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Submitted on 30 Jun 2009
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Diffeomorphic Framework : From the Discontinuous
Image Matching Problem to a Stochastic Growth Model
François-Xavier Vialard
To cite this version:
François-Xavier Vialard. Hamiltonian Approach to Shape Spaces in a Diffeomorphic Framework : From the Discontinuous Image Matching Problem to a Stochastic Growth Model. Mathematics [math]. École normale supérieure de Cachan - ENS Cachan, 2009. English. �tel-00400379�
P Q L2(S1, R) L2(Tm, R) H = L2(T m, Rd) P× Q Fs Hs
large deformation diffeomorphisms
+
large deformation diffeomorphisms
• • • large de-formation diffeomorphisms • Fs L2
Fs× F s
•
p≥ 1 U Rd V ⊂ Xp 0(U, Rd) Cp U V CV |v|p, ≤ CV|v|V . V p ≥ 1 p− L2([0, 1], V ) L1([0, 1], V ) |v|L1 ≤ |v|L2 L1 L2 Lp([0, 1], V ) Lp([0, 1], χp0)
Ifv∈ L1([0, 1], V ) then its flow is defined: if t∈ [0, 1]
(
∂tφv0,t(x) = vt(φ0,t(x))
φv0,0(x) = Id .
For all times, t∈ [0, 1], φvs,t:= φv0,t◦ [φv0,s] 1is aCpdiffeomorphism. The application
(t, x)∈ [0, 1]2× U 7→ φv0,t(x)∈ U
is Lipschitz in both variables, more precisely:
Lip(φv0,t)≤ ecV vL1,
|φv0,t− φv0,s| ≤
Z t s
cV|vr|V dr .
The derivatives ofφvs,tverify the integral equation fork≤ p:
dkxφvs,t(x) = Id + Z t
s
dk[vr◦ φvs,t(x)] dr ,
and there exists constantsC, C independent on v such that for all s, t∈ [0, 1], |φvs,t|p, ≤ CeC
′cV v L1.
L1
v
Letu, v∈ L1([0, 1], V ) be two time dependent vector fields, then
|φu0,t− φv0,t| ≤ cV|u − v|L1ecV vL1.
Ifp≥ 2, a bound on the derivative is given by
|dφu0,t− dφv0,t| ≤ C |u − v|L1ecV uL1,
withC depending continuously on|u|L1 and|v|L1.
Ifvn →L2([0,1],V ) v, then dkφvnt,s → dkφvt,s for the uniform convergence on
every compact uniformly int, s∈ [0, 1] for k ≤ p.
Ifun ⇀L2([0,1],V ) u (weak convergence), then φunt,s → φut,sfor the uniform convergence
on every compact uniformly int, s∈ [0, 1]. If K ⊂ Rda compact set, then
sup (t,s,x) [0,1]2 K|φ un t,s(x)− φut,s(x)| →n + 0 . p ≥ 2 L2([0, 1], V ) L1([0, 1], V ) GV GV :={φv0,1| v ∈ L2([0, 1], V )} . vt v1 t GV
Introducingd(ψ1, ψ2) := inf{|v|L2| ψ1= φv0,1◦ψ2}, (GV, d) is a complete
L2([0, 1], V ) L1([0, 1], V ) GV L2 L1 φ∈ GV d(ψ1◦ φ, ψ2◦ φ) = d(ψ1, ψ2) . v GV
large deformation diffeomorphisms
Letψ1, ψ2 ∈ GV be two diffeomorphisms, then there existsv∈ L2([0, 1], V )
such thatψ1= ψ2◦ φv0,1, and|v|L2 = d(ψ1, ψ2).
vt
φ Rn v
Rn Adφv = (dφ v)◦ φ 1
Let(ut, vt) ∈ L2([0, 1], V ) be two time dependent vector fields, and denote
byφε0,tthe flow generated by the vector fieldut+ εvt, then we have:
∂εε=0φε0,1(x) = Z 1 0 [dφt,1]φ0,t(x)vt(φ0,t(x))dt = dφ0,1( Z 1 0 Adφt,0(vt) dt) . ∂εε=0φ0,1◦ (φε0,1) 1 =− Z 1 0 Adφt,1(vt) dt . Proof. At ∈ Rn At(φ0,t(x)) = ∂εε=0φ ε 0,t(x) d dtAt(φ0,t(x)) = dut(At(φ0,t(x))) + vt(φ0,t(x)), L LutAt= d du u=0[dφ0,t+u] 1(A t+u(φt+u(x))) = [dφ0,t] 1(vt(φ0,t(x))). V
V C1 |v|1, ≤ |v|V V U x∈ U M δx: v∈ V 7→ v(x) ∈ Rd, |v(x)| ≤ M |v|V V V K : V 7→ V , L : V 7→ V x ∈ U p ∈ Tx = Rd Tx x Tx x Rd V k(., x)p hk(., x)p, viV =hv(x), pi . Kδxp δxp k k : (y, x)∈ U × U 7→ k(y, x) ∈ L(Tx, Ty) , k(y, x) : p∈ Tx = k(y, x)p∈ Ty. • k(x, y) = k(y, x) • k (xi)i [1,n] n (pi)i [1,n] n Rd | n X i=1 k(., xi)pi|2V ≥ 0 ⇔ X i,j pjk(xj, xi)pi ≥ 0 .
For any positive matrix k on U × U there exists a uniquely determined
(vi)i N
V k(y, x) =Pi Nvi(y).vi(x)
v(y).v(x) : p∈ Tx 7→ hv(x), piv(y) ∈ Ty.
V
LetV a RKHS of vector fields on U , then there is an equivalence between • V ⊂ C0(U, Rd),
• k is continuous on U × U.
Proof. k f ∈ V
x, y∈ U p∈ Rd
|hf(x), pi − hf(y), pi| = |hk(., x)p − k(., y)p, fiV|
≤ |pk(x, x)p + pk(y, y) − 2pk(x, y)p|12|f|V , f V ⊂ C0(U, Rd) k(., x)p (vi)i N k(x, x) =X i N vi(x).vi(x), vi pk(x, x)p k U× U f = k(., y)p |k(x, y) − k(x , y )| ≤ |k(x, y) − k(x , y)| + |k(x , y) − k(x , y )| ,
Ifk has continuous derivatives, ∂m+nk(x, y)
∂mx∂ny
m, n∈ [1, r] ,
P i [1,n]k(., xi)pi = 0 v ∈ V Pi [1,n]hv(xi), pii = 0 pi = 0∀i ∈ [1, n] V V x := (x1, . . . , xn) n p = (p1, . . . , pn) k(x) : Tx1× . . . × Txn 7→ Tx1× . . . × Txn k(x) = [k(xi, x)p]i [1,n]
Letk be a positive definite kernel. If x is a group of n distinct points in Rdand u∈ [Rd]na vector, there exists a unique vector fieldv∈ V such that
v(xi) = ui∀i ∈ [1, n],
and which minimizes the norm onV among the vector fields in V verifying . More-over, v(.) = X i [0,n] k(., xi)pi, such thatk(x)p = u. Proof. W u W = u + W0, W0 = Vect. {w | hk(., xi)pi, wi = 0 ; pi ∈ Rd, i∈ [1, n]} u u W0 u∈ Vect{k(., xi)pi| pi∈ Rd, i∈ [1, n]} W0 k(x) Hm(U, Rd) m Rd k(x− y) d = 1 µ R k k(x) = Z R eitxdµ(x) , k(x, y) = k(x− y)
d
k(|x − y|)Id k
d
Suppose thatf : R+ 7→ R+ is a continuous function, then
the two following assertions are equivalent:
• the function x ∈ Rn7→ f(|x|) is a positive kernel,
• the function t ∈ R+ 7→ f(√t) is the Laplace transform of a positive finite Borel
measure on R+. k(x, y) = e |x−y|2σ2 σ k(x, y) = 1 1+|x−y|2σ2 k(x, y) = e |x−y| σ 0 D(Id, φ)2+ H(φ) , H
GV L2([0, 1], V ) V v ∈ L2([0, 1], V ) R1 0 |ut|2V dt φ0,1 = ψ φε0,t ε ∂tφε0,t = vtε◦ φε0,t, ∂εφε0,t= wtε◦ φε0,t. ∂ε,tφε0,t = ∂t,εφε0,t, ∂εvtε= ∂twεt+ dwtε(vtε)− dvtε(wεt). u, v adu(v) = [u, v] , ∂εvtε= ∂twεt − advε tw ε t. ε = 0 Z 1 0 (Lvt, ∂εvt) dt = Z 1 0 (Lvt, ∂twtε− advε tw ε t) dt = 0 . w0= w1 = 0 Z 1 0 (d dtLvt+ advtLvt, wt) dt = 0 , ad ad wt φε0,t w V w0 = w1= 0 d dtLvt+ advtLvt= 0 . advtw = ds s=0d Adφt,t+sv φt,s ut Adψv := dψ(v◦ ψ 1) ψ d dt[Adφ0,tLvt] = 0 .
n Rd x= (x1, . . . , xn) y= (y1, . . . , yn) J(u) = Z 1 0 |ut| 2 V dt + d(φ0,1.x, y)2, d Rnd φ0,1 u ∈ L2([0, 1], V ) φ.x φ x φ.x := (φ(x1), . . . , φ(xn)) x Rd y u ∈ L2([0, 1], V ) min{E(v) = Z 1 0 |vt| 2 V dt| φv0,1.x = φu0,1.x}. x y n x x n
The action of the group is transitive among the group ofn distinct points
if n > 1. For n = 1, two groups of points are in the same orbit if and only if the
permutation σ ordering x in increasing order: xσ(1) < xσ(2) < . . . < xσ(n) is the permutation ordering y in increasing order.
Proof. x n xi 6= xj i6= j φ0,1 φ(xi) 6= φ(xj) φ0,1 C1 c : [0, 1] 7→ [Rd]n c(0) = x c(1) = y ci(t)6= cj(t) t ∈ [0, 1] i, j ∈ [1, n] 1 n v ∈ L2([0, 1], V ) φt.x = c(t) v vt(.) = k(., c(t))p(t) p(t) [Rd]n k(c(t), c(t))p(t) = dtdc(t) k(c(t), c(t))
p(t) C1 d dtc i(t) = v t(ci(t)) i∈ [0, n] , φ0,t.x = c(t) t∈ [0, 1]
The vector field introduced in the above proof is the unique minimizer for the norm onL2([0, 1], V ) of the vector fields v verifying .
