O N THE R ANK -O NE T HEOREM FOR BV FUNCTIONS Annalisa Massaccesi - - PowerPoint PPT Presentation

▶
o n the r ank o ne t heorem for bv
SMART_READER_LITE
LIVE PREVIEW

O N THE R ANK -O NE T HEOREM FOR BV FUNCTIONS Annalisa Massaccesi - - PowerPoint PPT Presentation

Jun 17, 2023 278 likes •605 views

State-of-the-art Proof Generalization to Heisenberg groups O N THE R ANK -O NE T HEOREM FOR BV FUNCTIONS Annalisa Massaccesi Warwick, 13 - 07 - 2017 Joint works with Don Vittone Annalisa Massaccesi O N THE R ANK -O NE T HEOREM FOR BV

slide-1
SLIDE 1

State-of-the-art Proof Generalization to Heisenberg groups

ON THE RANK-ONE THEOREM FOR BV

FUNCTIONS

Annalisa Massaccesi Warwick, 13 - 07 - 2017

Joint works with Don Vittone

Annalisa Massaccesi ON THE RANK-ONE THEOREM FOR BV FUNCTIONS

slide-2
SLIDE 2

State-of-the-art Proof Generalization to Heisenberg groups

1

State-of-the-art

2

Proof

3

Generalization to Heisenberg groups

Annalisa Massaccesi ON THE RANK-ONE THEOREM FOR BV FUNCTIONS

slide-3
SLIDE 3

State-of-the-art Proof Generalization to Heisenberg groups

Section 1 State-of-the-art

Annalisa Massaccesi ON THE RANK-ONE THEOREM FOR BV FUNCTIONS

slide-4
SLIDE 4

State-of-the-art Proof Generalization to Heisenberg groups

THE RANK-ONE THEOREM

  • Definition. An integrable function u : Ω ⊂ Rn → Rm has (locally)

bounded variation if Du = (D1u, . . . , Dnu) is a Radon measure. We decompose Du = Dau + Dsu = ∇uL n + σu|Dsu| Theorem (Alberti, 1993). If u ∈ BV(Ω; Rm), then rank(σu(x)) = 1 for |Dsu|-a.e. x ∈ Ω . Applications to (vector) variational problems and systems of PDEs.

Annalisa Massaccesi ON THE RANK-ONE THEOREM FOR BV FUNCTIONS

slide-5
SLIDE 5

State-of-the-art Proof Generalization to Heisenberg groups

THE RANK-ONE THEOREM

  • Definition. An integrable function u : Ω ⊂ Rn → Rm has (locally)

bounded variation if Du = (D1u, . . . , Dnu) is a Radon measure. We decompose Du = Dau + Dsu = ∇uL n + σu|Dsu| Theorem (Alberti, 1993). If u ∈ BV(Ω; Rm), then rank(σu(x)) = 1 for |Dsu|-a.e. x ∈ Ω . Applications to (vector) variational problems and systems of PDEs.

Annalisa Massaccesi ON THE RANK-ONE THEOREM FOR BV FUNCTIONS

slide-6
SLIDE 6

State-of-the-art Proof Generalization to Heisenberg groups

THE RANK-ONE THEOREM

  • Definition. An integrable function u : Ω ⊂ Rn → Rm has (locally)

bounded variation if Du = (D1u, . . . , Dnu) is a Radon measure. We decompose Du = Dau + Dsu = ∇uL n + σu|Dsu| Theorem (Alberti, 1993). If u ∈ BV(Ω; Rm), then rank(σu(x)) = 1 for |Dsu|-a.e. x ∈ Ω . Applications to (vector) variational problems and systems of PDEs.

Annalisa Massaccesi ON THE RANK-ONE THEOREM FOR BV FUNCTIONS

slide-7
SLIDE 7

State-of-the-art Proof Generalization to Heisenberg groups

LITERATURE AND REMARKS

The rank-one property for BV functions was (1988) conjectured by E. De Giorgi and L. Ambrosio; (1993) proved by G. Alberti (see also Alberti-Csörnyei-Preiss); (2016) proved G. De Philippis and F. Rindler (consequence of a more general statement on A -free measures); (2016) proved by A. M. and D. Vittone.

