Growing Sand Piles on a table with side walls. Luigi De Pascale - - PowerPoint PPT Presentation

Introduction Optimal transport at a.e. t Discrete source: f = k

j=1 cj δyj

More general source: f ∈ M+(Ω)

Growing Sand Piles on a table with side walls.

Luigi De Pascale (Pisa, Italy), Chloé Jimenez (Brest, France).

Optimal Transport in the Applied Science, RICAM Sand Piles

Introduction Optimal transport at a.e. t Discrete source: f = k

j=1 cj δyj

More general source: f ∈ M+(Ω) The model The main results Some References

Plan

1

Introduction The model The main results Some References

2

Optimal transport at a.e. t Duality in Optimal Transport The Optimal Transport map General results

3

Discrete source: f = k

j=1 cjδyj 4

More general source: f ∈ M+(Ω)

Optimal Transport in the Applied Science, RICAM Sand Piles

Introduction Optimal transport at a.e. t Discrete source: f = k

j=1 cj δyj

More general source: f ∈ M+(Ω) The model The main results Some References

A Sand Pile on a table with walls

Space variable: x ∈ Rd (d > 1), Time variable: t ∈ [0, T], derivative (D, ∂t), spatial divergence: div Ω ⊂ Rd, convex, open and bounded: the table , f ∈ M+(Ω) a bounded Radon measure: the source, g : ∂Ω → R+ l.s.c. : the height of the wall , u : Ω × [0, T] → R: the profile of the pile, the standing layer,

Optimal Transport in the Applied Science, RICAM Sand Piles

Introduction Optimal transport at a.e. t Discrete source: f = k

j=1 cj δyj

More general source: f ∈ M+(Ω) The model The main results Some References

A Sand Pile on a table with walls

The slope is bounded: |Du| ≤ 1 (Dµtu(·, t)µt): the flux of the rolling layer projected on Rd at time t. It follows the steepest descent Du. Be carefull! µt can be not better than a measure so that Du(·, t) × µt could have no meaning!

Optimal Transport in the Applied Science, RICAM Sand Piles

Introduction Optimal transport at a.e. t Discrete source: f = k

j=1 cj δyj

More general source: f ∈ M+(Ω) The model The main results Some References

A Sand Pile on a table with walls

The slope is bounded: |Du| ≤ 1 (Dµtu(·, t)µt): the flux of the rolling layer projected on Rd at time t. The sand rolls only when the slope is maximal: |Dµtu(·, t)| = 1 µt a.e. x.

Optimal Transport in the Applied Science, RICAM Sand Piles

Introduction Optimal transport at a.e. t Discrete source: f = k

j=1 cj δyj

More general source: f ∈ M+(Ω) The model The main results Some References

The System

At some point, the sand will start to fall from the walls νt ∈ M+(Ω) represents how much sand fall from witch point. The System (PDE) ∂tu − div(Dµtuµt) = f−ν, in Rd×]0, T[, (E) |Du| ≤ 1 a.e., |Dµtu(·, t)| = 1 µt a.e. x, (I) Initial condition: u(·, 0) = 0, (B) Boundary Condition: 0 ≤ u(x, t) ≤ g(x) ∀x ∈ ∂Ω a.e. t ∈]0, T[, u(x, t) = g(x) νt-a.e. x, a.e. t ∈]0, T[. Additional condition: u ≥ 0 and non-decreasing.

Optimal Transport in the Applied Science, RICAM Sand Piles

Introduction Optimal transport at a.e. t Discrete source: f = k

j=1 cj δyj

More general source: f ∈ M+(Ω) The model The main results Some References

The System

At some point, the sand will start to fall from the walls. νt ∈ M+(Ω) represents how much sand fall from witch point. The System

Primal-Dual

(PDE) ∂tu − div(Dµtuµt) = f−ν, in Rd×]0, T[, (E) |Du| ≤ 1 a.e., |Dµtu(·, t)| = 1 µt a.e. x, (I) Initial condition: u(·, 0) = 0, (B) Boundary Condition: 0 ≤ u(x, t) ≤ g(x) ∀x ∈ ∂Ω a.e. t ∈]0, T[, u(x, t) = g(x) νt-a.e. x, a.e. t ∈]0, T[. Additional condition: u ≥ 0 and non-decreasing.