A minimizerv of the inexact matching verifies the following geodesic equation, vt(.) =− n X i=1 k(., φ0,t(xi))[dφt,0]φ0,t(xi)(pi) ,
with [pi]i [1,n] a vector defined on the cotangent space of[qi(0)]i [1,n] called the initial
momentum. Proof. ut Z 1 0 hvt , uti + n X i=1 h(dφ0,1(φ0,1(xi)− yi)), Z 1 0 [Adφt,0ut](xi) dti = 0 . Z 1 0 X hk(., φ0,t(xi))[dφt,0]φ0,t(xi)(pi), utiV dt , pi = dφ0,1(φ0,1(xi)− yi)) u∈ L2([0, 1], V ) Lvt= n X i=1 Kδφ0,t(xpi(t) i) pi(t) = [dφt,0]φ0,t(xi)(pi) I 7→ I ◦ φ1,0, I U ⊂ Rd χS S χS◦ φ1,0 = χφ 0,1(S)
I0 Itarg J(v) = Z 1 0 |vt| 2 V dt + Z U H(I0◦ φ1,0, Itarg)dµ , µ U H I0 C1 Itarg ∈ L (U) H(x, y) =|x − y|2 Z U
2(I0◦ φ1,0− Itarg)h∇I0, ∂εφε0,1idµ .
It := I0◦ φt,0 vt+ εut ∂ε Z U (I1◦ (φ0,1◦ φε1,0)− Itarg)2dµ =− Z U (I1− Itarg)h∇I1, Z 1 0 Adφ t,1(ut)i dµ . f, g ∈ L2(U, Rd) ψ Z Uhf, Ad ψgi dµ = Z Uhdψ f ◦ ψ, giJac(ψ)dµ . ∇(I ◦ ψ) = dφ (∇I ◦ φ) . ∆ = (I0◦ φ1,0− Itarg) Z U
2(I0◦ φ1,0− Itarg)h∇I0, Adφt,1(ut)idµ =
Z
U
pth∇It, utiV ,
pt= Jac(φt,1)∆◦ φt,1
IfI0 isC1 andItarg ∈ L (U), a minimizer of the functional with
H(x, y) =|x − y|2verifies the geodesic equation
vt(.) =
Z
U
k(., x)∇It(x)pt(x)dµ(x) ,
withpt= Jac(φt,1)∆◦ φt,1and∆ = (I0◦ φ1,0− Itarg).
It
I I ◦ φ1,0 I
x0 ∈ U vt x0 x1 ∈ U
I0 = χS S C1 S H Itarg H1 U V Jt= Z V χU ◦ φt1dx = µ(V ∩ φt(U )), µ Jt V We have, ∂t t=0+µ(V ∩ φt(U )) = Z ∂UhX, ni˜1V (X)dHn 1,
with ˜1V(X)(y) = limǫ 0+1V¯(y + ǫX), if the limit exists and 0 elsewhere. And we
denote byn the outer unit normal to ∂U .
H Im(U ) = BV (U )∩ L (U )
B
LetH be a locally Lipschitz function from R2to R+,(f, g)∈ B×B be two
images,X be a Lipschitz time dependent vector field C1in space andφ
tbe its associated
flow. We define the functional
Jt(f, g) =
Z
U
V U
X U
then, under the additional assumption that H is C1 in the first variable (the derivative w.r.t. to such a variable being denoted by∇1H), we have
∂t=0+Jt=
Z
h∂1H(f (x), gX0(x)),−X0idx ,
where∂1H(f, gX0) is a part of the BV derivative of H(f, g), defined by
∂1H(f (x), l) =∇1H(f (x), l)(∇f(x) + Dcf (x)) + jH(x)Hn 1
x
Jf, withjH(x) = (H (f+(x), l)− H (f (x), l)) νf(x). • First reduction X0 o(t) • Second reduction F (x, y) = xy Im(U ) ∇1F f C1 |Jt(f, g)− J0(f, g)| ≤ Lip1(F ) Z U|f ◦ φ 1 t − f|dx ≤ Lip1(F )||X|| Z t 0 Z U|∇(f ◦ φ 1 s )|dx ds .Lip1(F ) F {(f(x), g(y))|(x, y) ∈ U2} f ∈ BV
[0, t0]
X 1({0})
xy t≤ t0
g∈ Im(U ) f ∈ BV (U) g∈ L (U)
|Jt(f, g)− J0(f, g)| ≤ Ct||g|| ||f||BV. g BV f f g • Third reduction F (x, y) = xy g Im(U ) x X(x)6= 0 Rn 0 Jt =R R nf ◦ ψ(x − tv) g ◦ ψ Jac(ψ)dx . ψ V g◦ ψ Jac(ψ) Im(U ) BV
The equation can be rewritten as
vt(.) = k(., xt)pt,
and we have,
(
˙pt=−pt∂1k(xt, xt)pt,
which are the Hamiltonian equations for the landmark matching: with
H(p, q) = 1
2pk(x)p,
the equations can be rewritten
( ˙pt=−∂xH , ˙xt= ∂pH . Proof. vt(.) = n X i=1 k(., φ0,t(xi))[dφt,0]φ0,t(xi)(pi) . [pt]i = [dφt,0]φ0,t(xi)(pi) ∂tdφ0,t = dvt.dφ0,t, ∂t[dφt,0]φ0,t = ∂tdφ0,t1 =−dvt.dφ0,t1 , dvt(.) =Pni=1∂1k(., φ0,t(xi))[pt]i f ∈ C0 (U ) d dt < pt, f >L2=− < pt,h∇f, vti >L2 . ( ˙ It=h∇It, vti = ∂pH(p, q) , ˙pt=< pt,h∇., vti >L2=−∂IH(p, q) ,
H
H(I, p) = 1 2
Z
U U
pt(x)∇I(x)k(x, y)pt(y)∇I(y) dxdy .
L2(S1, R2) I 7→ η0,1◦ I ◦ φ1,0, η (st)∈ L2([0, 1], S) S [0, 1] [0, 1] J(vt, st) = λ 2 Z 1 0 kvtk 2 Vdt + β 2 Z 1 0 kstk 2 Sdt + Z M H(η0,1◦ I0◦ φ1,0(x), Itarg(x))dx, β, λ N
˙pt=−∂qH + εdBt, ˙qt= ∂pH , pt qt C1 vt(x) = k(x, qt)pt. • • L2(S1, R) Fs
L2
L1
A Lipschitz open domain is an open subset U of Rn connected, bounded and nonempty such that for everyx∈ ∂U there exist
• an affine orthonormal basis B = (e1(x), . . . , en(x)) of Rninx,
• a Lipschitz function w defined on Vect(e1(x), . . . , en 1(x)) = Rn 1,
such that ifz∈ Rnis described by its coordinates in B,(z
1, . . . , zn) then
U∩ B(x, ǫ) = {z ∈ B(x, ǫ)|zn> w(z− znen)}.
A Lipschitz domain is a subsetD of Rnif there exists an open domainU
such that
U ⊂ D ⊂ ¯U . U
(M )
Iff ∈ Im(M), there exists a partition of M in Lipschitz domains (Ui)i [0,k]
for an integerk≥ 0, and the restriction fUi is Lipschitz.
The extension theorem of Lipschitz function in Rnenables to consider that on eachUi,fUi is the restriction of a Lipschitz function defined on Rn.
The most simple example is a piecewise constant function,f =Pki=1ai1Ui
withai ∈ R.
A partition satisfying the definition 3 is not uniquely defined and in fact it does not seem possible to reach a uniqueness condition with an additional assumption such as a partition of minimal length. Obviously such a minimal partition will contain the jump set of the function.
Moreover, let us consider the equivalence relation on the domain of definition of the piecewise Lipschitz functionf defined by x ∼ y if there exists a connected open set on
which the restriction of f is continuous. Then it gives a partition in open sets of the
domain, but this partition is not a Lipschitz one.
M M L2 Z U|I 0◦ φ1,0− Itarg|2dµ = Z U (I0◦ φ1,0)2− 2ItargI0◦ φ1,0+ Itarg2 dµ .
φ1,0
I2 0
(I0 ◦ φ1,0)2 = I02 ◦ φ1,0
Jt=RUItargI0◦ φ1,0dµ
LetU, V be two bounded Lipschitz domains of Rn. Let X be a Lipschitz
vector field on Rn andφt be the associated flow. Finally, letg and f be Lipschitz real
functions on Rn. Consider the following quantity depending ont,
Jt=
Z
φt(U )
f ◦ φt1g1Vdµ,
whereµ is the Lebesgue measure, then ∂t t=0+Jt= Z U−h∇f, Xig1V dµ + Z ∂UhX, nifg˜1V (X)dHn 1.
with ˜1V(X)(y) = limǫ 0+1V¯(y + ǫX), if the limit exists, 0 elsewhere. And n denotes
the outer unit normal on∂U .
C1
X Rn
g
LetU, V be C1 domains of Rn,X be a Lipschitz vector field and φt be
the associated flow. Letf and g be respectively Lipschitz and continuous real functions
on Rn. Consider the following quantity depending ont,
Jt=
Z
φt(U )
f ◦ φt1g1Vdµ,
whereµ is the Lebesgue measure, then ∂t t=0+Jt= Z U−h∇f, Xig1V dµ + Z ∂UhX, nifg˜1V (X)dHn 1.
If x ∈ ∂U ∩ ∂V with U, V two C1 open sets in Rn, then there exists ψ a
C1 diffeomorphism of Rn,W a neighborhood of x and k a C1function defined on Rn 1 such that
1. ψ(x) = 0, 2. ψ(U ∩ W ) = {(z, t) ∈ Rn 1× R|t > 0} ∩ B(0, 1), 3. ψ(V ∩ W ) = {(r, s) ∈ Rn 1× R|s > k(r)} ∩ B(0, 1) or ψ(V ∩ W ) = {(r, s) ∈ Rn 1× R|s < k(r)} ∩ B(0, 1). Proof. W U C1 C1 φ ψ(V ) C1 v H 0 ψ(V ) ∂ψ(V ) C1 f H (e1, . . . , en) Rn hv, eni 6= 0 A ∈ L(Rn) A(v) = en (e1, . . . , en 1) = Ker(A− Id) p1 H p2 (e1, . . . , en 1) hv, eni 6= 0 p2 H : H 7→ Ker(A − Id) z = p1(z) + λ(z)en z = p2(z) + α(z)n z λ, α∈ (Rn) A(z) = p 1(p2(z)) + (λ(p2(z)) + α(z))en x = p2(z) A◦ ψ(∂V ) = {(x, λ(p21(x)) + f◦ p21(x))|x ∈ p2(H∩ W )}. f ◦ p21 C1 A◦ φ hn, eni = 0 H C1
Letφtbe the flow of the vector field LipschitzX on Rn(withkXk bounded
on Rn). LetV ={(x, z) ∈ Rn 1× R|z > w(x)} be an open set with w a C1function,
and we introducewt(x) = inf{z ∈ barR|(x, z) ∈ φt(V )}.