  • Remark. For SBV functions, i.e., Dsu = (u+ − u−)νRH n−1R with R

rectifiable set, the theorem is way easier.

Annalisa Massaccesi ON THE RANK-ONE THEOREM FOR BV FUNCTIONS

slide-8
SLIDE 8

State-of-the-art Proof Generalization to Heisenberg groups

LITERATURE AND REMARKS

The rank-one property for BV functions was (1988) conjectured by E. De Giorgi and L. Ambrosio; (1993) proved by G. Alberti (see also Alberti-Csörnyei-Preiss); (2016) proved G. De Philippis and F. Rindler (consequence of a more general statement on A -free measures); (2016) proved by A. M. and D. Vittone.

  • Remark. For SBV functions, i.e., Dsu = (u+ − u−)νRH n−1R with R

rectifiable set, the theorem is way easier.

Annalisa Massaccesi ON THE RANK-ONE THEOREM FOR BV FUNCTIONS

slide-9
SLIDE 9

State-of-the-art Proof Generalization to Heisenberg groups

Section 2 Proof

Annalisa Massaccesi ON THE RANK-ONE THEOREM FOR BV FUNCTIONS

slide-10
SLIDE 10

State-of-the-art Proof Generalization to Heisenberg groups

USEFUL FACTS

u : Ω → R has bounded variation if and only if the subgraph Eu := {(x, t) ∈ Ω × R : t < u(x)} has finite perimeter in Ω × R. This is equivalent to χEu ∈ BV(Ω × R). If u ∈ BV(Ω), then DχEu is rectifiable, more precisely DχEu = νEuH n∂∗Eu , where ∂∗Eu is the (rectifiable) reduced boundary and νEu is the inner normal.

See Miranda - several papers in the 60s; Giaquinta, Modica, Souˇ cek - Cartesian currents in the calculus of variations; Federer - Geometric Measure Theory.

Annalisa Massaccesi ON THE RANK-ONE THEOREM FOR BV FUNCTIONS

slide-11
SLIDE 11

State-of-the-art Proof Generalization to Heisenberg groups

USEFUL FACTS

u : Ω → R has bounded variation if and only if the subgraph Eu := {(x, t) ∈ Ω × R : t < u(x)} has finite perimeter in Ω × R. This is equivalent to χEu ∈ BV(Ω × R). If u ∈ BV(Ω), then DχEu is rectifiable, more precisely DχEu = νEuH n∂∗Eu , where ∂∗Eu is the (rectifiable) reduced boundary and νEu is the inner normal.

See Miranda - several papers in the 60s; Giaquinta, Modica, Souˇ cek - Cartesian currents in the calculus of variations; Federer - Geometric Measure Theory.

Annalisa Massaccesi ON THE RANK-ONE THEOREM FOR BV FUNCTIONS

slide-12
SLIDE 12

State-of-the-art Proof Generalization to Heisenberg groups

USEFUL FACTS

u : Ω → R has bounded variation if and only if the subgraph Eu := {(x, t) ∈ Ω × R : t < u(x)} has finite perimeter in Ω × R. This is equivalent to χEu ∈ BV(Ω × R). If u ∈ BV(Ω), then DχEu is rectifiable, more precisely DχEu = νEuH n∂∗Eu , where ∂∗Eu is the (rectifiable) reduced boundary and νEu is the inner normal.

See Miranda - several papers in the 60s; Giaquinta, Modica, Souˇ cek - Cartesian currents in the calculus of variations; Federer - Geometric Measure Theory.

Annalisa Massaccesi ON THE RANK-ONE THEOREM FOR BV FUNCTIONS

slide-13
SLIDE 13

State-of-the-art Proof Generalization to Heisenberg groups

USEFUL REMARK

in Heisenberg

  • Remark. Define

V := {p ∈ ∂∗Eu : (νEu(p))n+1 = 0} . It is possible to see that νEu(x, t) = σu(x) H n-a.e. (x, t) ∈ V For instance: u(x) =

  • Cantor staircase

x ∈ [0, 1] x ∈ ]1, 2] u(x)

