Optimal transportation, curvature flows, and related a priori - PDF document

Optimal transportation, curvature flows, and related a priori estimates Alexander Kolesnikov Moscow 2010 1

Curvature flow Shrinking family of (convex) surfaces ∂A t , where every point x ∈ ∂A t is moving in the direction of the normal n with the speed depending on the curvature of ∂A at x x = − K ( x ) · n ( x ) . ˙ Here K ( x ) is the (Gauss) curvature of ∂A t 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 ϕ , satis- fying 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 ...) 2

Transportational 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 = ρ 0 dx be a probability measure on convex set A , ν = ρ 1 dx be a probability measure on B R = { x : | x | ≤ R } . There exist a function ϕ with convex sublevel sets { ϕ ≤ t } and a mapping T : A → B R such that ν = µ ◦ T − 1 and T has the form T = ϕ ∇ ϕ |∇ ϕ | . The level sets of ϕ are moving according to a (generalized) Gauss curvature flow x = − t d − 1 ρ 1 ( T ) ˙ ρ 0 ( x ) K ( x ) · n( x ) , (1) where t = ϕ ( x ). Scaling: For every n consider another measure ν n = ν ◦ S − 1 with S n ( x ) = x | x | n . n Let ∇ W n be the optimal transportation pushing forward µ to ν n . Set T n = S − 1 n ◦ ∇ W n . Define a new potential function ϕ n by 1 n + 2 ϕ n +2 W n = . n Then T is the limit of T n , where ∇ ϕ n T n = ϕ n n +1 . n |∇ ϕ n | and T n pushes forward µ to ν . Remark : There exist a unique mapping of this type. 3

Motivation for this study: Two different ways of proving the classical isoperimetric inequality 1) Transportation proof (M. Gromov) A ⊂ R d B r = { x : | x | ≤ r } — ball in R d with vol( A ) = vol( B r ) T = ∇ V : A → B r — optimal transportation of H d | A to H d | B r Change of variables formula det D 2 V = 1 Arithmetic-geometric inequality: 1 ≤ ∆ V d . � det D 2 V dx ≤ 1 � vol( A ) = ∆ V dx d A A = 1 � n A , ∇ V � d H d − 1 ≤ r � d H d − 1 ( ∂A ) . d ∂A The isoperimetric inequality follows from vol( A ) = vol( B r ) = c d r d . 4

2) Geometric flows (P. Topping) Let A t be a family of convex sets such that ∂A t evolve according to the Gauss curvature flow: x ( t ) = − K ( x ( t )) · n ( x ( t )) ˙ Here x ( t ) ∈ ∂A r − t , n — outer normal, K — Gauss curvature of ∂A r − t , A s 1 ⊂ A s 2 for s 1 ≥ s 2 . Existence: K. Tso (Chou), 1975. a) Evolution of the volume: ∂ � K d H d − 1 = −H d − 1 ( S d − 1 ) = − κ d ∂t vol( A t ) = − (2) ∂A t (by Gauss-Bonnet theorem). Volume decreases with a constant speed. b) Evolution of the surface measure: ∂ � ∂t H d − 1 ( ∂A t ) = − KH d H d − 1 , H — mean curvature ∂A t √ d − 1 H Arithmetic-geometric inequality: K ≤ d − 1 . ∂ � d ∂t H d − 1 ( ∂A t ) ≤ − ( d − 1) d − 1 d H d − 1 . K ∂A t H¨ older inequality: d − 1 d H d − 1 � d − 1 � 1 � �� K d H d − 1 ≤ d d � d . H d − 1 ( ∂A t ) κ d = K ∂A t ∂A t � 1 ∂t H d − 1 ( ∂A t ) ≤ − ( d − 1) ∂ � d − 1 . H d − 1 ( ∂A t ) (3) d d − 1 κ d The isoperimetric inequality follows by comparison arguments from (2) and (3). 5

