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Optimal transportation, curvature flows, and related a priori estimates

Alexander Kolesnikov

Moscow 2010

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Curvature flow Shrinking family of (convex) surfaces At, where every point x A on the curvature of A at x x = -K (x) · n(x). Here K (x) is the (Gauss) curvature of At at x. Approaches to the curvature flows · 1) Solving a parabolic nonlinear equation with smooth data ( R.S. Hamilton, R. Huisken ... ) · 2) Consider surfaces as level sets of a potential function , satisfying a nonlinear parabolic equation in viscosity sence (L.C. Evans, J. Spruck, Y.G. Chen, Yo. Giga, S. Goto ...) · 3) Singular limits (H.M. Soner, L. Ambrosio ...)
t

is moving in the direction of the normal n with the speed depending

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Transp ortational approach The Gauss flows can be obtained from the optimal transportation by a certain scaling procedure. One has to construct a "parabolic" version of the optimal transportation. (V. Bogachev, A. Kolesnikov) Let µ = 0 dx be a probability measure on convex set A, = 1 dx be a probability measure on BR = {x : |x| R}. There exist a function with convex sublevel sets { t} and a mapping T : A BR such that = µ T . T = | | curvature flow x = -td where t = (x). Scaling: For every n consider another measure
- n = Sn 1 with Sn(x) = x|x|n. -1 -1

and T has the form

The level sets of are moving according to a (generalized) Gauss 1 ( T ) K (x) · n(x), 0 ( x ) (1)

Let

Wn be the optimal transportation pushing forward µ to n.

- Set Tn = Sn 1

Wn. Define a new potential function n by 1 Wn = n+2. n+2 n Then T is the limit of Tn, where n Tn = n n. n+1 | n | Remark: There exist a unique mapping of this type.
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and Tn pushes forward µ to .


Motivation for this study: Two different ways of proving the classical isoperimetric inequality 1) Transp ortation pro of (M. Gromov) AR T=
d

Br = {x : |x| r} -- ball in Rd with vol(A) = vol(Br ) V : A Br -- optimal transportation of Hd|A to Hd|Br Change of variables formula det D2V = 1 Arithmetic-geometric inequality: 1 V . d

1 V dx dA A 1 r = nA, V dHd-1 Hd-1( A). d A d The isoperimetric inequality follows from vol(A) = det D2V dx vol(A) = vol(Br ) = cdrd.

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2) Geometric flows (P. Topping) Let At be a family of convex sets such that At evolve according to the Gauss curvature flow: x(t) = -K (x(t)) · n(x(t)) Here x(t) A A
r-t r -t

, n -- outer normal, K -- Gauss curvature of

, As1 As2 for s1 s2.

Existence: K. Tso (Chou), 1975. a) Evolution of the volume: vol(At) = - t K dH
At d-1

= -H

d-1

(S

d-1

) = -d

(2)

(by Gauss-Bonnet theorem). Volume decreases with a constant speed. b) Evolution of the surface measure: H t
d-1

( A t ) = -
At

K H dH

d-1

, H -- mean curvature
d -1

Arithmetic-geometric inequality: d-1 H ( At) -(d - 1) t HЁ older inequality: d =
At



K

H d-1

. .

K
At

d d-1

dH

d-1

K dH H t

d-1


At

K

d d -1

dH

d-1

d-1 d

H

d-1

( A t )

1 d

.

d-1

( A t ) -

(d - 1)
d d-1

H

d-1

( A t )

1 d-1

.

(3)

d The isoperimetric inequality follows by comparison arguments from (2) and (3).

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Change of variables formula

Change of variables for the optimal transp ortation (R. McCann) If where V is the optimal transportation of µ to , then µ-almost every2 det Da V =

0 . 1 ( V ) 2 Here Da V is the second Alexandrov derivative of V .

Main difficulty: potential is not Sobolev, but only BV. The second derivatives of do exist only in directions orthogonal to
| |

.

Change of variables for T Theorem The following change of variables formula holds for µ-almost all x: 0 . K |Da|d-1 = 1 (T ) Here K is the Gauss curvature of the corresponding level set and Da is the absolutely continuous component of D.

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Reverse mapping Take x Br with |x| = t. Let H be the support function of At = { t}. H (v ) = sup x, v ,
xAt

S ( x) = T n=
x |x|

-1

( x) = H · n +

S d-1

H

,

S d-1

-- spherical gradient.

Variants of the parab olic maximum principle 1) Let f be a twice continuously differentiable function on a convex set A Rd. Then there exists a constant C = C (d) depending only on d such that sup f (x) sup f (x) + C (d)
xA x A Cf

| f |K dx.

where Cf are contact points of the level sets {f = t} with the convex envelopes of {-f t}. 2) Maximum principle: (d = 2) For every smooth f defined on : {0 < R0 r R1, } with | - | < one has: sup f C
1,

· sup f + C
p

2, f

|fr (f + f )| dx, r

where f : {fr 0, f + f 0} and p is the parabolic boundary of .

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Regularity results 1) Sob olev estimates for Theorem: Let d = 2, =
CR r

· IBR , . | |
p+1

T =

Assume that T pushes forward µ to . Then C
p,R A

| |

p+1

dµ
A

µ

µ

dµ +
A

1+p µ dH1. Kp

( Proof: change of variables formula, integration by parts). 2) Uniform estimates for Theorem: There exists a universal constant p > 0 such that sup | | C1(M ) sup | | + C2(M )
A A

provided M = sup
µ Lp (µ)

,

-1 µ Lp (µ)

, | µ|

-1 µ Lp (µ)

< .

( Proof: Sobolev estimates of + a parabolic analog of the Alexandrov maximum principle).

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Problem: What kind of flows can b e constructed by mass-transp ortational metho ds? Assume for simplicity that d = 2. Let F (r, ) be a smooth function. Consider a mapping of the type T = F (, n) · n, where has level convex subsets and n =
| |

. One has

det DT = | |F Fr (, n)K. In particular, assume that F depends on in the following way
r

F (r, n) = where H (t, n) = sup (support function).

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0

- g (Hr 1(s, n))Hr (s, n) ds,

xAt

x, n is the corresponding dual potential

Then, assuming that T pushes forward µ = µ(x)dx to |BR , one has the following change of variables formula µ = g (| |)K. Since | |-1 is the speed of level sets At in the direction of the inward normal, one gets that At are moving according to the following curvature flow: x=- 1 . g -1(µ/K )

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Examples of flows of this typ e 1) Power-Gauss curvature flows x = -K p · n. (geometry, computer vision) 2) Logarithmic Gauss curvature flows x = - log K · n (Minkowsky-type problems) Main difficulty: One needs more regularity of . In particular, it is natural to expect that is Sobolev (not only BV).

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