Then, there exists ε > 0 such that wtis defined in R fort ∈] − ε, ε[ and is C1 in both
variables. The partial derivative is
∂t t=0wt(x) =−h∇w(x), p1(X(x, w(x)))i + p2(X(x, w(x))),
M 0 A(M) H A Proof. F (x, t) = p1(φt(x, w(x)))− x0, C1 ∂ xF (x0, 0) = IdRn−1 x0 ∈ Rn 1 C1 x0 : t 7→ x0(t) F (x, t) = 0 ⇔ x = x0(t) (x0, 0) x0(t) ∂t t=0xt = −p1(X0(x0, w(x0))). wt(x) = p2(φt(xt, w(xt))), t7→ wt(x0) C1 x0 ∂t|t=0wt(x0) =−h∇w(x0), p1(X0(x0, w(x0)))i + p2(X(x0, w(x0))).
In the first lemma, one can suppose thatV is a Lipschitz open set. However
our proof can not be extended if we assume the open sets to be Lipschitz domains and the change of coordinates to be a Lipschitz homeomorphism. As there exist bi-Lipschitz homeomorphisms that do not preserve the Lipschitz regularity of domains, the result even seems to be false in this case.
The second lemma can be extended to Lipschitz regularity for the function w. The
gen-eralization of the proof could use an implicit function theorem with Lipschitz regularity which can be found in [PS03] and some additional details. The final result is the almost everywhere differentiability ofwt.
Proof. f U W x∈ ∂U ∂V ∩ W = ∅ C1 Z (f 1U)◦ φt1(g1V)dx = Z (f 1U)◦ ψ ◦ ˜φt1(g1V)◦ ψJac(ψ)dx, ˜ φt = ψ◦ φt◦ ψ 1 g C1 W x0 ∈ ∂U ∩ ∂V U ∩ W = {(x, y) ∈ Rn 1× R|y > w(x)} w C1 Rn 1 V ∩ W = {(x, y) ∈ Rn 1× R|y > 0} y < 0 f g W Jt= Z Rn−1 Z + [wt(x)]+ f◦ φt1(x)g(x)dx, [wt(x)]+ wt(x) wt ∂U φt wt wt(x) = inf{z|(x, z) ∈ φt(V )} ∂t t=0wt(x) =−h∇w(x), p1(X(x, w(x)))i + p2(X(x, w(x))), p1, p2 Rn 1 R x7→ [x]+ Rn ∂t t=0+[wt]+(x) =−h∇w(x), p1( ˜X(x, w(x)))i + p2( ˜X(x, w(x))), ˜ X = ˜1V(X)X w(x) > 0 w(x) < 0 w 1({0}) ∇w = 0 ∂t=0+Jt= Z Rn−1 Z + [wt(x)]+−h∇f, Xidx + Z Rn−1 f (w(x))g(w(x)) (h∇w(x), p1(X(x, w(x)))i − p2(X(x, w(x)))) dx . Z Rn−1 f (w(x))g(w(x)) (h∇w(x), p1(X(x, w(x)))i − p2(X(x, w(x)))) dx = Z ∂UhX, nifg˜1V (X)dHn 1,
C1
LetU be a bounded Lipschitz domain, for any ǫ > 0 there exists V a C1
domain such that,S = U \ ¯V ∪ V \ ¯U , is a rectifiable open set verifying: µ(S) < ǫ
Hn 1(∂S) < ǫ.
g
Proof of the semi-differentiation lemma 1
Jt lim sup t 0 | 1 t(Jt(U )− J0(U ))| ≤ (µ(U) (f )|X| + CH n 1(∂U )|f| )|g| , C |Jt− J0 t | ≤ | Jt− Jt1 t | + | Jt1− J0 t |, Jt1 =RUf ◦ φt1g1Vdx |J 1 t − J0 t | ≤ Z U 1 |t||f ◦ φ 1 t − f||g| dx ≤ Z U (f )|φ 1 t (x)− x| |t| |g| dx . limt 0|φ −1 t (x) x t | = |X(x)| lim t 0| Jt1− J0 t | ≤ Z U (f )|g| |X| dx . |Jt− J 1 t t | ≤ 1 |t| Z Rn|1φt(U )− 1U||f| |g| dx. U t (φt) ≤ 2 |Xt| < M
µ(∆(U, φt(U ))≤ t max(2, M)nHn 1(∂U ),
∆(U, V ) = U\ V ∪ V \ U s0 > 0
∆(U, φs0(U ))⊂ Ψ([0, s0]×∂U)
Ψ (Ψ)≤ max(2, M) Hn([0, s0]× ∂U) = s0Hn 1(∂U )
Hn(Ψ([0, s0]× ∂U)) ≤ s0max(2, M )nHn 1(∂U ) .
1 |t| Z Rn|1φt(U )− 1U||f| |g| dx ≤ CH n 1(∂U ) |f| |g| , C = max(2, M )n U Jt(U ) = Rφt(U )f ◦ φt1g1Vdx C1 ∂t=0+Jt(U )=. Z U− < ∇f, X > g1V dµ + Z ∂U < X, n > f g˜1V(X)dHn 1. Uε C1 U Jt(U )− Jt(Uε) = Jt(U\ Uε)− Jt(Uε\ U) , lim sup t 0 1 |t||Jt(U )− Jt(Uε)− J0(U )− J0(Uε)| = lim sup t 0 1 |t||Jt(U\ Uε)− J0(U\ Uε)− (Jt(Uε\ U) − J0(Uε\ U))| ≤ lim sup t 0 1 |t||Jt(U\ Uε)− J0(U\ Uε)| + lim supt 0 1 |t||Jt(Uε\ U) − J0(Uε\ U)| ≤ ε( (f )|X| + C|f| )|g| . | lim sup t 0+ Jt(U )− J0(U ) t − ∂s=0+Js(Uε)| ≤ ε( (f )|X| + C|f| )|g| . ∂s=0+Js(U ) |∂s=0+Js(U )− ∂s=0+Js(Uε)| ≤ ε( (f )|X| + C|f| )|g| , | lim sup t 0+ Jt(U )− J0(U )
t − ∂s=0+J(U )| ≤ |∂t=0+J(U )− ∂t=0+J(Uε)|
+ ε( (f ) +|f| )|g| |X| ≤ 2ε( (f )|X| + C|f| )|g| . U V Vε µ( ¯V \ Vε) + µ( ¯Vε\ V ) < ε Hn 1(∂( ¯V \ Vε)) + Hn 1(∂( ¯Vε\ V )) < ε Jt(V )=. Z φt(U ) f◦ φt1g1V dx .
y = φt(x) V U X X Jt= Z φ−1t (V ) g◦ φtf 1UJac(φt)dx ,
limt 0|Jac(φt) 1t | = |div(X)| ≤ nLip(X)
V
(M )
Let(f, g) ∈ Im(M)2be two images,X be a Lipschitz vector field on Rn
andφtbe the associated flow. Let us define:
Jt=
Z
M
f ◦ φt1(x)g(x)dµ(x),
then the semi-differentiation ofJtis
∂t t=0+Jt=
Z
M−h∇f, Xigdx −
Z
(f+− f )˜ghνf, XidHn 1.
with˜gX(x) := limt 0+g(φt(x)) if the limit exists and if not, ˜gX(x) = 0.
Proof: f f = Pki=1f 1x Ui (Ui)i=1,...,k
f g ∂t t=0+Jt=− Z MhDf, Xi˜g, Df SBV g˜ µ a.e. ˜g = g Jf SBV SBV BV L2 C1 φt
BV g BV g Hn 1 (g+, g , ν) g g+ g ν Jg B BV Rn BVc(Rn) BV (Rn) B = BVc(Rn)∩ L (Rn) Rn Rn
IfX is a vector field on Rnandg a BV function, we define g
X bygX(x) =
g(x) if x /∈ Jg. OnJg, we define Hn 1a.e.
• gX(x) = g+(x) ifhν(x), X(x)i > 0,
• gX(x) = g (x) ifhν(x), X(x)i < 0,
• else hν(x), X(x)i = 0 and gX(x) = g
−(x)+g+(x) 2 .
Hence,gX lies inBV (Rn)× L1(Jg; Hn 1).
In order to make use of change of variables formulas, the action by a diffeo-morphismψ is given by (g◦ ψ)X ◦ ψ 1= gdψ(X ψ−1). X Hn 1 gX C1 S1 S2 Lipp BV SBV B X(t, x) : R× Rn 7→ Rn x t Xt(x) = X(t, x) Xt t (t, x) ∈ R × Rn 7→ φ t(x) ∈ Rn X X X0 X X
LetX be a Lipschitz time dependent vector field and φtbe its associated
BV functions with compact support. We define the functional Jt(f, g) = Z Rn f◦ φt1(x)g(x)dx , then we have ∂t=0+Jt= Z gX0(x)h−X0(x), ∂f (dx)i,
where∂f stands for the distributional derivative of f , which is a finite vector measure.
f g
L2 R
U|f ◦ φt1− g|2dµ
LetX be a Lipschitz time dependent vector field C1in space andφtbe its
associated flow,(f, g)∈ B × B with B = BVc(Rn)∩ L (Rn). We define the functional:
Jt(f, g) = Z Rn|f ◦ φ 1 t (x)− g(x)|2dx , then we have ∂t=0+Jt=−2 Z (f (x)− gX0(x)h∂f, −X0idx .
In this equation, the notationf stands for the precise representative of f i.e. at a
discon-tinuity pointf (x) = f+(x)+f2 −(x), which can be also written
∂t=0+Jt= 2 Z Rn (f (x)− g(x))h∇f, −X0idx + Z Jf (f+(x) + f (x)− 2gX0(x))hj(f), −X0idHn 1.
Let H be a locally Lipschitz function H : R2 7→ R and C1 in the first variable, (f, g) ∈ B × B be two functions (with B = BVc(Rn)∩ L (Rn) ), X be a
Lipschitz time dependent vector fieldC1 in space andφtits associated flow. We define
the functional Jt(f, g) = Z Rn H(f◦ φt1(x), g(x))dx , then we have ∂t=0+Jt= Z h∂1H(f (x), gX0(x)),−X0idx ,
where∂1H(f, gX0) is a part of the BV derivative of H(f, g), defined by
∂1H(f (x), l) = ∂xH(f (x), l)(∇f(x) + Dcf (x)) + jH(x)Hn 1
x
Jf,withjH(x) = (H (f+(x), l)− H (f (x), l)) νf(x) (the derivative of H w.r.t. to the first
variable is denoted by∂xH).
The notation ∂1H(f, gX0) is not the usual one and can be understood as
follows: letf be a BV function and for any l ∈ R, the function x → H(f(x), l) is a BV
function as the composition of a Lipschitz function with a BV function. Hence we denote its distributional derivative∂1H(f (x), l).
f g H H˜ H˜ Rn H = ˜H (f )× (g) H Rn L2 Lp p > 1 p = 1 Gα : (x, y) 7→ |x − y|α R2 α ∈]0, 1[ BV BV α < 1 H(x, y) = x12, f (t) = ( 1 n2 t∈ [2n+11 ,2n1 ], 0 J0(f, f ) = 0 n0 t Jt(f, f ) = Z 1 0 p |f(x − t) − f(x)|dx ≥ 2t X n n0 1 n, ∂t=0+Jt=Pn=1 1 n = +∞ SBV (U ) f g Jt(f, g)− J0(f, g) L f g L gX
X X Rn X C1 L X • • f g SBV BV X • • U U g gn Lipp(U ) Hd 1− (gn)X gX
X Hd 1 G U d(G, ∂U ) > 0 SBVG(U ) ={f ∈ SBV (U )|J(f) ⊂ G} SBV (U ) G SBVG,δ(U ) = {f ∈ SBV (U)||Dsf|(U \ G) < δ} SBV (U ) G Lipp(U ) SBVG(U ) Vect(SBVG(U ), G) SBV (U )
Letf be a function in SBVG(U ): there exists a sequence (un)n N⊂ Lipp(U )
such that
un→ f in BV and (un)X → fX pointwisely Hd 1− a.e.