Optimal Transport in the Applied Science, RICAM Sand Piles

Introduction Optimal transport at a.e. t Discrete source: f = k

j=1 cj δyj

More general source: f ∈ M+(Ω) The model The main results Some References

The System

At some point, the sand will start to fall from the walls. νt ∈ M+(Ω) represents how much sand fall from witch point. The System (PDE) ∂tu − div(Dµtuµt) = f − ν, in Rd×]0, T[, (E) |Du| ≤ 1 a.e., |Dµtu(·, t)| = 1 µt a.e. x, (I) Initial condition: u(·, 0) = 0, (B) Boundary Condition: 0 ≤ u(x, t) ≤ g(x) ∀x ∈ ∂Ω a.e. t ∈]0, T[, u(x, t) = g(x) νt-a.e. x, a.e. t ∈]0, T[. Additional condition: u ≥ 0 and non-decreasing.

Optimal Transport in the Applied Science, RICAM Sand Piles

Introduction Optimal transport at a.e. t Discrete source: f = k

j=1 cj δyj

More general source: f ∈ M+(Ω) The model The main results Some References

The meaning of the equation

Regularity of the solution (u, µ, ν): u ∈ L∞(0, T, W 1,∞(Ω)), ∂tu ∈ L∞(0, T, M(Ω)), µ ∈ L∞(0, T, M+(Ω)), ν ∈ L∞(0, T, M+(∂Ω)), Nota Bene: then u ∈ C([0, T]; L2(Ω)) and (I) makes sense. Meaning of (PDE): ∂tu − div(Dµtuµt) = f − ν, in Rd×]0, T[, d dt

• Rd u(·, t)ϕ(·) dx +
• Dµtu · Dϕ dµt =
• Ω

ϕdft −

• ∂Ω

ϕ dνt, in D′(0, T).

Optimal Transport in the Applied Science, RICAM Sand Piles

Introduction Optimal transport at a.e. t Discrete source: f = k

j=1 cj δyj

More general source: f ∈ M+(Ω) The model The main results Some References

The meaning of the equation

Regularity of the solution (u, µ, ν): u ∈ L∞(0, T, W 1,∞(Ω)), ∂tu ∈ L∞(0, T, M(Ω)), µ ∈ L∞(0, T, M+(Ω)), ν ∈ L∞(0, T, M+(∂Ω)), Nota Bene: then u ∈ C([0, T]; L2(Ω)) and (I) makes sense. Meaning of (PDE): for all ϕ ∈ D(Rd), it means ∂tu − div(Dµtuµt) = f − ν, in Rd×]0, T[, d dt

• Rd u(·, t)ϕ(·) dx +
• Dµtu · Dϕ dµt =
• Ω

ϕdf −

• ∂Ω

ϕ dνt, in D′(0, T).

Optimal Transport in the Applied Science, RICAM Sand Piles

Introduction Optimal transport at a.e. t Discrete source: f = k

j=1 cj δyj

More general source: f ∈ M+(Ω) The model The main results Some References

Main Theorems

Theorem 1 Let u ∈ L∞(]0, T[, W 1,∞(Ω)). Then u is a solution of the system (together with some µ and ν) iff u satisfies (I) and is a solution, a.e. t ∈]0, T[ of: max{f−∂tu(·, t), vM(Ω),Cb(Ω) : v ∈ Lip1(Ω), 0 ≤ v ≤ g on ∂Ω}. In other words: if (I) is satisfied u is a solution of the system ⇔ f − ∂tu(·, t) ∈ ∂I∞(u(·, t)) a.e t.

Optimal Transport in the Applied Science, RICAM Sand Piles

Introduction Optimal transport at a.e. t Discrete source: f = k

j=1 cj δyj

More general source: f ∈ M+(Ω) The model The main results Some References

Main Theorems

Theorem 1 Let u ∈ L∞(]0, T[, W 1,∞(Ω)). Then u is a solution of the system (together with some µ and ν) iff u satisfies (I) and is a solution, a.e. t ∈]0, T[ of: max{f−∂tu(·, t), vM(Ω),Cb(Ω) : v ∈ Lip1(Ω), 0 ≤ v ≤ g on ∂Ω}. If the system admits a solution (u, µ, ν), then

u is unique and, for every choice of ν, µ is also unique, u(·, t) is differentiable µt-a.e. and Dµtu = Du µt-a.e.

