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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

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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

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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

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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 X1, . . . , Xm which we complete it to a basis (X1, . . . , Xn) of lie by choosing for every k = 2, . . . , r a basis of Vk.

Giovanna Citti (Universit` a di Bologna) Total variation October 11, 2014 3 / 54

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Carnot groups

Let G Lie group with dimension n is a Carnot group = V1 ⊕ V2 ⊕ ... ⊕ Vr where [V1, Vj] = Vj+1, if j = 1, ..., r − 1, and [Vk, Vr] = 0, k = 1, ..., r. (X1, ..., Xn) stratified basis of containing a basis of Vk for every k deg(X) = k if X ∈ Vk horizontal gradient ∇Xf = (X1f · · · Xmf ) Divergence: divXv =

m

  • i=1

Xivi

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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 vanishing for graphs: ν0 =

(−1,∇0u)

√

1+|∇0u|2

h0 =

  • 1 + |∇0u|2div(ν0) =
  • 1 + |∇0u|2Au.

where Au =

m

  • i,j=1
  • δij −

XiuXju 1 + |∇0u|2

  • XiXju

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∂uǫ ∂t = hǫ =

n

  • i=1

X ǫ

i

X ǫ

i uǫ

Wǫ

  • =

n

  • i,j=1

aǫ

ij(∇ǫuǫ)X ǫ i X ǫ j uǫ

(1) for x ∈ and t > 0, with uǫ(x, 0) = ϕ(x), hǫ is the mean curvature of the graph of uǫ(·, t) and W 2

ǫ = 1 + |∇ǫuǫ|2 = 1 + n

  • i=1

(X ǫ

i uǫ)2 and aǫ ij(ξ) = W −1 ǫ

  • δij −

ξiξj 1 + |ξ|2

  • ,

for all ξ ∈ Rn.

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  • ∂tuǫ = hǫ

in Q = ×(0, T) uǫ = ϕ

  • n ∂pQ.

(2) Here ∂pQ = (×{t = 0}) ∪ (∂ × (0, T)) denotes the parabolic boundary of Q.

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|x|2r! =

r

  • k=1

mk

  • i=1

|xi|

2r! d(i) , and d0(x, y) = |y−1x|.

(3) The ball-box theorem in [?] states that there exists A = A(G, σ0) such that for each x ∈ G, A−1|x| ≤ d0(x, 0) ≤ A|x|. If x ∈ G and r > 0, we will denote by B(x, r) = {y ∈ G | d0(x, y) < r} as well as the pseudo-distance dG,ǫ(x, y) = Nǫ(y−1x) with N2

ǫ (x) =

  • d(i)=1

x2

i + min

  • r
  • i=2

(

  • d(k)=i

x2

k)

1 i , ǫ−2

d(i)≥2

x2

i

  • .

(4)

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Stability of the homogenous structure as ǫ → 0

If G is a Carnot group, dǫ is the distance function associated to σǫ, we will denote Bǫ(x, r) = {y ∈ G|dǫ(x, y) < r}. There is a constant C independent of ǫ such that for every x ∈ G and r > 0, |Bǫ(x, 2r)| ≤ C|Bǫ(x, r)|.

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Theorem

There exists constants CΛ > 0 depending on G, σ0, Λ but independent of ǫ such that for each ǫ > 0, x ∈ G and t > 0 one has C −1

Λ

e−CΛ

dǫ(x,0)2 t

|Bǫ(0, √t)| ≤ Γǫ,A(x, t) ≤ CΛ e

− dǫ(x,0)2

CΛt

|Bǫ(0, √t)|. (5) For s ∈ and k−tuple (i1, . . . , ik) ∈ {1, . . . , m}k there exists Cs,k > 0 depending only on k, s, G, σ0, Λ such that |(∂s

t Xi1 · · · XikΓǫ,A)(x, t)| ≤ Cs,kt−s−k/2 e − dǫ(x,0)2

CΛt

|Bǫ(0, √t)| (6) for all x ∈ G and t > 0. For any A1, A2 ∈ MΛ, s ∈ and k−tuple (i1, . . . , ik) ∈ {1, . . . , m}k there exists Cs,k > 0 depending only on k, s, G, σ0, Λ such that |(∂s

t Xi1 · · · XikΓǫ,A1)(x, t) − ∂s t Xi1 · · · XikΓǫ,A2)(x, t)| ≤

(7)

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Definition

For every ǫ > 0 and ¯ x, ¯ x0 ∈ ¯ G define ¯ dǫ(¯ x, ¯ x0) =

m

  • i=1

|vǫ

i |+ m

  • i=1

|wǫ

i |+ n

  • i=m+1

min(|wǫ

i |, |wǫ i |1/d(i))+ n

  • i=m+1

|vǫ

i |1/d(i)

For ǫ = 0 and ¯ x, ¯ x0 ∈ ¯ G define ¯ d0(¯ x, ¯ x0) =

n

  • i=1

(|w0

i |1/d(i) + |v0 i |1/d(i))

We will denote by ¯ Bǫ and ¯ B0 the corresponding metric balls.

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Lemma

There exists constants CΛ > 0 depending on G, σ0, Λ but independent of ǫ such that for each ǫ > 0, ¯ x ∈ ¯ G and t > 0 one has C −1

Λ

e−CΛ

¯ dǫ(¯ x,0)2 t

|¯ Bǫ(0, √t)| ≤ ¯ Γǫ,A(¯ x, t) ≤ CΛ e

−

¯ dǫ(¯ x,0)2 CΛt

|¯ Bǫ(0, √t)|. (9) For s ∈ and k−tuple (i1, . . . , ik) ∈ {1, . . . , m}k there exists Cs,k > 0 depending only on k, s, G, σ0, Λ such that |(∂s

t X ǫ i1 · · · X ǫ ik ¯

Γǫ,A)(¯ x, t)| ≤ Cs,kt−s−k/2 e

−

¯ dǫ(¯ x,0)2 CΛt

|¯ Bǫ(0, √t)| (10) for all ¯ x ∈ ¯ G and t > 0. Moreover, as ǫ → 0 one has X ǫ

i1 · · · X ǫ ik∂s t ¯

Γǫ,A → Xi1 · · · Xik∂s

t ¯

ΓA (11) uniformly on compact sets, in a dominated way on all ¯ G.

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Lemma

Let uǫ ∈ C 3(Q) be a solution to (??) and denote v0 = ∂tu, vi = X r

i u for

i = i, . . . , n. Then for every h = 0, . . . , n one has that vh is a solution of ∂tvh = X ǫ

i (aijXjvh) = aǫ ij(∇ǫuǫ)X ǫ i X ǫ j vh + ∂ξkaǫ ij(∇ǫu)X ǫ i X ǫ j uǫX ǫ kvh, (12)

where aǫ

ij(ξ) =

1

  • 1 + |ξ|2
  • δij −

ξiξj 1 + |ξ|2

  • .

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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 hǫ = 0.

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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 uǫ ∈ 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

∂×(0,T)

|∇ǫuǫ| ≤ sup

∂×(0,T)

|∇1uǫ| ≤ C. (13)

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