Moreover, iff ∈ SBVG,δ(U ), then there exists a sequence (un)n N⊂ Lipp(U ) such that
lim sup
n ||f − un||BV ≤ 2δ and (un
)X → fX pointwisely Hd 1− a.e. outside Jf \ G.
Proof. f ∈ SBVG(U ) (V1, V2) U G ⊂ ∂V1∩ ∂V2 := Γ Rn Rn U ∇G ∇∂U d(G, ∂U ) > 0 ∂U f ∈ SBVG(U ) fi= fV i ∈ W 1,1(V i) i = 1, 2 W1,1 W1,1 W1,1(Ω) L1(∂Ω, Hd 1) W1,1 L1 V1 V2 Γ un Lipp(U ) ⊂ SBV (U) BV f BV V1 V2
W1,1 L1 f un Γ ||(u+n − un)− (f+− f )||1=|| |u(1)n − u(2)n | − |f(1)− f(2)| ||1 ≤ ||u(1)n − f(1)||1+||u(2)n − f(2)||1 (1) (2) u f Γ + − L1 W1,1 un f BV un Lipp(U ) BV Hd 1− g g Hd 1− (un)X gX Γ Γ un g (un)X gX Γ (un)X gX un g Γ Γ X un g L1 Γ (d− 1)− f ∈ SBVG(U ) U W1,1 BV Jf BV un f Jun⊂ G J(un)⊂ G Ω = V1 Ω = V2
Letf ∈ SBV (Ω) be a function with jump set denoted by Jf: then we may
obtain by convolution a sequence of smooth functionun on Ω, such that lim supn|f −
un|BV ≤ 2|Dsf| and un→ f pointwisely Hd 1−a.e. outside Jf.
Proof. ρn
limn||f ∗ ρn− f||L1 = 0
|∂(f ∗ ρn)− ∂f|(U) = |(Daf )∗ ρn+ (Dsf )∗ ρn− ∂f|(U)
BV
limn|∂(f ∗ ρn)|(U) = |∂f|(U) lim||(Daf )∗ ρn− Daf||L1 = 0
L1 lim||(Dsf )∗ ρ n||L1 =|Dsf|(U) lim sup n |∂(f ∗ ρn)− ∂f|(U) ≤ limn ||(D af )∗ ρ n− Daf||L1 + lim n ||(D sf )∗ ρ n||L1 +|Dsf|(U) = 2|Dsf|(U). un = f ∗ ρn Hd 1− f Jf f fX un (un)X
For any function f in SBV (U ) there exists a sequence (un)n ⊂ Lipp(U )
such thatun→ f in BV. Moreover, for any finite measure µ << Hd 1this sequence may
be chosen so that(un)X → fX pointwiselyµ−a.e.
Proof. f (Gi)i N δ > 0 N P RGi: i>N|Dsf| ≤ δ Gi ∩ Gj = ∅ (i, j) ∈ [0, N]2 i6= j G H G H˜ ˜ H G∩ ˜H =∅ Hd 1({H \ ˜H}) (ψ0, ψ1, . . . , ψN) i ≥ 1 ψi = 1 Gi ψi = 0 Gk k6= i, 1 ≤ k ≤ N f =PNi=0f ψi fi = f ψi SBV fi Gi |Dsfi|(U \ Gi) =RU Giψi|Dsf| = R U SNj=1Gjψi|Dsf| PN i=0 R U SNj=1Gjψi|Dsf| = PN i=0|Dsfi|(U\SNj=1Gj)≤ δ fi u(i)n ∈ Lipp(U ) un Lipp(U ) lim sup n ||un− f||BV ≤ 2δ, un→ f Hd 1− a.e. [ j>N Gj. un ||un − f||BV < 3δ δ f BV
σ− Hd 1 µ N Hd 1Sj>NGj µSj>NGj δ un d(un, g) < 2δ d d(f, g) = inf{ε : µ({|f − g| > ε}) < ε}
Fort≤ t0, ifg∈ L (U) and f ∈ BV (U), we have
|Jt(f, g)− J0(f, g)| ≤
Z
U|f ◦ (φ 1
t )− f| |g| dx ≤ Ct||g|| ||f||BV
for a constantC which only depends on the vector field X and on t0.
On a subsetA⊂ U, the same result is true for the functional Jt(A; f, g):
Z
A|f ◦ (φ 1
t )− f| |g| dx ≤ Ct||X||L∞(ACt)||g|| ||f||BV,
where Aεis{x ∈ U : d(x, A) < ε} and C is again a constant which only depends on
the vector fieldX and on t0.
Analogously, if on the contraryf ∈ L (U) and g ∈ BV (U), then we have |Jt(f, g)− J0(f, g)| ≤ Ct||f|| ||g||BV. Proof. g∈ L f ∈ C1 Z U|f ◦ (φ 1 t )− f| |g| dx ≤ Z t 0 Z U|∇f ◦ φs| |X ◦ φs| |g| dx ds = Z t 0 Z U|∇f| |X| |g ◦ (φ s) 1| |Jac(φs) 1| dx ds s (φs) 1 |J(φs) 1| ||X|| sup s [0,t]||J(φs ) 1|| ||g|| Z |∇f|. A⊂ U Z t 0 Z A|∇f ◦ φ s||X ◦ φs| |g| dx ds ≤ Z t 0 Z ACt|∇f||X| |g ◦ (φ s) 1||Jac(φs) 1| dx ds
ACt ||X|| ≤ C f ∈ BV (U) (fk)k f L1 ||fk||BV → ||f||BV Jt(f, g) = Z U f ◦ φt1gdx = Z U f g◦ φtJac(φt)dx = Z U f g◦ φtdx + Z U f g◦ φt(Jac(φt)− 1)dx. J0(f, g) X −X Ct||f|| ||g||BV ||f|| ||g||1||Jac(φt)− 1|| |Jac(φt)− 1| ≤ Ct ||g||1 ≤ ||g||BV. t
Ifψtdenotes the usual flow associated to a time dependent vector fieldX =
X(t, x) (that we suppose continuous in time and C1 in space) andφtthe flow associated
to the (constant in time) vector fieldX0= X(0,·), then we have
Z U f◦ (φt1)− f ◦ (ψt1) g dx = o(t). Proof. χt:= φt1◦ ψt Y χt ˙ χt= Y (t, χt) Y χt Z U f◦ (φt1)− f ◦ (ψt1) g dx = Z U f ◦ (φt1◦ ψt)− f g◦ ψt|Jac(ψt)| dx ≤ Z U Z t 0 |∇f ◦ χs||Y ◦ χs||g ◦ ψt| Jac(ψt ) dx
≤ t||g||L∞||Jacψ||L∞||Jacχ 1||L∞||f||BV||Y ||L∞(U [0,t]).
||Y ||L∞(U [0,t]) → 0 φt ψt X C1 χ∈ C1 Y ∈ C0 Y (0,·) = 0 limt 0||Y ||L∞(U [0,t])= 0 ˙ χt= ∂(φt1) ∂t +∇x(φ 1 t )· X(t, ψt).
t = 0 φt1 = φ t φ
φ0= id
Y (0,·) = ˙χt t=0=−X0+ Id· X(0, ·) = 0.
Ifg∈ Lipp(U ) and f ∈ SBV (U), then the semi-differentiation result is true.
Proof. fk ∈ Lipp(U ) f BV (U ) f = fk+ rk ||rk||BV → 0 Jt(f, g) = Jt(fk, g) + Jt(rk, g) lim sup t 0 Jt(f, g)− J0(f, g) t ≤ Z Uh∂fk ,−Xi gXdx + C||g|| ||rk||BV lim inf t 0 Jt(f, g)− J0(f, g) t ≥ Z Uh∂fk , Xi gXdx− C||g|| ||rk||BV. k Z Uh∂f k, Xi gXdx→ Z Uh∂f, Xi g Xdx. fk f XgX Jt(f, g)
Ifg∈ SBV (U) ∩ L (U) and f ∈ SBV (U) ∩ L (U), then the
differenti-ation result is true.
Proof. f g BV gk Lipp(U ) (Jt(f, g − gk)− J0(f, g − gk))/t R U∂f· X(gk)X R U∂f · XgX |∂f|− (gk)X gX
Ifg∈ SBV (U) ∩ L (U) and f ∈ SBV (U), then the differentiation result
Proof. g f BV fk = Hk◦ f Hk Hk(z) = z |z| ≤ k − 1, 0 ≤ Hk≤ 1; |Hk(z)| ≤ k ∨ |z|, Hk ∈ C1 fk → f BV (U ) f g SBV (U ) BV Lipp(U ) SBV (U ) BV (U ) f g
Suppose thatf ∈ BV (U) and g ∈ C (U), then the differentiation result is
true. Proof. Jt(g) = Z U f (x) g◦ φt(x) Jac(φt)dµ(x), ∂t=0Jt(g) = Z U fh∇g, Xi + g (∇ · X) dx, ∂t=0Jt(g) = Z U [f∇ · (gX)] dx, ∂t=0Jt(g) = − Z h∂f, gXi. g
Suppose thatf ∈ BV (U) and g ∈ C0(U ), then the differentiation result is
true. Proof. g gk C ||gk− g|| → 0 ∂f W1,1
f = fc+ fs
fc fs f
(f, g)
Suppose thatf = fc+ fsandg = gc+ gswithfc, gc ∈ BV (U) ∩ C0(U )
andfs, gs∈ SBV (U) ∩ L (U), then the differentiation result is true.