Optimal Transport in the Applied Science, RICAM Sand Piles

Introduction Optimal transport at a.e. t Discrete source: f = k

j=1 cj δyj

More general source: f ∈ M+(Ω) The model The main results Some References

Main Theorems

Theorem 2 The system admits a solution (u, µ, ν). Moreover: ∂tu ∈ L2(Ω×]0, T[).

Optimal Transport in the Applied Science, RICAM Sand Piles

Introduction Optimal transport at a.e. t Discrete source: f = k

j=1 cj δyj

More general source: f ∈ M+(Ω) The model The main results Some References

References

L Prigozhin, 1996, table without walls He showed existence and uniqueness of u for f ∈ (L4(]0, T[, W 1,4(Ω))′ and: u is a solution of the system ⇔ f − ∂tu(·, t) ∈ ∂I∞(u(·, t)) a.e t.

• G. Aronsson and Cie

1972 G. Aronsson : introduction of a model for f = k

i=1 ciδxi, link with an ODE,

1996 G. Aronsson, L. Evans, Y. Wu (Ω = Rd): Lp-approximation (p-Laplacian), f smooth,

Optimal Transport in the Applied Science, RICAM Sand Piles

Introduction Optimal transport at a.e. t Discrete source: f = k

j=1 cj δyj

More general source: f ∈ M+(Ω) The model The main results Some References

• L. Evans, M. Feldman, R. Gariepy

1997 L. Evans, M. Feldman, R. Gariepy (Ω = Rd): Lp-approximation, f Lipschitz, table without boundaries, link with optimal transport. Equilibrium in the open table problem 2004 P. Cannarsa, P. Cardaliaguet and C. Sinestrari: convergence toward equilibrium (source in Lp) Study of the Equilibrium:

• G. Bouchitté, G. Buttazzo, P. Seppecher (1997), G.

Bouchitté, G. Buttazzo (2001),

• P. Cannarsa, P. Cardaliaguet (2004), P. Cannarsa, P.

Cardaliaguet, G. Crasta, E. Giorgeri (2005), P. Cannarsa,

• P. Cardaliaguet, E. Giorgeri (2007),
• G. Crasta, A. Malusa (2007).

Optimal Transport in the Applied Science, RICAM Sand Piles

Introduction Optimal transport at a.e. t Discrete source: f = k

j=1 cj δyj

More general source: f ∈ M+(Ω) The model The main results Some References

References

Open Table, f ∈ L1(0, T, M(Ω)) 2013 N. Igbida : new weak formulation. Table with arbitrary high walls 2008 G. Crasta, S. Finzi Vita. Forthcoming 2014 G. Crasta, A. Malusa.

Optimal Transport in the Applied Science, RICAM Sand Piles

Introduction Optimal transport at a.e. t Discrete source: f = k

j=1 cj δyj

More general source: f ∈ M+(Ω) Duality in Optimal Transport The Optimal Transport map General results

Plan

1

Introduction The model The main results Some References

2

Optimal transport at a.e. t Duality in Optimal Transport The Optimal Transport map General results

3

Discrete source: f = k

j=1 cjδyj 4

More general source: f ∈ M+(Ω)

Optimal Transport in the Applied Science, RICAM Sand Piles

Introduction Optimal transport at a.e. t Discrete source: f = k

j=1 cj δyj

More general source: f ∈ M+(Ω) Duality in Optimal Transport The Optimal Transport map General results

• Duality. Let ρ ∈ M(Ω).

Duality: We have min(P) = max(D) with: (D) max{ρ, v : v ∈ Lip1(Ω), 0 ≤ v(x) ≤ g(x) on ∂Ω}, (P) min γ ∈ M+(¯ Ω × ¯ Ω), ν ∈ M(∂Ω) π1

♯ γ = ρ−+ν+

π2

♯ γ = ρ++ν−

• ¯

Ωׯ Ω

|x − y|dγ(x, y)+

• ∂Ω

g(x)dν+(x)

• min(P) =

min

ν∈M(∂Ω){W1(ρ−+ν+, ρ++ν−)+

• ∂Ω

g dν+}

Optimal Transport in the Applied Science, RICAM Sand Piles

Introduction Optimal transport at a.e. t Discrete source: f = k

j=1 cj δyj

More general source: f ∈ M+(Ω) Duality in Optimal Transport The Optimal Transport map General results

Duality

Duality: We have min(P) = max(D) with: (D) max{ρ, v : v ∈ Lip1(Ω), 0 ≤ v(x) ≤ g(x) on ∂Ω}, (P) min γ ∈ M+(¯ Ω × ¯ Ω), ν ∈ M(∂Ω) π1

