every r >0. Then kD2Φk <∞ with 0<ε< γ.
Wε,1(Q) 2
In particular, applying the fractional Sobolev embedding theorem (see, e.g.,
[Ad, Ch. V]) we get
Corollary 4. Under assumptions of Theorem 2, for every ε>0,
d
kD2Φk∈Ld−γ2+ε.
loc
3
Remark 5. Notethatincorollary4wedonotassumethatV isboundedfrom
below.
2. Auxiliary results
Below we will use the following function space (see, e.g., [St]).
Let Λp,q, 0<α<2, be the space of functions with the finite norm
α
kf(x+t)+f(x−t)−2f(x)k q 1
kfkΛpα,q =kfkp+hZRd (cid:0) |t|d+αq p(cid:1) dtiq,
where k·k is the Lp-norm with respect to the Lebesgue measure. We will apply
p
the following equivalence result from [St, Ch.5, Sec. 5, Propos. 8’].
Lemma 6. For α>1 the norm kfkΛpα,q is equivalent to
(k∇f(x+t)−∇f(x)k )q 1
kfk + p dt q.
p hZRd |t|d+(α−1)q i
In particular, if α>1 and kfkΛpα,q <∞, then f admits the Sobolev derivatives.
LetusrecallthateveryconvexfunctionΦonRd admitsthefollowingtwotypes
of second derivatives.
Definition. We say that the measure µ is the distributional derivative of
ev
a convex function Φ along unit vectors e,v if the following integration by parts
formula holds for any test function ξ:
ξ dµ =− ∂ ξ ∂ Φ dx.
Z ev Z e v
We set
∂ Φ:=µ .
ev ev
Definition. The absolutely continuous part (∂ Φ) of ∂ Φ is called the
ev a ev
second Alexandrov derivative of Φ along e,v. Clearly,
∂ Φ≥(∂ Φ) ≥0
ee ee a
in the sense of measures.
Let us denote by D2Φ the matrix consisting on these absolutely continuous
a
parts. We will apply the following result of R. McCann from [McCann].
Theorem 7. Forµ-almostallxthefollowing changeofvariables formulaholds
e−V(x) =detD2Φ(x)·e−W(∇Φ(x)).
a
We will need the following lemma from [Ko1]. In fact, this is a generaliza-
tion of the well-known above-tangent lemma which has numerous applications in
probability and gradient flows of measures with respect to the Kantorovichmetric
(see [Vi1], [Vi2],[AGS], [BoKo2]). The prooffollowsdirectly fromthe changeof
variables and integration by parts.
4
Lemma 8. Assume that W is twice continuously differentiable and
D2W ≥K·Id
for some K ∈R, p≥0. Then
δ V(δ Φ)pdµ
Z y y
K K
≥ |∇Φ(x+y)−∇Φ(x)|2(δ Φ)p dµ+ |∇Φ(x−y)−∇Φ(x)|2(δ Φ)p dµ
2 Z y 2 Z y
+p ∇δ Φ,(D2Φ)−1∇δ Φ (δ Φ)p−1 dµ,
Z D y a y E y
where δ V =V(x+y)+V(x−y)−2V(x).
y
Corollary 9. It follows easily from Lemma 8 that inequality
δ V(δ Φ)pdµ≥p ∇δ Φ,(D2Φ)−1∇δ Φ (δ Φ)p−1 dµ
Z y y Z D y a y E y
holds for any log-concave measure ν. In particular, this holds for the restriction of
Lebesgue measure 1 λ| on a convex subset A. In this case W is a constant on
λ(A) A
A and W(x)=+∞ if x∈/ A.
Remark 10. Let us assume that V is twice differentiable and y =te for some
unit vector e. Dividing by t2p+2 and passing to the limit we obtain
(2) V Φp dµ ≥ K kD2Φ·ek2Φp dµ
Z ee ee Z ee
+ p h(D2Φ)−1∇Φ ,∇Φ iΦp−1 dµ.
Z ee ee ee
Now it is easy to get some of the results of [Ko1] from (2). In particular,
applying integration by parts for the left-hand side and Ho¨lder inequalities, one
can easily obtain that
p+1
KkΦ2 k ≤ kV2k .
ee Lp(µ) 2 e Lp(µ)
Itisworthmentioningthat(2)gives,infact,anaprioriestimateforderivatives
of Φ up to the third order (due to the term p h(D2Φ)−1∇Φ ,∇Φ iΦp−1 dµ).
ee ee ee
R
We denote by ∆ Φ the absolutely continuous part of the distributional Lapla-
a
cian of Φ.
Lemma 11. Under Assumptions A and B, there exists C such that
d
∆ Φ(x+y) dµ≤ e−V(x) d ∂ Φ(x+y) ≤C
Z a Z xixi
Xi=1 (cid:2) (cid:3)
uniformly in y ∈Rd.