π

x Eu

νEu

1 and Du = H

log 2 log 3 C 1 3 − δ1 = π♯(νEuH 1V). Annalisa Massaccesi ON THE RANK-ONE THEOREM FOR BV FUNCTIONS

slide-14
SLIDE 14

State-of-the-art Proof Generalization to Heisenberg groups

USEFUL REMARK

in Heisenberg

  • Remark. Define

V := {p ∈ ∂∗Eu : (νEu(p))n+1 = 0} . It is possible to see that (Dsu, 0) = π♯ (νEuH nV) . For instance: u(x) =

  • Cantor staircase

x ∈ [0, 1] x ∈ ]1, 2] u(x)

π

x Eu

νEu

1 and Du = H

log 2 log 3 C 1 3 − δ1 = π♯(νEuH 1V). Annalisa Massaccesi ON THE RANK-ONE THEOREM FOR BV FUNCTIONS

slide-15
SLIDE 15

State-of-the-art Proof Generalization to Heisenberg groups

PROOF OF THE RANK-ONE THM.

W.l.o.g. m = 2 and u = (u1, u2). The polar vector σu is multiple of (σ1, σ2). If σi(x) = 0 for some i, there is nothing to prove. Otherwise x = π(p1) = π(p2) for some pi ∈ Vi and σi|Dsui| = π♯

  • νEuiH nVi
  • i = 1, 2

with Vi ⊂ Ni ∪ ∞

j=1 Σj i, where Σj i are surfaces.Then

(σ1(x), 0) = νEu1(p1) = ±νΣ

j1 1 (p1)

(σ2(x), 0) = νEu2(p2) = ±νΣ

j2 2 (p2) Annalisa Massaccesi ON THE RANK-ONE THEOREM FOR BV FUNCTIONS

slide-16
SLIDE 16

State-of-the-art Proof Generalization to Heisenberg groups

PROOF OF THE RANK-ONE THM.

W.l.o.g. m = 2 and u = (u1, u2). The polar vector σu is multiple of (σ1, σ2). If σi(x) = 0 for some i, there is nothing to prove. Otherwise x = π(p1) = π(p2) for some pi ∈ Vi and σi|Dsui| = π♯

  • νEuiH nVi
  • i = 1, 2

with Vi ⊂ Ni ∪ ∞

j=1 Σj i, where Σj i are surfaces.Then

(σ1(x), 0) = νEu1(p1) = ±νΣ

j1 1 (p1)

(σ2(x), 0) = νEu2(p2) = ±νΣ

j2 2 (p2) Annalisa Massaccesi ON THE RANK-ONE THEOREM FOR BV FUNCTIONS

slide-17
SLIDE 17

State-of-the-art Proof Generalization to Heisenberg groups

PROOF OF THE RANK-ONE THM.

W.l.o.g. m = 2 and u = (u1, u2). The polar vector σu is multiple of (σ1, σ2). If σi(x) = 0 for some i, there is nothing to prove. Otherwise x = π(p1) = π(p2) for some pi ∈ Vi and σi|Dsui| = π♯

  • νEuiH nVi
  • i = 1, 2

with Vi ⊂ Ni ∪ ∞

j=1 Σj i, where Σj i are surfaces.Then

(σ1(x), 0) = νEu1(p1) = ±νΣ

j1 1 (p1)

(σ2(x), 0) = νEu2(p2) = ±νΣ

j2 2 (p2) Annalisa Massaccesi ON THE RANK-ONE THEOREM FOR BV FUNCTIONS

slide-18
SLIDE 18

State-of-the-art Proof Generalization to Heisenberg groups

PROOF OF THE RANK-ONE THM.

W.l.o.g. m = 2 and u = (u1, u2). The polar vector σu is multiple of (σ1, σ2). If σi(x) = 0 for some i, there is nothing to prove. Otherwise x = π(p1) = π(p2) for some pi ∈ Vi and σi|Dsui| = π♯

  • νEuiH nVi
  • i = 1, 2

with Vi ⊂ Ni ∪ ∞

j=1 Σj i, where Σj i are surfaces.Then

(σ1(x), 0) = νEu1(p1) = ±νΣ

j1 1 (p1)

= ← − H n-a.e. by the following lemma (σ2(x), 0) = νEu2(p2) = ±νΣ

j2 2 (p2) Annalisa Massaccesi ON THE RANK-ONE THEOREM FOR BV FUNCTIONS

slide-19
SLIDE 19

State-of-the-art Proof Generalization to Heisenberg groups

GEOMETRIC LEMMA

in Heisenberg

  • Lemma. Let Σ1 and Σ2 be C1-hypersurfaces in Rn+1, set

T :=   p ∈ Σ1 : ∃ q ∈ Σ2 with

  • π(p) = π(q)
  • (νΣ1(p))n+1 = (νΣ2(q))n+1 = 0
  • νΣ1(p) = νΣ2(q)

   . Then H n(T) = 0.