♯ γ = ρ−+ν+

π2

♯ γ = ρ++ν−

• ¯

Ωׯ Ω

|x − y|dγ(x, y)+

• ∂Ω

g(x)dν+(x)

• min(P) =

min

ν∈M(∂Ω){W1(ρ−+ν+, ρ++ν−)+

• ∂Ω

g dν+}

Optimal Transport in the Applied Science, RICAM Sand Piles

Introduction Optimal transport at a.e. t Discrete source: f = k

j=1 cj δyj

More general source: f ∈ M+(Ω) Duality in Optimal Transport The Optimal Transport map General results

Duality

Duality again We have max(D) = min(D′) (D′) min σ ∈ M(Ω, Rd) ν ∈ M(∂Ω) −divσ = ρ−ν in Rd d|σ| +

• gdν+(x)
• .

Optimal Transport in the Applied Science, RICAM Sand Piles

Introduction Optimal transport at a.e. t Discrete source: f = k

j=1 cj δyj

More general source: f ∈ M+(Ω) Duality in Optimal Transport The Optimal Transport map General results

Primal-Dual Optimality condition (D) − (D′)

System

u, ν and σ are optimal iff                            −div(Dµu × µ) = ρ−ν in Rd σ = Dµuµ, |σ| = µ, |Dµu| = 1 µ-a.e.x, 0 ≤ u ≤ g on ∂Ω, u(x) = 0 ν−-a.e.x, u(x) = g(x) ν+-a.e.x.

Optimal Transport in the Applied Science, RICAM Sand Piles

Introduction Optimal transport at a.e. t Discrete source: f = k

j=1 cj δyj

More general source: f ∈ M+(Ω) Duality in Optimal Transport The Optimal Transport map General results

Transport Density

• G. Bouchitté, G. Buttazzo

For any γ optimal for (P), we can build an optimal µ by setting: µ, ϕM(Ω),Cb(Ω) :=

• Ω2

1 ϕ((1 − s)x + sy)|y − x| dsdγ(x, y) for all ϕ ∈ Cb(Ω).

Optimal Transport in the Applied Science, RICAM Sand Piles

Introduction Optimal transport at a.e. t Discrete source: f = k

j=1 cj δyj

More general source: f ∈ M+(Ω) Duality in Optimal Transport The Optimal Transport map General results

Primal-Dual Optimality condition (D) − (P) Let (γ, ν, u) admissible. Then (γ, ν, u) is optimal iff u(y) − u(x) = |x − y| γ-a.e.x, u(x) = 0 ν−-a.e.x, u(x) = g(x) ν+-a.e.x. Decomposition of γ: γ = γii + γbi + γib + γbb ρ+ = π2

♯ [γii + γbi],

ρ− = π1

♯ [γii + γib],

ν− = π2

♯ [γib + γbb],

ν+ = π1

♯ [γbi + γbb].

Optimal Transport in the Applied Science, RICAM Sand Piles

Introduction Optimal transport at a.e. t Discrete source: f = k

j=1 cj δyj

More general source: f ∈ M+(Ω) Duality in Optimal Transport The Optimal Transport map General results

"Transport cost":

• Ω×Ω

|x − y|dγ +

• gdν+

=

• Ω×Ω

|x − y|dγii +

• ∂Ω×Ω

|x − y|dγib +

• ∂Ω×Ω

|x − y| + g(x) dγbi +

• ∂Ω×∂Ω

|x − y| + g(x) dγbb Lemma If (D) has a solution u ≥ 0, it exists (γ, ν) optimal for (P) such that γ = γii + γbi and ν = ν+.

Optimal Transport in the Applied Science, RICAM Sand Piles

Introduction Optimal transport at a.e. t Discrete source: f = k

j=1 cj δyj

More general source: f ∈ M+(Ω) Duality in Optimal Transport The Optimal Transport map General results

We get rid of γbb: As (γ, ν) is optimal then (γ − γbb, ν − π1

♯ γbb + π2 ♯ γbb) "costs

more" that is:

• ∂Ω×∂Ω

|x − y| + g(x) dγbb ≤ 0 so that x = y and g(x) = 0 γbb-a.e.(x,y) and (γ − (γ)bb, ν − π1

♯ γbb + π2 ♯ γbb) is still optimal.