Proof. The inequality ∆ Φ(x+y) dµ≤ d e−V(x) d ∂ Φ(x+y) is
a i=1 xixi
clear in view of the fact thatRthe singular part ofP∂ ΦR is nonne(cid:2)gative. Moreov(cid:3)er,
xixi
5
we get
d d
e−V(x) d ∂ Φ(x+y) ≤c e−V1(x) d ∂ Φ(x+y)
Z xixi 1 Z xixi
Xi=1 (cid:2) (cid:3) Xi=1 (cid:2) (cid:3)
=c h∇Φ(x+y),∇V (x)ie−V1(x) dx≤c R |∇V (x)|e−V1(x) dx
1Z 1 1 Z 1
≤c |∇V (x)| dµ.
2Z 1
(cid:3)
Finally, we will use the fractional Sobolev embedding theorem (see [Ad, Ch.
V]).We formulateitinthe followingformgivenin[MSh](see also[BBM], [KL]).
Theorem 12. Let p>1, 0<s<1 and sp<d. Then for every u∈Ws,p(Rd)
0
one has
s(1−s)
kukp ≤c(d,p) kukp ,
Lq(Rd) (d−sp)p−1 W0s,p(Rd)
where q =dp/(d−sp).
3. Proof of Theorem 2
Let us apply Lemma 8 with p=1. We have
δ V ·δ Φ dµ≥ ∇δ Φ,(D2Φ)−1∇δ Φ dµ.
ZRd y y ZRdD y a y E
Taking into account Lemma 11, we obtain kD2Φk dµ < ∞. Then Cauchy
Rd a
inequality yields R
2
kD2Φk dµ· δ V ·δ Φ dµ≥ |∇δ Φ|dµ .
ZRd a ZRd y y (cid:16)ZRd y (cid:17)
By the Ho¨lder inequality, for every p,q≥1, 1 + 1 =1, we have
p q
δ V ·δ Φ dµ≤ (δ V) ·δ Φ dµ≤k(δ V) k ·kδ Φk
Z y y Z y + y y + Lp(µ) y Lq(µ)
Rd Rd
Let us now estimate kδ Φk . Note that |∇Φ|≤R, and hence δ Φ≤2R|y|.
y Lq(µ) y
Letusalsomentionthatt→Φ(x+ty)isaone-dimensionalconvexfunctionfor
afixedx. Therefore,m =∂2Φ(x+ty)isanonnegativemeasureonR1. Moreover,
y tt
1
δ Φ= h∇Φ(x+sy)−∇Φ(x−sy),yi ds
y Z
0
1 s
= dm ds.
Z Z y
0 −s
Therefore, we have
kδ Φkq = (δ Φ)q dµ≤(2R|y|)q−1 δ Φ dµ
y Lq(µ) Z y Z y
1 s
=(2R|y|)q−1 e−V(x)d ∂2Φ(x+ty) dt ds.
Z Z Z tt
0 −s (cid:2) (cid:3)
6
It follows from Lemma 11 that
kδ Φkq ≤C|y|1+q.
y Lq(µ)
Hence,
2
(cid:16)ZRd|∇δyΦ|dµ(cid:17) ≤ZRdδyV ·δyΦ dµ≤Ck(δyV)+kLp(µ)|y|1+q1.
Now we divide this inequality by |y|d+2+γ and integrate it over a bounded subset
Q⊂Rd. By condition (1), we get
1 2
(3) |∇Φ(x+y)+∇Φ(x−y)−2∇Φ(x)| dµ dy <∞.
ZQ |y|d+2+γ(cid:16)ZRd (cid:17)
Let us take a smooth compactly supported function ξ ≥0. We will show that
(4)
1 2
|ξ(x+y)∇Φ(x+y)+ξ(x−y)∇Φ(x−y)−2ξ(x)∇Φ(x)|dx dy <∞.
ZRd |y|d+2+γ(cid:16)ZRd (cid:17)
Let us split this integral in two parts:
··· dx= ··· dx+ ··· dx=I +I .
Z Z Z 1 2
Rd B1 B1c
To estimate the second part, we note that
|ξ(x+y)∇Φ(x+y)+ξ(x−y)∇Φ(x−y)−2ξ(x)∇Φ(x)| dx≤4R |ξ(x)| dx.
Z Z
Rd Rd
Thus, I <∞.
2
LetusestimateI . Itfollowsfromestimate(3)andcondition(3)ofthetheorem
1
that
1 2
|∇Φ(x+y)+∇Φ(x−y)−2∇Φ(x)|ξ(x) dx dy <∞.