Σ1 x Rn p q R

νΣ2(q)

Σ2

νΣ1(p) π

Annalisa Massaccesi ON THE RANK-ONE THEOREM FOR BV FUNCTIONS

slide-20
SLIDE 20

State-of-the-art Proof Generalization to Heisenberg groups

PROOF OF THE GEOMETRIC LEMMA

  • Proof. Define the C1-hypersurfaces ˜

Σi := {(p, t1, t2) : (p, ti) ∈ Σi} ⊂ Rn+2, with R2n+2 = Rn × R × R, thus ν˜

Σ1(p, t1, t2) = ((νΣ1(p, t1))1, . . . , (νΣ1(p, t1))n+1, 0)

ν˜

Σ2(p, t1, t2) = ((νΣ2(p, t2))1, . . . , (νΣ1(p, t2))n, 0, (νΣ2(p, t2))n+1) .

Put R :=

  • (p, t1, t2) ∈ ˜

Σ1 ∩ ˜ Σ2 :

  • ν˜

Σ1(p, t1, t2)

  • n+1 =
  • ν˜

Σ2(p, t1, t2)

  • n+2 = 0

ν˜

Σ1(p, t1, t2) = ±ν˜ Σ2(p, t1, t2)

  • We can notice that

by transversality, there exists ˜ R n-dimensional C1 surface with R ⊂ ˜ R; π(R) = T by construction.

Annalisa Massaccesi ON THE RANK-ONE THEOREM FOR BV FUNCTIONS

slide-21
SLIDE 21

State-of-the-art Proof Generalization to Heisenberg groups

PROOF OF THE GEOMETRIC LEMMA

  • Proof. Define the C1-hypersurfaces ˜

Σi := {(p, t1, t2) : (p, ti) ∈ Σi} ⊂ Rn+2, with R2n+2 = Rn × R × R, thus ν˜

Σ1(p, t1, t2) = ((νΣ1(p, t1))1, . . . , (νΣ1(p, t1))n+1, 0)

ν˜

Σ2(p, t1, t2) = ((νΣ2(p, t2))1, . . . , (νΣ1(p, t2))n, 0, (νΣ2(p, t2))n+1) .

Put R :=

  • (p, t1, t2) ∈ ˜

Σ1 ∩ ˜ Σ2 :

  • ν˜

Σ1(p, t1, t2)

  • n+1 =
  • ν˜

Σ2(p, t1, t2)

  • n+2 = 0

ν˜

Σ1(p, t1, t2) = ±ν˜ Σ2(p, t1, t2)

  • We can notice that

by transversality, there exists ˜ R n-dimensional C1 surface with R ⊂ ˜ R; π(R) = T by construction.

Annalisa Massaccesi ON THE RANK-ONE THEOREM FOR BV FUNCTIONS

slide-22
SLIDE 22

State-of-the-art Proof Generalization to Heisenberg groups

PROOF OF THE GEOMETRIC LEMMA

  • Proof. Define the C1-hypersurfaces ˜

Σi := {(p, t1, t2) : (p, ti) ∈ Σi} ⊂ Rn+2, with R2n+2 = Rn × R × R, thus ν˜

Σ1(p, t1, t2) = ((νΣ1(p, t1))1, . . . , (νΣ1(p, t1))n+1, 0)

ν˜

Σ2(p, t1, t2) = ((νΣ2(p, t2))1, . . . , (νΣ1(p, t2))n, 0, (νΣ2(p, t2))n+1) .

Put R :=

  • (p, t1, t2) ∈ ˜

Σ1 ∩ ˜ Σ2 :

  • ν˜

Σ1(p, t1, t2)

  • n+1 =
  • ν˜

Σ2(p, t1, t2)

  • n+2 = 0

ν˜

Σ1(p, t1, t2) = ±ν˜ Σ2(p, t1, t2)

  • We can notice that

by transversality, there exists ˜ R n-dimensional C1 surface with R ⊂ ˜ R; π(R) = T by construction.