Proof. Jt(fc, gc) Jt(fc, gs) Jt(fs, gc) Jt(fs, gs) R Uh∂f, Xi gX, f g Jt(fs, gs) Jt(fs, gc) Jt(fc, gc) Jt(fc, gs) fc gs Jt(fc, gs) = Z U fc◦ φt1gsdx = Z U fc gs◦ φtJac(φt)dx = Z U fcgs◦ φtdx + Z U fcgs(Jac(φt)− 1)dx + Z U fc(gs◦ φt− gs)(Jac(φt)− 1)dx. X −X RUh∂gs, fci t R Ufcgs(∇ · X) dx Z U fc(gs◦ φt− gs)(Jac(φt)− 1)dx ≤Ct Z U|fc| |gs◦φt−gs| dx ≤ Ct 2 ||fc|| ||gs||BV, t = 0 d dtJt(fc, gs) = Z Uh∂g s, fci + Z U fcgs(∇ · X) . R h∂fc,−Xi(gs)X = R h∂fc,−Xigs (gs)X gs ∂fc (d− 1)− fc gsX BV
The same techniques of the last proofs could be used to prove a statement such as the following: if the differentiation result is true forf ∈ BV and g belonging to
a certain functional class S, then the same result stays true ifg belongs to the closure of S for the uniform convergence.
gX BV gX BV g BV SBV U R U g∈ BV (U) gX gX(x0) = limx x+ 0 g(x) X(x0) > 0, limx x− 0 g(x) X(x0) < 0, g(x0) X(x0) = 0.
Suppose thatf, g ∈ BV (U) (U ⊂ R): then the differentiation result is
true. Proof.
The set of functions onU which are uniform limits of BV functions is the
following vector spaceRL(U ):
Proof. xh x Ph|f(xh+1)− f(xh)| f (xh) RL(U ) f ∈ RL(U) ε > 0 x ∈ U Vε,x =]aε,x, bε,x[ x f ]aε,x, x[ ]x, bε,x[ ε U U f ε g ||f − g|| < ε g BV RL(U )
Suppose thatf ∈ BV (U) and g ∈ RL(U): then the differentiation result
is true. Proof. f BV (U ) g (gk)k BV RL(U ) g gX d • • BV
If for eachx ∈ U such that X(x) 6= 0, there exists a neighborhood V of x
such that the result is true forJt(V ; f, g), with
Jt(V ; f, g) =
Z
V
f◦ φt1(x)g(x)dx,
Zε = {y ∈ U; |X(y)| ≥ ε} Zc ε U Zε Zεc t≤ t0 Jt(Zεc; f, g)− J0(Zεc; f, g) t ≤C(ε + L||X|| 2t), L X Zc ε t||X|| |X| tL||X|| t t→ 0 lim sup t 0 Jt(U ; f, g)− J0(U ; f, g) t − Z Uh∂f, −Xig X ≤ Z Uh∂f, −Xig X − Z Zεh∂f, −Xig X + Cε. ε→ 0 RZε∂f· XgX → R X=0 ∂f· XgX =RU∂f· XgX z0 ∈ U X(z0)6= 0 C1 φt(x) = x + tν z0 R× Rn
IfX is a C1vector field on Rnandx
0∈ Rnis a point such thatX(x0)6= 0,
then there exist a neighborhoodV of x0, a neighborhoodU of 0, a vector v∈ Rn, and a
C1 diffeomorphismψ : U 7→ V such that for any x ∈ U we have
φt◦ ψ(x) = ψ(x + tv)
fort such that x + tv∈ U.
Proof. x0 = 0 X(0) = v = e1 (e1, . . . , en) Rn ψ(x1, . . . , xn) = φx1(0, x2, . . . , xn) , ψ φt C1 ψ 0 0 ψ C1 φt◦ ψ(x1, . . . , xn) = φt◦ φx1(0, x2, . . . , xn) = φt+x1(0, x2, . . . , xn) = ψ(x + te1) .
(X◦ φt◦ ψ) = ∇ψ(x + tv) · v .
Let(f, g)∈ BV (U) be two functions, X be a C1vector field on Rnand
φtbe the associated flow. Set
Jt=
Z
U
f◦ φt1(x)g(x)dx,
then the differentiation ofJtgives:
∂t t=0+Jt= Z < ∂f,−X > gX. Proof. Jt Jt= Z Rn f ◦ ψ(x − tv) g ◦ ψ Jac(ψ)dx. h = g◦ ψ Jac(ψ) BV Jac(ψ) H = ν ν x = (x , h) Jt− J0 = Z R n (f ◦ ψ(x − tν) − f ◦ ψ(x))h(x)dx , = Z H=Rn−1 Z R (f ◦ ψ(x + (h − t)ν) − f ◦ ψ(x + hν))h(x + hν)dx dh . dx δjt(x) = Z R f◦ ψ(x + (h − t)ν) − f ◦ ψ(x + hν) t h(x + hν)dh. |δjt(x)| ≤ Z R|∂(f ◦ ψ) ν|khk , Z H Z R|∂(f ◦ ψ) ν| ≤ |∂(f ◦ ψ)|(Rn) < +∞ δjt lim t 0+δjt(x) = Z R −∂(f ◦ ψ)ν(g◦ ψ)νJac(ψ) , (g ◦ ψ)ν Rn (g◦ ψ)ν Hd 1 ∂t=0+Jt = Z H Z R −∂(f ◦ ψ)ν(g◦ ψ)νJac(ψ) , = Z h∂(f ◦ ψ), −νi(g ◦ ψ)νJac(ψ) , = Z h∂f, −(∇ψ) ◦ ψ 1· νi(g ◦ ψ)ν ◦ ψ 1.
φ = v(g◦ ψ)νJac(ψ)
∂t=0+Jt=
Z
h∂f, −X0igX0.
If f ∈ BV (U) and ψ is a diffeomorphism of U, then, for any bounded
measurable functionφ : U → Rd, we have
Z Uh∂(f ◦ ψ), φi = Z Uh∂f, (Dψ)· φ Jacψ ◦ ψ 1i. Proof. f φ Z Uh∂(f ◦ ψ), φi = Z Uh(∇f) ◦ ψ, (Dψ) · φidx. x = ψ 1(y) f ∈ C1 f ∈ BV f fn ∂fn ⇀ ∂f φ∈ C0(U ; Rd) φ φ φn φ H(f, g) (x, y) = xy
IfH is a Lipschitz function in two variables, we denote by
Lip1(H) = inf{M ∈ R+|∀(x, x , y) ∈ R3|H(x, y) − H(x , y)| ≤ M|x − x |1}.
Exchanging the two variables, we define Lip2(H) as well.
i = 1, 2 i(H)≤ (H)
H f, g ∈ B
Ifg∈ L (Rn) with compact support and f ∈ B, then if 0 ≤ t ≤ t0
|Jt(f, g)− J0(f, g)| ≤ CtLip1(H)kfkBV,
Proof. g∈ L f ∈ C1 |Jt(f, g)− J0(f, g)| ≤ 1(H) Z U|f ◦ φ 1 t − f|dx ≤ 1(H)||X|| Z t 0 Z U|∇(f ◦ φ 1 s )|dx ds. |Jt(f, g)− J0(f, g)| ≤ c 1(H)||X|| sup s [0,t]||J(φs ) 1|| Z |∇f|. f ∈ BV
Remark that the Lipschitz constant Lip1(H) can be restricted on Im(f )×
Im(g) and that will be used in the next proof.
Proof of Theorem 15: H(x, y) = xy B fkgh H(x, y) = xkyh ∂t=0 +Jt= Z hD(fk)(gh)X0(x),−X0idx = Z hD(fk)(gX0(x))h,−X0idx = Z h∂1H(f (x), gX0(x)),−X0idx . ∂1H(f (x), l) x → H(x, l) f H(0, y) = 0 y ∈ R H H(x, y)− H(0, y) ∂t=0+Jt ∂xH(x, y) Pε (f )× (g) |∂xH− Pε| ,K < ε K (f )× (g) ⊂ K K R2 f g L H(0, y) = 0 Pε Qε ∂xQε = Pε 1,K(H − Qε) < ε 1,K K Jt(H)=. Z Rn H(f ◦ φt1(x), g(x))dx .
Qε Qε | lim sup t 0+ Jt(H)− J0(H) t − ∂s=0+Js(Qε)| ≤ CεkfkBV | lim sup x 0+ Jt(H)− J0(H) t − ∂t=0+Jt(H)| ≤| lim supt 0+ Jt(H)− J0(H) t − ∂s=0+Js(Qε)| +|∂s=0+Js(Qε)− ∂t=0+Jt(H)| . |∂s=0+Js(Qε)− ∂t=0+Jt(H)| . ∂t=0+Jt(H) H |∂t=0+Jt(Qε− H)| ≤ 1,K(Qε− H)|f|BV|X| ≤ ε|f|BV|X| , ε vt+ εut ∂εφ1◦ (φε1) 1 ε ∂εφε0,1 = dφε0,1( Z 1 0 Adφε t,0(ut) dt) . p ≥ 2 t∈ [0, 1] u∈ V Adφε t,1(u) = [dφ ε t,1]φε 1,t(u◦ φ ε 1,t) C1 φε t,1 C2 ε U ∂εφ1◦ (φε1) 1 1−
large deformation diffeomorphisms I η◦ I η (η, φ) E(η, φ) = D(Id, (η, φ))2+ 1 σ2kη ◦ I0◦ φ 1− I targk2L2, I0 Itarg σ D V S M Im(M ) := SBV (M )∩ L (M )
Let I0, Itarg ∈ Im(M ) be respectively the initial image and the target
image,H be a locally Lipschitz function H : R2 7→ R and C1 in the first variable and
V, S two 2−admissible spaces of vector fields respectively on M and R.
Let J: L2([0, 1], V × S) 7→ R be the functional defined by,
J(vt, st) = λ 2 Z 1 0 kv tk2Vdt + β 2 Z 1 0 ks tk2Sdt + Z M H(η0,1◦ I0◦ φ1,0(x), Itarg(x))dx,
withφ0,tandη0,tthe flows of the vector fieldsvt∈ L2([0, 1], V ) and st∈ L2([0, 1], S).
mini-mizer, there exists(pa, pb, pc)∈ L1(M, Rn)× L1(JI0, Rn)× L1(M, R) such that: βst= Z M pc(y)d[ηt,1]It 0(y)kS(I t 0(y), .)dµ(y), λvt= Z M kV(φ0,t(x), .)[dφ0,t]x1 (pa(x))dµ(x) + Z JI0 kV(φ0,t(x), .)[dφ0,t]x1 (pb(x))dHn 1(x), with: Itt′ = η0,t′ ◦ I0◦ φt,0,
andJI0 the jump set ofI0. More precisely for the(pa, pc) we show, we have the equation:
pa(x) +∇xI01 pc(x) = 0. β = 0 L2 pb = 0 pa = 0 (x0 = 0, x1, . . . , xn 1, xn = 1) ∈ [0, 1] ]0, 1[ R n− 1 (x1, . . . , xn 1) R Rn 1 S
Proof of the theorem: L2([0, 1], V × S)
λ 2 R1 0 kvtk2Vdt + β 2 R1 0 kstk2Sdt (vn, sn)n N L2([0, 1], V × S) (v, s) (vn, sn) ⇀ n (v, s) .
