TTotal variation flow in the Subelliptic Heisenberg group Giovanna Citti October 11, 2014 Giovanna Citti (Universit` a di Bologna) Total variation October 11, 2014 1 / 54
Overview Motivation of the problem: the description of the visual cortex - C-. Sarti Journal of Math Vision 2006 Heisenberg group of dimension 1 - Capogna, C.-, Manfredini Indiana U. Math. Journal 2009 Higher dimension Heisenberg group - Capogna, C.-, Manfredini Crelle Journal 2010 Giovanna Citti (Universit` a di Bologna) Total variation October 11, 2014 2 / 54
Motivation of the problem: The visual cortex Let G be an analytic and simply connected Lie group w stratification = V 1 ⊕ V 2 ⊕ ... ⊕ V r , where [ V 1 , V j ] = V j +1 , if j = 1 , . . . , r − 1, and [ V k , V r ] = 0, k = 1 , . . . , r . The metric structure is given by assuming that one has a left invariant positive definite form We fix a orthonormal horizontal frame X 1 , . . . , X m which we complete it to a basis ( X 1 , . . . , X n ) of lie by choosing for every k = 2 , . . . , r a basis of V k . Giovanna Citti (Universit` a di Bologna) Total variation October 11, 2014 3 / 54
Carnot groups Let G Lie group with dimension n is a Carnot group = V 1 ⊕ V 2 ⊕ ... ⊕ V r where [ V 1 , V j ] = V j +1 , if j = 1 , ..., r − 1, and [ V k , V r ] = 0, k = 1 , ..., r . ( X 1 , ..., X n ) stratified basis of containing a basis of V k for every k deg ( X ) = k if X ∈ V k horizontal gradient ∇ X f = ( X 1 f · · · X m f ) Divergence: m � div X v = X i v i i =1 Giovanna Citti (Universit` a di Bologna) Total variation October 11, 2014 4 / 54
Curvature of a graph The notion has been obtained with different techniques: [D.Danielli, N. Garofalo, Nhieu, prepr. 2001] [N. Sherbakova, 2006 ]. [Stroffolini Manfredi] [Garofalo Pauls], [RItore rosales] [Ritore Galli], [Cheng, Hwang, Malchiodi, Yang] u : G → R Normal to the graph. It is the horizontal projection of the normal - never ( − 1 , ∇ 0 u ) √ vanishing for graphs: ν 0 = 1+ |∇ 0 u | 2 � � 1 + |∇ 0 u | 2 div ( ν 0 ) = 1 + |∇ 0 u | 2 Au . h 0 = where m � X i uX j u � � Au = δ ij − X i X j u 1 + |∇ 0 u | 2 i , j =1 Giovanna Citti (Universit` a di Bologna) Total variation October 11, 2014 5 / 54
n n � X ǫ ∂ u ǫ i u ǫ � � � X ǫ a ǫ ij ( ∇ ǫ u ǫ ) X ǫ i X ǫ ∂ t = h ǫ = = j u ǫ (1) i W ǫ i =1 i , j =1 for x ∈ and t > 0, with u ǫ ( x , 0) = ϕ ( x ), h ǫ is the mean curvature of the graph of u ǫ ( · , t ) and n � � ξ i ξ j ǫ = 1 + |∇ ǫ u ǫ | 2 = 1 + i u ǫ ) 2 and a ǫ W 2 � ( X ǫ ij ( ξ ) = W − 1 δ ij − , ǫ 1 + | ξ | 2 i =1 for all ξ ∈ R n . Giovanna Citti (Universit` a di Bologna) Total variation October 11, 2014 6 / 54
� ∂ t u ǫ = h ǫ in Q = × (0 , T ) (2) u ǫ = ϕ on ∂ p Q . Here ∂ p Q = ( ×{ t = 0 } ) ∪ ( ∂ × (0 , T )) denotes the parabolic boundary of Q . Giovanna Citti (Universit` a di Bologna) Total variation October 11, 2014 7 / 54
m k r 2 r ! | x | 2 r ! = � � d ( i ) , and d 0 ( x , y ) = | y − 1 x | . | x i | (3) k =1 i =1 The ball-box theorem in [ ? ] states that there exists A = A ( G , σ 0 ) such that for each x ∈ G , A − 1 | x | ≤ d 0 ( x , 0) ≤ A | x | . If x ∈ G and r > 0, we will denote by B ( x , r ) = { y ∈ G | d 0 ( x , y ) < r } as well as the pseudo-distance d G ,ǫ ( x , y ) = N ǫ ( y − 1 x ) with r � � 1 � � � i , ǫ − 2 � N 2 x 2 x 2 x 2 ǫ ( x ) = i + min ( k ) . (4) i d ( i )=1 i =2 d ( k )= i d ( i ) ≥ 2 Giovanna Citti (Universit` a di Bologna) Total variation October 11, 2014 8 / 54
Stability of the homogenous structure as ǫ → 0 If G is a Carnot group, d ǫ is the distance function associated to σ ǫ , we will denote B ǫ ( x , r ) = { y ∈ G | d ǫ ( x , y ) < r } . There is a constant C independent of ǫ such that for every x ∈ G and r > 0, | B ǫ ( x , 2 r ) | ≤ C | B ǫ ( x , r ) | . Giovanna Citti (Universit` a di Bologna) Total variation October 11, 2014 9 / 54