ZB1 |y|d+2+γ(cid:16)ZRd (cid:17)
Therefore, it is enough to show that
1
I = |∇Φ(x+y) ξ(x+y)−ξ(x)
3 ZB1 |y|d+2+γ(cid:16)ZRd (cid:0) (cid:1)
2
+∇Φ(x−y) ξ(x−y)−ξ(x) | dx dy <∞.
(cid:0) (cid:1) (cid:17)
Since |y|≤1 and ξ is compactly supported, there exists R >0 such that
0
1
I ≤ ∇Φ(x+y) ξ(x+y)−ξ(x)
3 ZBR0 |y|d+2+γ(cid:16)ZBR0 (cid:12)(cid:12) (cid:0) (cid:1)
2
+∇Φ(x−y) ξ(x−y)−ξ(x) dx dy <∞.
(cid:0) (cid:1)(cid:12) (cid:17)
Using smoothness conditions on ξ, we obtain (cid:12)
∇Φ(x+y) ξ(x+y)−ξ(x) +∇Φ(x−y) ξ(x−y)−ξ(x)
(cid:12) (cid:12)
(cid:12) (cid:0) (cid:1) (cid:0) (cid:1)(cid:12)
(cid:12) = ∇Φ(x+y)−∇Φ(x−y) ξ(x+y)−ξ(x) (cid:12)
(cid:12)
(cid:12)(cid:0) (cid:1)(cid:0) (cid:1)
(cid:12) +∇Φ(x−y) ξ(x+y)+ξ(x−y)−2ξ(x)
(cid:12)
(cid:0) (cid:1)(cid:12)
≤C R |y|2+|y||∇Φ(x+y)−∇Φ(x−y)| . (cid:12)
0
(cid:16) (cid:17)
7
Thus, it is sufficient to show that
1 2
(5) |∇Φ(x+y)−∇Φ(x−y)| dx dy <∞.
ZBR |y|d+γ(cid:16)ZBR (cid:17)
To prove (5), we use the representation
1 d
|∇Φ(x+y)−∇Φ(x−y)| dx= d ∂ Φ (x+sy) (s)·y dx.
ZBR ZBR(cid:12)(cid:12)Z−1Xi=1 (cid:2) s ei i i(cid:12)(cid:12)
(cid:12)
The latteris boundedby C|y| ∆Φ. This immediately implies thatI <∞and
B2R 3
therefore (4) is proved. R
This means that |ξ·∇Φ| <∞ for every smooth compactly supported ξ.
Λ1,2
1+γ/2
Using smoothness conditions on ξ and boundedness of ∇Φ, we get from Lemma 6
that D2Φ has no singular parts and
1 2
(6) |D2Φ(x+y)−D2Φ(x)| dx dy <∞
ZBr |y|d+γ(cid:16)ZBr (cid:17)
for any B .
r
Finally, applying the Cauchy–Schwarzinequality, we obtain for every δ >0
1 2
|D2Φ(x+y)−D2Φ(x)| dx dy
(cid:16)ZBr|y|d+γ/2−δ/2 ZBr (cid:17)
1 2 |y|δ
≤ |D2Φ(x+y)−D2Φ(x)| dx dy· dy <∞, ∀B
ZBr |y|d+γ(cid:16)ZBr (cid:17) ZBr |y|d r
Changing variables implies
|D2Φ(z)−D2Φ(x)|
dx dz <∞
Z Z |z−x|d+ε
Q Q
for every 0<ε< γ and bounded Q. The proof is now complete. (cid:3)
2
4. Remarks on improved integrability
d
ByapplyingTheorem2wegetabetter (local)integrabilityofkD2Φk(Ld−γ2+ε
insteadofL1);seeCorollary4. Thiscanbeusedtoimprovetheestimatesobtained
in Theorem 2.
We assume for simplicity that we transport measures with periodical densities
by a periodical optimal mapping
T(x)=x+∇ϕ(x),
where ϕ is periodical. Equivalently, one can consider optimal transportation of
probabilitymeasuresontheflattorusT. Thenonecanrepeattheabovearguments
and obtain the same estimates which become global.
In general, it can be shown that the assumption kD2ϕk∈Lr(T) implies
|D2ϕ(x+y)−D2ϕ(x)|r2+r1
dx dy <∞
ZTZT |y|d+r+r1(γ+r−q1)−ε
8
for every ε and kD2ϕk∈Lr′(T) with any r′ satisfying
2dr
r′ < r+1 .
d− r (γ+ r−1)
r+1 q
Starting with r = 1 and iterating this process one can obtain a sequence r such
0 n
that kD2ϕk∈Lrn−ε for every rn and ε>0
2drn
r =1, r = rn+1 .
0 n+1 d− rn (γ+ rn−1)
rn+1 q
One has kD2ϕk∈Lr−ε, where r =lim r solves the equation
n n
x2+x(q(γ−d)−1)+qd=0
with r >1.
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