n≥ 0 mn=. sup x M, l [0,1] n d(φvn1,0(x), φv1,0(x)) + d(η0,1sn(l), η0,1s (l))o lim N mn= 0 . DHn=. Z M|H(η sn 0,1◦ I0◦ φvn1,0(x), I (x))− H(η0,1s ◦ I0◦ φv1,0(x), I (x))|dx ≤ Z M|H(η sn 0,1◦ I0◦ φvn1,0(x), I (x))− H(η0,1s ◦ I0◦ φvn1,0(x), I (x))| | {z } E1(x) dx + Z M|H(η s 0,1◦ I0◦ φvn1,0(x), I (x))− H(ηs0,1◦ I0◦ φv1,0(x), I (x))| | {z } E2(x) dx . x∈ M E1(x)≤ 1(H)d(η0,1sn ◦ I0◦ φvn1,0, η0,1s ◦ I0◦ φvn1,0(x))≤ 1(H)mn. Z M E1(x)dx≤ 1(H)µ(M )mn µ ˜ I0 µ(D)≤ ε D=. {x ∈ M : ˜ I0(x)6= I0(x)} An= φ. vn0,1(M \ D) ∩ φv0,1(M \ D) , x∈ An E2(x) =|H(η0,1s ◦ ˜I0◦ φvn1,0(x), I (x))− H(ηs0,1◦ ˜I0◦ φv1,0(x), I (x))| ≤ 1(H) (η0,1s ◦ ˜I0)d(φvn1,0(x), φv1,0(x)) ≤ 1(H) (η0,1s ◦ ˜I0)mn Z M E2(x)dx≤ 1(H) (ηs0,1◦ ˜I0)µ(An)mn+ 2|H| µ(M \ An)
|H| = sup(x,y) [0,1]2|H(x, y)| M\ An ⊂ φvn0,1(D)∪ φv0,1(D)∪ ∂M
K=. | (φv0,1)| ∨supn 0| (φvn0,1)| vn L2([0, 1], V ) K Z M E2(x)dx≤ 1(H) (η0,1s ◦ ˜I0)µ(M )mn+ 4K|H| ε. DHn≤ 1(H)µ(M )mn+ 1(H) (η0,1s ◦ ˜I0)µ(M )mn+ 4K|H| ε lim sup DHn≤ 4K|H| ε . ε DHn→ 0 J v s s∈ L2([0, T ], S) s˜ s ∂˜sη0,1(x) = Z 1 0 [dηt,1]η0,t(x)s˜t(η0,t(x))dt . I1 = η0,1s ◦ I0◦ φv1,0 Z 1 0 [β < st, ˜st>S+ Z M
∂1H(I1(y), Itarg(y))[dηt,1]It
1(y)< kS(I t 1(y), .), ˜st>S dµ(y)]dt = 0, x = φ0,1(y) βst+ Z M
Jac(φ0,1)(x)[∂1H](I01(x), Itarg◦ φ0,1(x))[dηt,1]It
0(x)kS(I t
0(x), .)dµ(x) = 0.
pc(x) =−Jac(φ0,1)(x)[∂1H](I01(x), Itarg◦ φ0,1(x))
pc H [0, 1]2 jH(f (x), l) = (H(f+(x), l)− H(f (x), l))νf(x) vt ut δu =−R01Adφt,1(u) dt λ Z 1 0 hvt , utidt + Z
h∂1H(I1, Itarg), δui
+ Z
JI1hjH
f
H [0, 1]2 y∈ [0, 1]
|H(I1(x)+, y)− H(I1 (x), y)| ≤ Lip1(H)|I1+(x)− I1 (x)|.
L (JI1,|DsI1|) ⊂ L2(JI1,|DsI1|)
∂uJ= ∂uJ1+
Z
JI1hj
H(I1(x), [Itarg] δu(x))− jH(I1(x), Itarg(x)), δuidHn 1(x),
∂uJ1
R×L2(JI1,|DsI1|) H0 ={(∂uJ1, δu)|u ∈ L2([0, 1], V )} a = (∂uJ1, δu) b = (1, jH(I1(.), [Itarg] δu(.))− jH(I1(.), Itarg(.)))
∂uJ=ha, biH ≥ 0 .
B ={(1, jH(., [Itarg] δu(.))− jH(., Itarg(.)))|u ∈ L2([0, 1], V )} ,
y∈ [0, 1] (1, ˜j)∈ Conv(B) ∂uJ1+ Z JI1h˜j, δuidH n 1= 0. ˜j ˜j ∈ Conv(B)
∆H(x) =hjH(I1(x), [Itarg] δu(x))− jH(I1(x), Itarg(x)), νf(x)i .
x ∈ JI1 m(x) := min(∆H(I1(x), Itarg+ (x)), ∆H(I1(x), Itarg(x)))
M (x) := max(∆H(I1(x), Itarg+ (x)), ∆H(I1(x), Itarg(x)))
h˜j(x), νf(x)i ∈ [m(x), M(x)]
H(x, .)
x∈ JI1 {t ∈ R|∆H(x, t) = h˜j(x), νf(x)i}
H H : (x, y)7→ (x, ∆H(x, y))
˜
Itarg(x)∈ [Itarg(x), Itarg+ (x)]
˜j(x) = jH(I1(x), ˜Itarg(x))− jH(I1(x), Itarg(x)) x∈ JI1 7→ ˜ Itarg(x) L λLvt= Z Adφt,1∂1F I1, ˜Itarg ,
Adφu = Jac(φ)dφ (u◦ φ) λvt= Z k(., φ1,t(x))dφ1,t∂1H(I1(x), Itarg(x)), λvt= Z
Jac(φ1)(x)k(., φ0,t(x))dφt,0∂1[H(I01(x), ˜Itarg◦ φ0,1(x))],
λvt=
Z
Jac(φ1)(x)k(., φ0,t(x))[dφ0,t]x1 ∂1[H(I0v(x), ˜Itarg ◦ φ0,1(x))].
∂1H(I0v, ˜Itarg◦ φ0,1)
λvt+
Z
M
Jac(φ1)(x)k(., φ0,t(x))[dφ0,t]x1 ∇1H(I01(x), ˜Itarg ◦ φ0,1(x))dx
+ Z JI1 0 Jac(φ1)(x)k(., φ0,t(x))[dφ0,t]x1 j(x)dHn 1(x), j(x) := jH(I01(x), ˜Itarg ◦ φ0,1(x))vI1 0(x) I 1 0 = η0,1 ◦ I0 η0,1 JI0 = JI1 0 JI01 JI0 pb(x) := Jac(φ0,1)(x)j(x)
pa= Jac(φ0,1)∇1H(I01, ˜Itarg◦ φ0,1) =−∇I01pc,
pa+∇I01pc H I0, Itarg, pc, pb pa, pb, pc η0,t = Id + Z t 0 su◦ ηudu, βst(.) = Z M pc(y)d[ηt,0]It 0(y)kS(I t 0(y)), .)dµ(y), φ0,t = Id + Z t 0 vu◦ φudu, λvt(.) = Z M kV(., φ0,t(x))[dφ0,t]x1 (pa(x))dµ(x) + Z JI0 kV(., φ0,t(x))[dφ0,t]x1 (pb(x))dHn 1(x)dt. v s
ForT sufficiently small, the system of equations with
(pa, pb, pc)∈ L1(M, Rn)× L1(JI0,|DsI0|) × L1(M, R)
has a unique solution if both RKHS (geometric and contrast) are2 admissible. (see 1.1.1
for the definition)
Proof: L2([0, T ], V × S) L2([0, T ], V× S) T > 0 Ξ : L2([0, T ], V × S) 7→ L2([0, T ], V × S) (v, s) 7→ (ξ(v), ξ(s)), ξ(v)t = Z M kV(., φ0,t(x))[dφ0,t]x1 (pa(x))dµ(x) + Z JI0 kV(., φ0,t(x))[dφ0,t]x1 (pb(x))dHn 1(x)dt, ξ(s)t = Z M pc(x)d[ηt,0]It 0(x)kS(I t 0(x)), .)dµ(x). L2([0, T ], V ) kξ(v)t− ξ(u)tk2 ≤ Mkv − ukL1[0,T ], kξ(v)t− ξ(u)tk2≤ M √ Tkv − ukL2[0,T ], kξ(v) − ξ(u)k2L2[0,T ] ≤ MT 3 2kv − ukL2[0,T ].
kξ(v)t− ξ(u)tk2 = < ξ(v)t, ξ(v)t− ξ(u)t>− < ξ(u)t, ξ(v)t− ξ(u)t>,
kξ(v)t− ξ(u)tk2 ≤ 2 max(| < ξ(v)t, ξ(v)t− ξ(u)t>|, | < ξ(u)t, ξ(v)t− ξ(u)t>|).
ν p(x) := pa(x)1x /JI0 + pb(x)1x JI0 < ξ(v)t, ξ(v)t− ξ(u)t> = Z Z [dφv0,t]x1 (p(x))[k(φv0,t(x), φv0,t(y))[dφv0,t]y1 (p(y)) − k(φv0,t(x), φu0,t(y))[dφu0,t]y1 (p(y))]dν(y)dν(x).
r L2([0, T ], V × S) B(r) kφu0,t− φv0,tk ≤ cVkv − ukL1[0,T ]exp(cVr √ T ), kdφu0,t− dφv0,tk ≤ C kv − ukL1[0,T ]exp(cVr √ T ), k[dφu0,t] 1− [dφv0,t] 1k ≤ kdφu0,t− dφv0,tk 1− kdφu 0,t− dφv0,tk , kdφu 0,t− dφv0,tk < 1 | < ξ(v)t, ξ(v)t− ξ(u)t>| ≤ Z Z [|dφv0,t]x1 (p(x))|
|[k(φv0,t(x), φv0,t(y))[dφv0,t]y1 (p(y))− k(φv0,t(x), φv0,t(y))[dφu0,t]y1 (p(y))]|
+|[k(φv0,t(x), φv0,t(y))[dφu0,t]y1 (p(y))− k(φv0,t(x), φu0,t(y))[dφu0,t]y1 (p(y))]|dν(x)dν(y).