Annalisa Massaccesi ON THE RANK-ONE THEOREM FOR BV FUNCTIONS

slide-23
SLIDE 23

State-of-the-art Proof Generalization to Heisenberg groups

PROOF OF THE GEOMETRIC LEMMA

Finally the Area Formula gives H n(T) = H n(π(R)) =

  • R

| det(dπ)TR| dH n = 0 because dπP : TPR → Tπ(P)Σ1 is not surjective if P ∈ R (because TPR ∋ (0, . . . , 0, 1)

dπP

→ (0, . . . , 0)).

Annalisa Massaccesi ON THE RANK-ONE THEOREM FOR BV FUNCTIONS

slide-24
SLIDE 24

State-of-the-art Proof Generalization to Heisenberg groups

Section 3 Generalization to Heisenberg groups

Annalisa Massaccesi ON THE RANK-ONE THEOREM FOR BV FUNCTIONS

slide-25
SLIDE 25

State-of-the-art Proof Generalization to Heisenberg groups

RANK-ONE THEOREM IN HEISENBERG GROUPS

Theorem (Don, M., Vittone). If Ω ⊂ Hn, with n ≥ 2, and u ∈ BVH(Ω; Rm), then rank(σu) = 1 |Dsu| − a.e. Vocabulary: in the Heisenberg groups Hn = R2n+1 there is a (non-involutive) distribution of vectorfields called “horizontal”, more specifically Xi :=

  • 1, 0, −yi

2

  • Yi :=
  • 0, 1, xi

2

  • i = 1, . . . , n .

we only care about the horizontal derivatives, so u ∈ BVH(Ω) if and

  • nly if DHu = (X1u, Y1u, . . . , Xnu, Ynu) is a Radon measure.

Annalisa Massaccesi ON THE RANK-ONE THEOREM FOR BV FUNCTIONS

slide-26
SLIDE 26

State-of-the-art Proof Generalization to Heisenberg groups

RANK-ONE THEOREM IN HEISENBERG GROUPS

Theorem (Don, M., Vittone). If Ω ⊂ Hn, with n ≥ 2, and u ∈ BVH(Ω; Rm), then rank(σu) = 1 |Dsu| − a.e. Vocabulary: in the Heisenberg groups Hn = R2n+1 there is a (non-involutive) distribution of vectorfields called “horizontal”, more specifically Xi :=

  • 1, 0, −yi

2

  • Yi :=
  • 0, 1, xi

2

  • i = 1, . . . , n .

we only care about the horizontal derivatives, so u ∈ BVH(Ω) if and

  • nly if DHu = (X1u, Y1u, . . . , Xnu, Ynu) is a Radon measure.

Annalisa Massaccesi ON THE RANK-ONE THEOREM FOR BV FUNCTIONS

slide-27
SLIDE 27

State-of-the-art Proof Generalization to Heisenberg groups

RANK-ONE THEOREM IN HEISENBERG GROUPS

Theorem (Don, M., Vittone). If Ω ⊂ Hn, with n ≥ 2, and u ∈ BVH(Ω; Rm), then rank(σu) = 1 |Dsu| − a.e. Vocabulary: in the Heisenberg groups Hn = R2n+1 there is a (non-involutive) distribution of vectorfields called “horizontal”, more specifically Xi :=

  • 1, 0, −yi

2

  • Yi :=
  • 0, 1, xi

2

  • i = 1, . . . , n .

we only care about the horizontal derivatives, so u ∈ BVH(Ω) if and

  • nly if DHu = (X1u, Y1u, . . . , Xnu, Ynu) is a Radon measure.

Annalisa Massaccesi ON THE RANK-ONE THEOREM FOR BV FUNCTIONS

slide-28
SLIDE 28

State-of-the-art Proof Generalization to Heisenberg groups

PROOF IN HEISENBERG GROUPS

The “useful facts” are still true (with more work): if u ∈ BVH(Ω; Rm) then the subgraph Eu := {(p, t) ∈ Ω × R : t < u(p)} has finite perimeter (see Don’s Ph.D. thesis); if Q = 2n + 2 is the homogeneous dimension, ˜ νEu :=

  • (νEu)1 , . . . , (νEu)2n
  • and V =
  • p ∈ ∂∗

HEu : (νEu)2n+1 = 0

  • ,

then (DHu, −L 2n+1Ω) = π♯(νEuS Q∂∗

HE)

and Ds

Hu = π♯(˜

νEuS QV) . Now we can run the same proof as above, but...