We get rid of ν− and γib: As −u(x) = |x − y| γib- a.e. (x, y) and u ≥ 0, we get γib = 0 and ν− = π2

♯ γib = 0.

Optimal Transport in the Applied Science, RICAM Sand Piles

Introduction Optimal transport at a.e. t Discrete source: f = k

j=1 cj δyj

More general source: f ∈ M+(Ω) Duality in Optimal Transport The Optimal Transport map General results

Some results

Take ρ = f − ∂tu(·, t) with t fixed. Almost Theorem 1 1) If (u, µ, ν) is a solution of the system then

u(·, t) a.e. t is a solution of: (Dt) max{f − ∂tu(·, t), vM(Ω),Cb(Ω) : v ∈ Lip1(Ω), 0 ≤ v ≤ g on ∂Ω}. (σt := Dµtuµt, νt) is a solution a.e. t of: (D′

t)

min σ ∈ M(Ω, Rd) ν ∈ M(∂Ω) −divσ = f − ∂tu(·, t) − ν in Rd d|σ| +

• gdν+(x)
• Optimal Transport in the Applied Science, RICAM

Sand Piles

Introduction Optimal transport at a.e. t Discrete source: f = k

j=1 cj δyj

More general source: f ∈ M+(Ω) Duality in Optimal Transport The Optimal Transport map General results

Some results

Almost Theorem 1 1) Let

u(·, t) a solution of (Dt) for a.e. t with u(·, 0) = 0 and u ≥ 0, (σt, νt) a solution of (D′

t) with νt ≥ 0,

then (u, µ, ν) is a solution of the sytem with: µt = |νt|. 2) In case 1) is satisfied, take γt such that (γt, νt) is optimal for (Pt), µt is unique and for all ϕ ∈ Cb(Ω): µt, ϕM(Ω),Cb(Ω) :=

• Ω2

1 ϕ((1−s)x+sy)|y−x| dsdγt(x, y) Moreover u(·, t) is differentiable µt-a.e. and Dµtu = Du µt-a.e.

Optimal Transport in the Applied Science, RICAM Sand Piles

Introduction Optimal transport at a.e. t Discrete source: f = k

j=1 cj δyj

More general source: f ∈ M+(Ω) Duality in Optimal Transport The Optimal Transport map General results

Some results

Almost Theorem 1 1) Let

u(·, t) a.e. t a solution of (Dt) with u(·, 0) = 0 and u ≥ 0, (σt, νt) a solution of (D′

t) with νt ≥ 0,

then (u, µ, ν) is a solution of the sytem with: µt = |νt|. 2) In case 1) is satisfied, take γt such that (γt, νt) is optimal for (Pt), µt is unique and for all ϕ ∈ Cb(Ω): µt, ϕM(Ω),Cb(Ω) :=

• Ω2

1 ϕ((1−s)x+sy)|y−x| dsdγt(x, y) Moreover u(·, t) is differentiable µt-a.e. and Dµtu = Du µt-a.e.

Optimal Transport in the Applied Science, RICAM Sand Piles

Introduction Optimal transport at a.e. t Discrete source: f = k

j=1 cj δyj

More general source: f ∈ M+(Ω) Duality in Optimal Transport The Optimal Transport map General results

Some results

Almost Theorem 1 1) Let

u(·, t) a solution of (Dt) for a.e. t with u(·, 0) = 0 and u ≥ 0, (σt, νt) a solution of (D′

t) with νt ≥ 0,

then (u, µ, ν) is a solution of the sytem with: µt = |νt|. 2) In case 1) is satisfied, take γt such that (γt, νt) is optimal for (Pt), µt is unique and for all ϕ ∈ Cb(Ω): µt, ϕM(Ω),Cb(Ω) :=

• Ω2

1 ϕ((1−s)x+sy)|y−x| dsdγt(x, y) Moreover u(·, t) is differentiable µt-a.e. and Dµtu = Du µt-a.e.

Optimal Transport in the Applied Science, RICAM Sand Piles

Introduction Optimal transport at a.e. t Discrete source: f = k

j=1 cj δyj

More general source: f ∈ M+(Ω) Duality in Optimal Transport The Optimal Transport map General results

Sketch of Proof of Dµtu = Du

What we know u(y, t) − u(x, t) = |x − y| γt-a.e.x µt, ϕ :=

• Ω2

1 ϕ((1−s)x +sy)|y −x| dsdγt(x, y) ∀ϕ ∈ Cb(Ω) We call transport ray any segment ]x, y[ with : u(y, t) − u(x, t) = |x − y|. µt is supported on transport rays, u(·, t) is differentiable inside transport rays and Du =

y−x |y−x|...

