Introduction Optimal transport at a.e. t Discrete source: f = � k j = 1 c j δ yj 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 The model Optimal transport at a.e. t The main results Discrete source: f = � k j = 1 c j δ yj Some References More general source: f ∈ M + (Ω) Plan Introduction 1 The model The main results Some References Optimal transport at a.e. t 2 Duality in Optimal Transport The Optimal Transport map General results Discrete source: f = � k j = 1 c j δ y j 3 More general source: f ∈ M + (Ω) 4 Optimal Transport in the Applied Science, RICAM Sand Piles
Introduction The model Optimal transport at a.e. t The main results Discrete source: f = � k j = 1 c j δ yj Some References More general source: f ∈ M + (Ω) A Sand Pile on a table with walls Space variable: x ∈ R d ( d > 1), Time variable: t ∈ [ 0 , T ] , derivative ( D , ∂ t ) , spatial divergence: div Ω ⊂ R d , 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 The model Optimal transport at a.e. t The main results Discrete source: f = � k j = 1 c j δ yj Some References More general source: f ∈ M + (Ω) A Sand Pile on a table with walls The slope is bounded: | Du | ≤ 1 ( D µ t u ( · , t ) µ t ) : the flux of the rolling layer projected on R d at time t . It follows the steepest descent Du . Be carefull! µ t can be not better than a measure so that Du ( · , t ) × µ t could have no meaning! Optimal Transport in the Applied Science, RICAM Sand Piles
Introduction The model Optimal transport at a.e. t The main results Discrete source: f = � k j = 1 c j δ yj Some References More general source: f ∈ M + (Ω) A Sand Pile on a table with walls The slope is bounded: | Du | ≤ 1 ( D µ t u ( · , t ) µ t ) : the flux of the rolling layer projected on R d at time t . The sand rolls only when the slope is maximal: | D µ t u ( · , t ) | = 1 µ t a.e. x . Optimal Transport in the Applied Science, RICAM Sand Piles
Introduction The model Optimal transport at a.e. t The main results Discrete source: f = � k j = 1 c j δ yj Some References More general source: f ∈ M + (Ω) The System At some point, the sand will start to fall from the walls ν t ∈ M + (Ω) represents how much sand fall from witch point. The System (PDE) ∂ t u − div ( D µ t u µ t ) = f − ν, in R d × ] 0 , T [ , (E) | Du | ≤ 1 a.e., | D µ t u ( · , t ) | = 1 µ t 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 ) ν t -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 The model Optimal transport at a.e. t The main results Discrete source: f = � k j = 1 c j δ yj Some References More general source: f ∈ M + (Ω) The System At some point, the sand will start to fall from the walls. ν t ∈ M + (Ω) represents how much sand fall from witch point. The System Primal-Dual (PDE) ∂ t u − div ( D µ t u µ t ) = f − ν, in R d × ] 0 , T [ , (E) | Du | ≤ 1 a.e., | D µ t u ( · , t ) | = 1 µ t 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 ) ν t -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 The model Optimal transport at a.e. t The main results Discrete source: f = � k j = 1 c j δ yj Some References More general source: f ∈ M + (Ω) The System At some point, the sand will start to fall from the walls. ν t ∈ M + (Ω) represents how much sand fall from witch point. The System (PDE) ∂ t u − div ( D µ t u µ t ) = f − ν, in R d × ] 0 , T [ , (E) | Du | ≤ 1 a.e., | D µ t u ( · , t ) | = 1 µ t 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 ) ν t -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 The model Optimal transport at a.e. t The main results Discrete