Change of variables formula Change of variables for the optimal transportation (R. McCann) If ∇ V is the optimal transportation of µ to ν , then µ -almost every- where ρ 0 det D 2 a V = ρ 1 ( ∇ V ) . Here D 2 a V is the second Alexandrov derivative of V . Main difficulty: potential ϕ is not Sobolev, but only BV. The second derivatives of ϕ do exist only in directions orthogonal to ∇ ϕ |∇ ϕ | . Change of variables for T Theorem The following change of variables formula holds for µ -almost all x : ρ 0 K | D a ϕ | ϕ d − 1 = ρ 1 ( T ) . Here K is the Gauss curvature of the corresponding level set and D a ϕ is the absolutely continuous component of Dϕ . 6

Reverse mapping Take x ∈ B r with | x | = t . Let H be the support function of A t = { ϕ ≤ t } . H ( v ) = sup � x, v � , x ∈ A t S ( x ) = T − 1 ( x ) = H · n + ∇ S d − 1 H n = x | x | , ∇ S d − 1 — spherical gradient. Variants of the parabolic maximum principle 1) Let f be a twice continuously differentiable function on a convex set A ⊂ R d . Then there exists a constant C = C ( d ) depending only on d such that � sup f ( x ) ≤ sup f ( x ) + C ( d ) |∇ f | Kdx. x ∈ A x ∈ ∂A C f where C f 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 Ω : { 0 < R 0 ≤ r ≤ R 1 , α ≤ θ ≤ β } with | β − α | < π one has: �� | f r ( f + f θθ ) | sup f ≤ C 1 , Ω · sup f + C 2 , Ω dx, r Ω ∂ p Ω Γ f where Γ f : { f r ≤ 0 , f + f θθ ≤ 0 } and ∂ p Ω is the parabolic boundary of Ω. 7

Regularity results 1) Sobolev estimates for ϕ Theorem: Let d = 2, ρ ν = C R r · I B R , T = ϕ ∇ ϕ |∇ ϕ | . Assume that T pushes forward µ to ν . Then ρ 1+ p � � � ∇ ρ µ p +1 � � � |∇ ϕ | p +1 dµ ≤ µ K p d H 1 . C p,R dµ + � � ρ µ � A A ∂A ( Proof: change of variables formula, integration by parts). 2) Uniform estimates for ϕ Theorem: There exists a universal constant p > 0 such that sup |∇ ϕ | ≤ C 1 ( M ) sup |∇ ϕ | + C 2 ( M ) A ∂A provided � � � ρ µ � L p ( µ ) , � ρ − 1 � |∇ ρ µ | ρ − 1 � � M = sup µ � L p ( µ ) , < ∞ . � µ L p ( µ ) ( Proof: Sobolev estimates of ϕ + a parabolic analog of the Alexandrov maximum principle). 8

Problem: What kind of flows can be constructed by mass-transportational methods? Assume for simplicity that d = 2. Let F ( r, θ ) be a smooth function. Consider a mapping of the type T = F ( ϕ, n ) · n, where ϕ has level convex subsets and n = ∇ ϕ |∇ ϕ | . One has det DT = |∇ ϕ | FF r ( ϕ, n ) K. In particular, assume that F depends on ϕ in the following way � � r g ( H − 1 F ( r, n ) = 2 r ( s, n )) H r ( s, n ) ds, 0 where H ( t, n ) = sup x ∈ A t � x, n � is the corresponding dual potential (support function). Then, assuming that T pushes forward µ = ρ µ ( x ) dx to λ | B R , one has the following change of variables formula ρ µ = g ( |∇ ϕ | ) K. Since |∇ ϕ | − 1 is the speed of level sets A t in the direction of the in- ward normal, one gets that A t are moving according to the following curvature flow: 1 x = − ˙ g − 1 ( ρ µ /K ) . 9

Examples of flows of this type 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 ϕ . In particular, it is natural to expect that ϕ is Sobolev (not only BV). 10