Optimal Transport in the Applied Science, RICAM Sand Piles

Introduction Optimal transport at a.e. t Discrete source: f = k

j=1 cj δyj

More general source: f ∈ M+(Ω)

Plan

1

Introduction The model The main results Some References

2

Optimal transport at a.e. t Duality in Optimal Transport The Optimal Transport map General results

3

Discrete source: f = k

j=1 cjδyj 4

More general source: f ∈ M+(Ω)

Optimal Transport in the Applied Science, RICAM Sand Piles

Introduction Optimal transport at a.e. t Discrete source: f = k

j=1 cj δyj

More general source: f ∈ M+(Ω)

Case f = k

j=1 cjδyj

If (u, µ, ν) is a solution of the system, then for a.e. fixed t: 1) u(·, t) ≥ 0 is a solution of (Dt), 2) take (γt, νt) optimal for (Pt) with νt ≥ 0, 3) try to guess the shape of u.

Optimal Transport in the Applied Science, RICAM Sand Piles

Introduction Optimal transport at a.e. t Discrete source: f = k

j=1 cj δyj

More general source: f ∈ M+(Ω)

Optimal Transport at a.e. t

γt has marginals ∂tu(·, t) + νt and f. Decomposition: γt = (γt)ii + (γt)bi, π1

♯ (γt)ii = ∂tu,

π1

♯ (γt)bi = νt.

Primal-Dual Constraints u(yj, t) − u(x, t) = |yj − x| u(yj, t) − u(x, t) ≤ |yj − x| (γt)-a.e.(x, yj) ∀(x, yj) u(x) = g(x) u(x) ≤ g(x) νt-a.e.x ∀x ∈ ∂Ω

Optimal Transport in the Applied Science, RICAM Sand Piles

Introduction Optimal transport at a.e. t Discrete source: f = k

j=1 cj δyj

More general source: f ∈ M+(Ω)

Optimal Transport at a.e. t

γt has marginals ∂tu(·, t) + νt and f. Decomposition: γt = (γt)ii + (γt)bi, π1

♯ (γt)ii = ∂tu,

π1

♯ (γt)bi = νt.

Primal-Dual Constraints u(x, t) = u(yj, t) − |yj − x| u(x, t) ≥ u(yj, t) − |yj − x| (γt)ii-a.e.(x, yj) ∀(x, yj) u(yj, t) = g(x) + |yj − x| u(yj, t) ≤ g(x) + |yj − x| (γt)bi-a.e.(x, yj) ∀(x, yj), x ∈ ∂Ω

Optimal Transport in the Applied Science, RICAM Sand Piles

Introduction Optimal transport at a.e. t Discrete source: f = k

j=1 cj δyj

More general source: f ∈ M+(Ω)

Shape of u

Lemma Let rj(t) = u(yj, t). Then:

• n (spt)(∂tu) ∪ (spt)(f) ∪ (spt)(νt) we have:

u(x, t) = (rj(t) + |x − yj|)1Aj(t)(x), Aj(t) := {x ∈ ¯ Ω : rj(t) + |x − yj| = max

n {rn(t) + |x − yn|, 0}},

0 ≤ rj(t) ≤ max{g(x) + |x − yj| : x ∈ ∂Ω} and rj(t) = g(x0) + |x0 − yj| = min{g(x) + |x − yj| : x ∈ ∂Ω} (γt)bi-a.e.(x0, yj).

Optimal Transport in the Applied Science, RICAM Sand Piles

Introduction Optimal transport at a.e. t Discrete source: f = k

j=1 cj δyj

More general source: f ∈ M+(Ω)

Shape of u

Lemma Let rj(t) = u(yj, t). Then:

• n (spt)(∂tu) ∪ (spt)(f) ∪ (spt)(ν+) we have:

u(x, t) = (rj(t) − |x − yj|)1Aj(t)(x), Aj(t) := {x ∈ ¯ Ω : rj(t) − |x − yj| = max

n {rn(t) − |x − yn|, 0}},

0 ≤ rj(t) ≤ min{g(x) + |x − yj| : x ∈ ∂Ω} and rj(t) = g(x0) + |x0 − yj| = min{g(x) + |x − yj| : x ∈ ∂Ω} (γt)bi-a.e.(x0, yj).