Annalisa Massaccesi ON THE RANK-ONE THEOREM FOR BV FUNCTIONS

slide-29
SLIDE 29

State-of-the-art Proof Generalization to Heisenberg groups

GEOMETRIC LEMMA IN HEISENBERG GROUPS

Take ˜ Σi := {(p, t1, t2) ∈ Hn × R × R : (p, ti) ∈ Σi, ti ∈ R} (which are hypersurfaces of class C1

H), i = 1, 2, and put

˜ R :=

  • (p, t1, t2) ∈ ˜

Σ1 ∩ ˜ Σ2 :

  • ν˜

Σ1(p, t1, t2)

  • 2n+1 =
  • ν˜

Σ2(p, t1, t2)

  • 2n+2 = 0

ν˜

Σ1(p, t1, t2) = ±ν˜ Σ2(p, t1, t2)

  • Lemma. The following facts hold:

π(˜ R) = T; H Q(˜ R ∩ Bρ) < ∞ for every ball centered at some point of ˜ R, with radius ρ carefully chosen; H Q(π(˜ R)) = 0. ˜ R is locally an intrinsic Lipschitz graph (Franchi, Serapioni, Serra Cassano).

Annalisa Massaccesi ON THE RANK-ONE THEOREM FOR BV FUNCTIONS

slide-30
SLIDE 30

State-of-the-art Proof Generalization to Heisenberg groups

GEOMETRIC LEMMA IN HEISENBERG GROUPS

Take ˜ Σi := {(p, t1, t2) ∈ Hn × R × R : (p, ti) ∈ Σi, ti ∈ R} (which are hypersurfaces of class C1

H), i = 1, 2, and put

˜ R :=

  • (p, t1, t2) ∈ ˜

Σ1 ∩ ˜ Σ2 :

  • ν˜

Σ1(p, t1, t2)

  • 2n+1 =
  • ν˜

Σ2(p, t1, t2)

  • 2n+2 = 0

ν˜

Σ1(p, t1, t2) = ±ν˜ Σ2(p, t1, t2)

  • Lemma. The following facts hold:

π(˜ R) = T; H Q(˜ R ∩ Bρ) < ∞ for every ball centered at some point of ˜ R, with radius ρ carefully chosen; H Q(π(˜ R)) = 0. ˜ R is locally an intrinsic Lipschitz graph (Franchi, Serapioni, Serra Cassano).

Annalisa Massaccesi ON THE RANK-ONE THEOREM FOR BV FUNCTIONS

slide-31
SLIDE 31

State-of-the-art Proof Generalization to Heisenberg groups

GEOMETRIC LEMMA IN HEISENBERG GROUPS

Take ˜ Σi := {(p, t1, t2) ∈ Hn × R × R : (p, ti) ∈ Σi, ti ∈ R} (which are hypersurfaces of class C1

H), i = 1, 2, and put

˜ R :=

  • (p, t1, t2) ∈ ˜

Σ1 ∩ ˜ Σ2 :

  • ν˜

Σ1(p, t1, t2)

  • 2n+1 =
  • ν˜

Σ2(p, t1, t2)

  • 2n+2 = 0

ν˜

Σ1(p, t1, t2) = ±ν˜ Σ2(p, t1, t2)

  • Lemma. The following facts hold:

π(˜ R) = T; H Q(˜ R ∩ Bρ) < ∞ for every ball centered at some point of ˜ R, with radius ρ carefully chosen; H Q(π(˜ R)) = 0. ˜ R is locally an intrinsic Lipschitz graph (Franchi, Serapioni, Serra Cassano).

Annalisa Massaccesi ON THE RANK-ONE THEOREM FOR BV FUNCTIONS

slide-32
SLIDE 32

State-of-the-art Proof Generalization to Heisenberg groups

THANKS FOR YOUR ATTENTION!

Annalisa Massaccesi ON THE RANK-ONE THEOREM FOR BV FUNCTIONS