Theorem There exists constants C Λ > 0 depending on G , σ 0 , Λ but independent of ǫ such that for each ǫ > 0 , x ∈ G and t > 0 one has − d ǫ ( x , 0)2 d ǫ ( x , 0)2 e − C Λ e C Λ t t C − 1 | B ǫ (0 , √ t ) | ≤ Γ ǫ, A ( x , t ) ≤ C Λ | B ǫ (0 , √ t ) | . (5) Λ For s ∈ and k − tuple ( i 1 , . . . , i k ) ∈ { 1 , . . . , m } k there exists C s , k > 0 depending only on k , s , G , σ 0 , Λ such that − d ǫ ( x , 0)2 t X i 1 · · · X i k Γ ǫ, A )( x , t ) | ≤ C s , k t − s − k / 2 e C Λ t | ( ∂ s | B ǫ (0 , √ t ) | (6) for all x ∈ G and t > 0 . For any A 1 , A 2 ∈ M Λ , s ∈ and k − tuple ( i 1 , . . . , i k ) ∈ { 1 , . . . , m } k there exists C s , k > 0 depending only on k , s , G , σ 0 , Λ such that | ( ∂ s t X i 1 · · · X i k Γ ǫ, A 1 )( x , t ) − ∂ s t X i 1 · · · X i k Γ ǫ, A 2 )( x , t ) | ≤ (7) Giovanna Citti (Universit` a di Bologna) Total variation October 11, 2014 10 / 54 2
Definition x 0 ∈ ¯ For every ǫ > 0 and ¯ x , ¯ G define m m n n ¯ � � � � | v ǫ | w ǫ min( | w ǫ i | , | w ǫ i | 1 / d ( i ) )+ | v ǫ i | 1 / d ( i ) d ǫ (¯ x , ¯ x 0 ) = i | + i | + i =1 i =1 i = m +1 i = m +1 x 0 ∈ ¯ For ǫ = 0 and ¯ x , ¯ G define n i | 1 / d ( i ) + | v 0 ¯ � ( | w 0 i | 1 / d ( i ) ) d 0 (¯ x , ¯ x 0 ) = i =1 We will denote by ¯ B ǫ and ¯ B 0 the corresponding metric balls. Giovanna Citti (Universit` a di Bologna) Total variation October 11, 2014 11 / 54
Lemma There exists constants C Λ > 0 depending on G , σ 0 , Λ but independent of ǫ x ∈ ¯ such that for each ǫ > 0 , ¯ G and t > 0 one has x , 0)2 ¯ x , 0)2 d ǫ (¯ ¯ d ǫ (¯ − e − C Λ e C Λ t t B ǫ (0 , √ t ) | ≤ ¯ C − 1 Γ ǫ, A (¯ x , t ) ≤ C Λ B ǫ (0 , √ t ) | . (9) Λ | ¯ | ¯ For s ∈ and k − tuple ( i 1 , . . . , i k ) ∈ { 1 , . . . , m } k there exists C s , k > 0 depending only on k , s , G , σ 0 , Λ such that x , 0)2 ¯ d ǫ (¯ − x , t ) | ≤ C s , k t − s − k / 2 e C Λ t t X ǫ i 1 · · · X ǫ i k ¯ | ( ∂ s B ǫ (0 , √ t ) | Γ ǫ, A )(¯ (10) | ¯ x ∈ ¯ for all ¯ G and t > 0 . Moreover, as ǫ → 0 one has X ǫ i 1 · · · X ǫ i k ∂ s t ¯ Γ ǫ, A → X i 1 · · · X i k ∂ s t ¯ Γ A (11) uniformly on compact sets, in a dominated way on all ¯ G. Giovanna Citti (Universit` a di Bologna) Total variation October 11, 2014 12 / 54
Lemma Let u ǫ ∈ C 3 ( Q ) be a solution to ( ?? ) and denote v 0 = ∂ t u, v i = X r i u for i = i , . . . , n. Then for every h = 0 , . . . , n one has that v h is a solution of ∂ t v h = X ǫ i ( a ij X j v h ) = a ǫ ij ( ∇ ǫ u ǫ ) X ǫ i X ǫ j v h + ∂ ξ k a ǫ ij ( ∇ ǫ u ) X ǫ i X ǫ j u ǫ X ǫ k v h , (12) where 1 ξ i ξ j � � a ǫ ij ( ξ ) = δ ij − . � 1 + | ξ | 2 1 + | ξ | 2 Giovanna Citti (Universit` a di Bologna) Total variation October 11, 2014 13 / 54
Lemma Let G be a step two Carnot group. If f : G → R is linear (in exponential coordinates) then for every ǫ ≥ 0 , the matrix with entries X ǫ i X ǫ j f is anti-symmetric, in particular every level set of f satisfies h ǫ = 0 . Giovanna Citti (Universit` a di Bologna) Total variation October 11, 2014 14 / 54
Let G be a Carnot group of step two, ⊂ G a bounded, open, convex (in the Euclidean sense) set and ϕ ∈ C 2 ( ¯ ). For ǫ > 0 denote by u ǫ ∈ C 2 ( × (0 , T )) ∩ C 1 ( ¯ × (0 , T )) the non-negative unique solution of the initial value problem ( ?? ). There exists C = C ( G , || ϕ | C 2 ( ) ) > 0 such that ¯ sup |∇ ǫ u ǫ | ≤ sup |∇ 1 u ǫ | ≤ C . (13) ∂ × (0 , T ) ∂ × (0 , T ) Giovanna Citti (Universit` a di Bologna) Total variation October 11, 2014 15 / 54
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