B(r) K(r, T ) := max(C , cV)r √ T exp(cVr √ T ) kdφu0,t− Idk ≤ 2K(r, T ), k[dφu0,t] 1k ≤ 1 1− 2K(r, T ). Mk ∈ R B(r) max(cV, C )kv − ukL1[0,T ]exp(cVr √ T ) < 1 2, 2 max(C , cV)kv − ukL1[0,T ]exp(cVr √ T )Mk 1 1− 2K(r, T )(kpak + kpbk) 2, 4K(r, T ) < 12 4 max(C , cV)kv − ukL1[0,T ]exp(cVr √ T )Mk(kpak + kpbk)2. max(C , cV)kv − ukL1[0,T ]exp(cVr √ T )Mk[ 1 1− 2K(r, T )] 2(kp ak + kpbk)2, 4K(r, T ) < 12 [1 2K(r,T )1 ]2 ≤ 2 2 max(C , cV)kv − ukL1[0,T ]exp(cVr √ T )Mk(kpak + kpbk)2.
kξ(v)t− ξ(u)tk2 ≤ 10 max(C , cV)kv − ukL1[0,T ]exp(cVr
√
r > 0 T > 0 Ξ
Letp1 := (p1a, p1b, p1c) and p2 := (p2a, p2b, p2c) two initial momentums in L1(M, Rn)× L1(JI0,|DsI0|) × L1(M, R) with w1 := (v1, s1) and w2 := (v2, s2) the
associated solutions to the fixed point problem. Then there existsK > 0 and T > 0 such
that |vt1− v2t|V ≤ K(kpa1− p2ak + kp1b − p2bk), |s1t − s2t|S ≤ Kkp1c− p2ck, ift < T . Proof. Ξpi i = 1, 2 wi i = 1, 2 Ξpi(wi) = wi kw1− w2k = kΞp1(w1)− Ξp2(w2)k ≤ kΞp1(w1)− Ξp1(w2)k + kΞp1(w2)− Ξp2(w2)k ≤ (Ξp1)kw1− w2k + kΞp1 p2(w2)k. T kΞp1 p2(w2)tk2≤ 20cVkw2kL1[0,T ]exp(cVr √ T )Mkkp1− p2k2, M02 = 20cVkw2kL1[0,T ]exp(cVr √ T )Mk k[w1]t− [w2]tk ≤ M0 1− (Ξp1)kp 1− p2k. T > 0 T = +∞
Constant speed curves in vector fields spaces
If the kernels (contrast and geometric) are admissible, we have
• If a vector field stis a solution of equation thenkstk2 is constant.
Proof: kstk2 kstk2S= Z Z M2 p(y )d[ηt,1]Is t(y′)kS(I s t(y ), Its(y))p(y)d[ηt,1]Is
t(y)dµ(y )dµ(y).
∂t(d[ηt,1]It
0(y)) =−d[st]I0t(y)d[ηt,1]I0t(y).
d[η0,1]I0 = d[ηt,1 ◦ η0,t]I0 d[st]x= Z M d[ηt,1]It 0(y)pc(y)∂1kS(x, I t 0(y))dµ(y). st st= 1 β Z M pc(y)d[ηt,0]It 0(y)kS(I t 0(y)), .)dµ(y) st∈ L2([0, T ], S) ∂t(d[ηt,1]It 0(y)) ∂tkstk2 =− Z Z M2p(y )dst(I t 0(y ))d[ηt,1]It 0(y′)kS(I t 0(y )), I0t(y)))p(y)d[ηt,1]It
0(y)dµ(y )dµ(y)
+ Z Z M2 p(y )d[ηt,1]It 0(y′)∂1kS(I t 0(y )), I0t(y)))st(I0t(y ))p(y)d[ηt,1]It
0(y)dµ(y )dµ(y) = 0.
ν JI0 I0 pt(x) = (d[φ0,t]x) 1(pa(x)1x /JI0 + pb(x)1x JI0), kvtk2= Z Z pt(x)kV(φ0,t(x), φ0,t(y))pt(y)dν(x)dν(y). ∂tpt(x) =−dφ0,t(x)vtpt(x) ∂tkvtk2 =− Z Z Z pt(x)∂kV(φ0,t(x), φ0,t(z))pt(z)kV(φ0,t(x), φ0,t(y))pt(y)dν(x)dν(y)dν(z) + Z Z Z
pt(x)∂kV(φ0,t(x), φ0,t(y))pt(y)kV(φ0,t(x), φ0,t(z))pt(z)dν(x)dν(y)dν(z) = 0.
The solution of proposition 10 is defined for all time (with RKHS2−admissible).
Proof. vt st
[0, Tmax] Tmax < +∞
limt Tmaxφ0,t := φ0,Tmax x φ0,t(x)
limt Tmaxdφ0,t(x) φ0,Tmax ν λvTmax+s(.) = Z M
kV(., y)[dφTmax,Tmax+s]y1 [dφ0,Tmax]φ1
Tmax,0(y)(pa(φTmax,0(y)))d(φ0,Tmax∗ µ)(y).
p(y) = [dφ0,Tmax]φ1
Tmax,0(y)(pa(φTmax,0(y)))
vTmax+s s ∈ [0, S] S > 0 vt t∈ [0, Tmax+ S] It 0 I0 pc(.)d[ηt,0]It 0(.) I0 pt= Jac(φt,1)∆◦ φt,1,
∆ = I1− Itarg ∆ H1 I0 Rn σ v Rn z t It(x) = I0(x− tv(x)) + tσ2z(x) + o(t) . σ2z U ⊂ Rk L2(U )
Rd
L2(U ) L2
(C1curves inL2(U )) Let J be an interval in R, σ be a positive real number
and W be the Hilbert space V × L2(U ) with V an admissible space of vector fields.
The inner product onW is defined byh(v, z), (v , z )iW =hv, v iV + σ2hz, z iL2(U )for
(v, z)∈ W and (v , z ) ∈ W .
The curve I : t ∈ J 7→ It ∈ L2(U ) is said to be C1 if there exists a continuous map
w : J ∋ t 7→ (vt, wt)∈ W such that
1. I ∈ C(J, L2(U, R)) for the L2topology,
2. for anyu∈ Cc (U, R), t7→ hIt, ui is C1and∂thIt, ui = σ2hzt, ui2+hIt, div(uvt)i2.
V (C1(U, Rn),kk
1, )
It∈ SBV (U)
∂thIt, ui = σ2hzt, ui2+hDIt, uvti.
LetI ∈ Im(U ) := SBV (U ) ∩ L (U ) be an image and V an admissible
RKHS of vector fields onU , then the map for any u∈ Cc (U, R)
Tu: W 7→ R
(v, z) 7→ hDI, vui + σ2hz, ui2
is linear continuous. Hence ker Tu ⊂ W is a closed subspace and introducing EI :=
∩u C∞ c (U )ker Tu, we define TIIm(U ) :={I } × W /EI. EI W EI W pI j W | ¯w|TI = inf{|w|W|p(w) = ¯w} w ∈ W p(w) = ¯w | ¯w|TI =|w|W
The geodesic distance between two imagesj0 andj1 inL2(U ) is defined by, d(J0, J1) = inf{ Z 1 0 |pI (dj dt)|Wdt| I ∈ C 1 pw([0, 1], L2(U )), I0 = j0, I1 = j1},
withCpw1 ([0, 1], L2(U )) is the set of piecewise C1curves inL2(U ) (as naturally defined
fromC1curves).
C1
Assume that the space of vector fieldV is compactly embedded in C01(U, Rd),
and letj0, j1 ∈ L2(U ) be two functions then there exists a C1curveI of length d(j0, j1)
withI0 = j0andI1= j1. (v, z) z C1 It= I0◦ φt,0+ σ2R0tzs◦ φvt,sds , zt= z0◦ φt,0Jac(φvt,0) , (vt, zt) = pI(dIdt) . I LetI ∈ L2be a function,
1. The operator∇I : DI 7→ L2(U, R) is defined by
DI :={v ∈ V |∃C, s.t. ∀u ∈ Cc (U, R)|hI, div(uv)i2| ≤ C|u|2} ,
and for anyv∈ DI,∇I.v is the unique element in L2(U, R) such that,
h∇Iv, ui2=hI, div(uv)i2
for anyu∈ Cc (U, R).
2. the adjoint operatorDI : DI 7→ V is defined by
DI :={u ∈ L2(U, R)|∃C, s.t. ∀v ∈ DI|h∇Iv, ui2| ≤ C|v|V} ,
and for anyu∈ DI,DI.u (or∇Iu) is the unique element in ¯DI such that,
hDIu, viV =hu, ∇I.vi2
(vt, zt) = p(dIdt) zt ∈ DIt vt− ∇Izt ∈ DIt v F= Z 1 0 |vs| 2 Vds + 1 σ2 Z U |I1− I0◦ φv1,0|2 R1 0 Jac(φv1,s) 1ds . I0 H1 I0 I1 Im(U ) E
The geodesic energy F(v) is semi-differentiable with respect to the vector
fieldv∈ L2([0, 1], V ) and we have for F
ε= F(v + εh), d dε ε=0+ Fε = 2 Z 1 0 hv t+ KDItzt, htiVdt + Z 1 0 2hKD(I0◦φvt,0)(z Adφ1,tH t −zt), htiVdt. withH =−R01Adφt,1htdt =R01(dφt,1ht)◦ φ1,tdt.
In the Lemma 16, the notationztAdφ1,tH stands for the quantity defined in the previous Chapter 2[zt]Adφ1,tH.
In the equation 3.28, we distinguish between the linear part and the non-linear part (re-spectively the first and the second term on the right hand side of the equality).
Proof. (ht)∈ L2([0, 1], V ) 2 Z 1 0 hvs , hsids. qtv qvt := Z t 0 Jac(φvt,s) 1ds , It= I0◦ φvt,0+ σ2ztqvt, H =− Z 1 0 Adφt,1htdt =− Z 1 0 (dφt,1ht)◦ φ1,tdt. Uε := σ12 R U I1 I0 φv+εh1,0 2 q1v+εhds d dεε=0+Uε= Z 1 0 Z h2D(I0◦ φv1,0), z1HAdφt,1hti − Z U σ2|z1|2 d dε[Jac(φ v+εh 1,s ) 1]dx dt.