source: f = � k j = 1 c j δ yj Some References More general source: f ∈ M + (Ω) The meaning of the equation Regularity of the solution ( u , µ, ν ) : u ∈ L ∞ ( 0 , T , W 1 , ∞ (Ω)) , ∂ t u ∈ L ∞ ( 0 , T , M (Ω)) , µ ∈ L ∞ ( 0 , T , M + (Ω)) , ν ∈ L ∞ ( 0 , T , M + ( ∂ Ω)) , Nota Bene: then u ∈ C ([ 0 , T ]; L 2 (Ω)) and (I) makes sense. Meaning of (PDE): ∂ t u − div ( D µ t u µ t ) = f − ν, in R d × ] 0 , T [ , d � � � � R d u ( · , t ) ϕ ( · ) dx + D µ t u · D ϕ d µ t = ϕ df t − ϕ d ν t , dt Ω ∂ Ω in D ′ ( 0 , T ) . Optimal Transport in the Applied Science, RICAM Sand Piles
Introduction The model Optimal transport at a.e. t The main results Discrete source: f = � k j = 1 c j δ yj Some References More general source: f ∈ M + (Ω) The meaning of the equation Regularity of the solution ( u , µ, ν ) : u ∈ L ∞ ( 0 , T , W 1 , ∞ (Ω)) , ∂ t u ∈ L ∞ ( 0 , T , M (Ω)) , µ ∈ L ∞ ( 0 , T , M + (Ω)) , ν ∈ L ∞ ( 0 , T , M + ( ∂ Ω)) , Nota Bene: then u ∈ C ([ 0 , T ]; L 2 (Ω)) and (I) makes sense. Meaning of (PDE): for all ϕ ∈ D ( R d ) , it means ∂ t u − div ( D µ t u µ t ) = f − ν, in R d × ] 0 , T [ , d � � � � R d u ( · , t ) ϕ ( · ) dx + D µ t u · D ϕ d µ t = ϕ df − ϕ d ν t , dt Ω ∂ Ω in D ′ ( 0 , T ) . Optimal Transport in the Applied Science, RICAM Sand Piles
Introduction The model Optimal transport at a.e. t The main results Discrete source: f = � k j = 1 c j δ yj Some References More general source: f ∈ M + (Ω) 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 − ∂ t u ( · , t ) , v � M (Ω) , C b (Ω) : v ∈ Lip 1 (Ω) , 0 ≤ v ≤ g on ∂ Ω } . In other words: if ( I ) is satisfied u is a solution of the system ⇔ f − ∂ t u ( · , t ) ∈ ∂ I ∞ ( u ( · , t )) a.e t . Optimal Transport in the Applied Science, RICAM Sand Piles
Introduction The model Optimal transport at a.e. t The main results Discrete source: f = � k j = 1 c j δ yj Some References More general source: f ∈ M + (Ω) 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 − ∂ t u ( · , t ) , v � M (Ω) , C b (Ω) : v ∈ Lip 1 (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 µ t -a.e. and D µ t u = Du µ t -a.e. Optimal Transport in the Applied Science, RICAM Sand Piles
Introduction The model Optimal transport at a.e. t The main results Discrete source: f = � k j = 1 c j δ yj Some References More general source: f ∈ M + (Ω) Main Theorems Theorem 2 The system admits a solution ( u , µ, ν ) . Moreover: ∂ t u ∈ L 2 (Ω × ] 0 , T [) . Optimal Transport in the Applied Science, RICAM Sand Piles
Introduction The model Optimal transport at a.e. t The main results Discrete source: f = � k j = 1 c j δ yj Some References More general source: f ∈ M + (Ω) References L Prigozhin, 1996, table without walls He showed existence and uniqueness of u for f ∈ ( L 4 (] 0 , T [ , W 1 , 4 (Ω)) ′ and: u is a solution of the system ⇔ f − ∂ t u ( · , t ) ∈ ∂ I ∞ ( u ( · , t )) a.e t . G. Aronsson and Cie 1972 G. Aronsson : introduction of a model for f = � k i = 1 c i δ x i , link with an ODE, 1996 G. Aronsson, L. Evans, Y. Wu ( Ω = R d ): L p -approximation ( p -Laplacian), f smooth, Optimal Transport in the Applied Science, RICAM Sand Piles
Introduction The model Optimal transport at a.e. t The main results Discrete source: f = � k j = 1 c j δ yj Some References More general source: f ∈ M + (Ω) L. Evans, M. Feldman, R. Gariepy 1997 L. Evans, M. Feldman, R. Gariepy ( Ω = R d ): L p -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 L p ) 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
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