Optimal Transport in the Applied Science, RICAM Sand Piles

Introduction Optimal transport at a.e. t Discrete source: f = k

j=1 cj δyj

More general source: f ∈ M+(Ω)

Shape of ∂tu

Lemma Let rj(t) = u(yj, t). Then: Assume for all j = 1, ...k, rj is derivable a.e. t ∈]0, T[. Then for a.e. t > 0 and a.e. x: ∂tu(x, t) =

k

• j=1

˙ rj(t)1Aj(t)(x) a.e. (x, t).

Optimal Transport in the Applied Science, RICAM Sand Piles

Introduction Optimal transport at a.e. t Discrete source: f = k

j=1 cj δyj

More general source: f ∈ M+(Ω)

Transport again

For small t γt = (γt)ii has marginals k

j=1 ˙

rj(t)1Aj(t) and k

j=1 cjδyj.

Using Primal-Dual condition: γt = k

j=1 ˙

rj(t)1Aj(t) ⊗ δyj. we get: ˙ rj(t)|Aj(t)| = cj ∀j = 1, ..., k. Lemma (ODE) There exist times (t1, ..., tk) ∈ Rk

+ and rj ∈ C1(0, tj) ∩ C([0, +∞))

which satisfy:

• rj(0) = 0,

˙ rj(t) =

cj |Aj(t)|

∀t ∈]0, tj[, rj(t) = minx∈∂Ω{g(x) + |x − yj|} ∀t > tj

Optimal Transport in the Applied Science, RICAM Sand Piles

Introduction Optimal transport at a.e. t Discrete source: f = k

j=1 cj δyj

More general source: f ∈ M+(Ω)

Case f = k

j=1 cjδyj

If (u, µ, ν) is a solution of the system, then: 1) u(·, t) is a solution of (Dt), 2) the function u has the shape: u(x, t) = (rj(t)−|x−yj|)1Aj(t)(x) with the previous definition of Aj, 3) the ODE must be satisfied for rj. On the other hand: 1) the ODE has a solution, 2) we associate to the rj solution some (u(·, t), σt, νt, γt), 3) (u(·, t), σt, νt, γt) is optimal for (Dt), (D′

t) and (Pt),

4) (u(·, t), µt, νt) is optimal for the system (with µt = |σt|.)

Optimal Transport in the Applied Science, RICAM Sand Piles

Introduction Optimal transport at a.e. t Discrete source: f = k

j=1 cj δyj

More general source: f ∈ M+(Ω)

Plan

1

Introduction The model The main results Some References

2

Optimal transport at a.e. t Duality in Optimal Transport The Optimal Transport map General results

3

Discrete source: f = k

j=1 cjδyj 4

More general source: f ∈ M+(Ω)

Optimal Transport in the Applied Science, RICAM Sand Piles

Introduction Optimal transport at a.e. t Discrete source: f = k

j=1 cj δyj

More general source: f ∈ M+(Ω)

Scheme of the proof of Theorem 2

Approximate f by f n discrete f n ≤ f. Make some estimate on un, νn

t , ∂tun, µn t :

un(·, t)∞ ≤ g∞ + diam(Ω), νn

t (∂Ω) ≤

• Ω

df(x), ∂tunL2(Ω×]0,T[) ≤ un(·, t)∞ ×

• Ω

df(x),

• Ω

dµn

t (x)

≤ diam(Ω) ×

• Ω

df(x). Pass to the limit and using again Bouchitté-Buttazzo, prove we get a solution.

Optimal Transport in the Applied Science, RICAM Sand Piles

Introduction Optimal transport at a.e. t Discrete source: f = k

j=1 cj δyj

More general source: f ∈ M+(Ω)

The L2-estimate of ∂tun:

With the notation of the previous section ∂tuL2(Ω×]0,T[) = T

• Ω

k

• j=1

(˙ rj(t))21Aj(t)(x) dtdx = T

k

• j=1

˙ rj(t) × ˙ rj(t)|Aj(t)|dt = T

k

• j=1

˙ rj(t)cjdt =

k

• j=1

cju(yj, T) ≤ u(·, t)∞

• Ω

f(x)dx.

Optimal Transport in the Applied Science, RICAM Sand Piles