u = φ1,t(x) zt= z1◦ φvt,1Jac(φvt,1) Z h2D(I0◦ φv1,0), z1HAdφv t,1hti = Z h2dφt,1D(I0◦ φv1,0)◦ φvt,1, Jac(φvt,1)z1H ◦ φt,1hti, = Z h2D(I0◦ φvt,0), Jac(φvt,1)zH1 ◦ φt,1hti, = Z h2D(I0◦ φvt,0), z Adφ1,tH t hti, =h2KD(I0◦ φvt,0)z Adφ1,tH t , htiV. I0, I1 ∈ (U ) zt∈ (U ) Jac(φv+εht,s ) 1 ∂ ∂tJac(φ v 0,t) = div(vt)◦ φv0,tJac(φv0,t). vt∈ L2([0, 1], V ) Jac(φv+εht,s ) 1 = exp − Z s t div(vu+ εhu)◦ φv+εht,u du . d dεJac(φ v+εh t,s ) 1) = Jac(φvt,s) 1 Z t s
div(hu)◦ φt,u− Jac(φvt,u) 1h∇φv t,uJacφ
v
t,u 1, hu◦ φvt,uidu
. Z 1 0 Z U σ2|z1|2 d dε[Jac(φ v+εh 1,s ) 1]dx dt = Z 1 0 hq v u|zu|2, div(hu)i2− h|zu|2,h∇quv, huii2du, = Z 1 0 −h2∇q v u|zu|2+ D[zu2]∇qvu, hui2du. d dε ε=0+Uε= Z 1 0 2hKD(I0◦φvt,0)z Adφ1,tH t , htiV +hK[2∇qtv|zt|2+ D[zt2]∇qtv], htiVdt. hDf, gi f, g ∈ SBV (U) hDf, g0i 0 g f g0 g Dzu2 = 2zuDzu DIt= D[I0◦ φvt,0] + σ2Dztqtv+ σ2zt∇qtv,
d dε ε=0+Uε= Z 1 0 2hKDItzt, htiVdt + Z 1 0 2hKD(I0◦ φvt,0)(z Adφ1,tH t − zt), htiVdt. i0 ◦ φvt,0 ztAdφ1,tH− zt Z0 JI0 I0 vt+ KDItzt+ KDs[I0◦ φvt,0]Z0◦ φvt,0Jac(φt,0) = 0. KDs[I0◦ φvt,0]Z0◦ φvt,0Jac(φt,0) = KAdφ∗ t,0(Z0D sI 0), vt− ∇Izt∈ DIt zt∈ DIt L (U )⊂ DIt
The geodesic equations onSBV (U ) are given by, It= I0◦ φt,0+ σ2R0tzs◦ φvt,sds , zt= z0◦ φt,0Jac(φvt,0) , vt+ KDItzt+ KAdφ∗ t,0(Z0D sI 0) = 0. (I0, z0, Z0) V 3− SBV (U ) H1(U ) SBV (U )
H1 0 1]x,y[ [x, y] ∈ R B(z, δ) =]x, y[ R z0 < z1 ∈ R d(z0, z1) r0 > 0 x, y ∈ R d(x, y) < r0 x y z1 ∈ B(z0, r0) v0 ψv 0 0,1(z0) = z1 |v0|2L2 . =R01|v0(t)|2Vdt |v|2 L2 v ∈ L2([0, 1], V ) ( φv0,1(z0) = z1 φv0,1(z0+ δ) = ψv00,1(z0+ δ) , δ > 0 d(z1, φv0,1(z0 + δ)) e cV vL2δ≤ d(z1, φv 0,1(z0+ δ))≤ ecV vL2δ . M 1]z0,z0+δ[ (M, z0, δ) M 1]z1,ψ0,1v0(z0+δ)[ F d(z0, z1) M > 0 M M
A minimizing geodesic betweenI0 = (M0, z0, δ0) and I1 = (M1, z1, δ1)
is said to be of photometric type if Supp(I0◦φ1,0)∩Supp(I1) =∅. Otherwise the geodesic
is said to be of geometric type.
I0, I1
F u0
σ = 1
|v0|2L2 2δM2 I0M = M 1]z0,z0+δ[ IM 1 = M 1]z1,ψ0,1v0(z0+δ)[ |v|2L2 + 2M2δ(1− γ|v0|L2)≤ F(v) = |v|2L2 + Z U |I1|2+|I0◦ φv1,0|2 R1 0 Jac(φv1,s) 1ds , γ 2− 1− c|v|L2 ≤ |Jac(φs,t)| ≤ 1 + c|v|L2, c |v|L2 ≤ ε |v0|L2 ≤ ε |v|2 L2 ≤ F(v) |I0◦ φv1,0|2 α≥ 0 |v|2 L2 = α|v0|2L2 β > 0 2M2δ = β|v0|2L2 α + β(1− γ|v0|L2)≤ min(1, β) . α≤ βγ|v0|L2 γ|v0|L2 ≤ 1 N N ∈ N α≤ β N , α + N α(1− 1 N)≤ 1 , α≤ 1 N . N d(φv0,1(z0+ δ), z1)≥ d(z0, z1) 2 . δ d((φv0,1(z0), z1)) < d(z0, z1) 2 . M M
P M I0 I1 P 0 M0 = sup. {M | ]0, M[⊂ P } 0 < M0 <∞ M0 lim M M0F(M ) = F(M0) F(M ) F I0M IM 1 M0 M0 Mn → M0 L2([0, 1], V ) v 0 F(v0)≤ lim inf n F(Mn) = F(M0) , F(v0) = F(M0) v0 2M02δ≤ |v0|2L2. |v0|L2 → 0 |v0|2L2 0 M0 0 f =X i=1 mi1]2i,2i+1 i4[ mi M0 mi = 1 |f|BV = 2 X i=1 mi ≤ 2 X i=1 1 i2 <∞ . f i f
mi1]2i,2i+1 i4[
R
I0
H1
Letǫ > 0 be a positive real number. If f is a piecewise Lipschitz function
on [a, b]⊂ R then there exists a neighborhood of 0 such that the exponential map expf
defined
expf : L (Jf)× SBV (U) ∩ L 7→ SBV (U)
(Z, g)7→ f ◦ φ1,0+ σ2
Z 1 0
gs◦ φvt,sds
(withφt,sandgtcharacterized by system ) is injective.
f Proof. wa = (Z. a, ga) wb = (Z. b, gb) φa φb vta vtb ε = min{|jf(x)| ; x ∈ R} ε > 0 | expf(twa)− expf(twb)◦ φa0,t|BV ≥ 2εH0(S) S =. {x ∈ jf| hva(x)− vb(x), jf(x)i 6= 0} t f v0a− v0b ∈ Df , Za = Zb wa, wb expf(twa) = expf(twb) va− vb f | expf(twa)− expf(twb)◦ φa0,t|2L2 ≈ Ct|ga− gb|2L2, C t→ 0 inf{|jf(x)| ; x ∈ Jf} ≥ ε > 0
Q P GV Q GV × Q 7→ Q (φ, q)7→ φ.q x= (x1, . . . , xn) φ.x = (φ(x1), . . . , φ(xn)) V × Q 7→ Q (v, q)7→ v.q v∈ L2([0, 1], V ) d dt[φ0,t.q] = vt.[φ0,t.q] , φ0,1.q = q + Z 1 0 vt.[φ0,t.q] dt . 1 2 Z 1 0 |vt| 2 V dt + G(φ0,1.q0, qtarg) , G v0 1 2 R1 0 |vt|2V dt φv0,1.q0= φv00,1.q0 inf12R01|vt|2dt q(0)∈ M0 q(1)∈ M1 M0, M1 Q qi ∈ Mi TqiMi i = 0, 1 M0 M1 S1
P Q q∈ Q v ∈ V 1 2|v|2V ˙q = v.q H(p, q, v) = (p, v.q)P Q− 1 2hv, viV . v V 7→ R v7→ (p, v.q)P Q V p, q∈ P × Q V V V p⋄ q (p, v.q)P Q=−(p ⋄ q, v)V∗ V . (Lv, v)V∗ V v p⋄ q + Lv = 0 , v =−K(p ⋄ q) . H(p, q) = (p⋄ q, Kp ⋄ q)V∗ V −1 2hv, viV = 1 2(p⋄ q, Kp ⋄ q) . ( ˙p =−∂qH(p, q) ˙q = ∂pH(p, q) , p(0)⊥ Tq(0), p(1)⊥ Tq(1). ∂qH(p, q) q 7→ (p, v.q) q ∈ Q δq 7→ ∂q(p, v.q)(δq) ∂q(p, v.q)(δq) V ∂q(p, v.q)(δq) =−(∂q(p⋄ q)(δq), v)V∗ V .
H(p, q) ∂qH(p, q) = (∂q(p⋄ q), K(p ⋄ q)) . V V Q = L2(S 1, R2) Q = L2(M, Rd, µ) M µ M = Sn n M = Tn n GV Q V × Q GV v∈ V (v, q) 7→ v ◦ q V Z Mhv(q(s)), p(s)idµ(s) ≤ |p|L 2|v ◦ q|L2 ≤ |p|L2|v| p µ(M ) . v dv [∂qv◦ q].δq = [dv ◦ q](δq) ∈ L2(M, Rd, µ) c0 c1 S1 ci i = 0, 1 w S1 ψt w dt t=0d φt(x) = w(x) {s 7→ w(s)ci(s)| w ∈ X (S1)} ⊂ Tci. pi⊥ Tci w S1 hpi(s), ci(s)i = 0 a.e. s ∈ S1. t 7→ It= I◦ φ0,t1 h∇I, vi ∈ L2(U ) I H1 H1 L2 I t 7→ It I JI JI
I It (M ) (U1, . . . , Un) M Σ0 = ∪ni=1∂Ui p : n Y i=1 W1, (M ) 7→ (M ) (Ii)i=1,...,n 7→ I = n X i=1 Ii1U i. It Σ0 Σ0 Q U c(U ) Q = (Qi)0 i r= (Σ, (Ii)1 i r)∈ L1(Σ0, M )× W1, (M )r, U = (v, s)∈ V × S , ˙ Q = f (Q, U ) = (v◦ Q0, (−h∇Qi, vi + s(Qi)) 1 i r) , c(U ) = λ2|v|2 V +β2|s|2S. F = L (Σ, Rn)× L1(M, R)r F P ∈ F H(P, Q) = min U Z Σ0hP0 (x), ˙Q0(x)idµΣ0(x) + r X i=1 Z M Pi(x) ˙Qi(x)dµ(x)− c(U) . U (u, v) (δv, δs)
λhv, δvi = Z Σ0hP 0, δv◦ Q0idµ Σ0 − r X i=1 Z M Pih∇Qi, δvidµ , βhs, δsi = r X i=1 Z M Piδs(Ii)dµ . λv(.) = Z Σ0 k(Q0(x), .)P0(x)dµΣ0(x)− r X i=1 Z M kV(x, .)Pi(x)∇Qi(x)dµ(x) , βs(.) = r X i=1 Z M kS(Qi(x), .)Pi(x)dµ . H(P, Q) = 1 2λ[ Z Σ0 Z Σ0
P0(x)kV(Q0(x), Q0(y))P0(y)dµΣ0(x)dµΣ0(y)
+ Z
M
Z
M
Pj(y)∇Qj(y)kS(y, x)∇Qi(x)Pi(x)dµ(x)dµ(y)
− 2 X 1 i r Z M Z Σ0
P0(y)kV(Q0(y), x)Pi(x)∇Qi(x)dµ(x)dµΣ0(y)]
+ 1 2β X 1 i,j r Z M Z M
Pj(y)kS(Qj(y), Qi(x))Pi(x)dµ(x)dµ(y) .
∀ i ∈ [1, r] ˙ Q0t = ∂P0H(Pt, Qt)(.) , ˙ Qit= ∂PiH(Pt, Qt)∀ i ∈ [1, r] , ˙ Pt0 =−∂Q0H(Pt, Qt), ˙ Pi t =−∂QiH(Pt, Qt)∀ i ∈ [1, r] . Ψ ∈ C0 (M, R) u ∈ C0 (M, Rn) ∀ i ∈ [1, r] ∂P0H(P, Q)(u) = Z Σ0hv ◦ Q
0(y), u(y)idµ Σ0(y) ,
∂PiH(P, Q)(Ψ) =
Z
M
Ψ(y) s(Qi(y))− hv(y), ∇Qi(y)idµ(y) ,
∂Q0H(P, Q)(u) =
Z
Σ0h[dv]Q
0(y)(u(y)), P0(y)idµΣ0(y) ,
∂QiH(P, Q)(Ψ) =
Z
M
