Singular perturbations of gradient flows and rate-independent evolution

Singular perturbations of gradient flows and

rate-independent evolution

Giuseppe Savar ´e

Dipartimento di Matematica “F. Casorati”, Universit `a di Pavia

CALCULUS OF VARIATIONS AND APPLICATIONS A conference to celebrate Gianni Dal Maso’s 65th birthday

January 29, 2020, SISSA, Trieste

In collaboration with Virginia Agostiniani, Riccarda Rossi

Outline

1 Rate-independent evolution and singular perturbation of gradient flows

2 Transversality conditions for the critical set

3 Compactness and variational characterization of the limit evolution

4 A useful tool: graph convergence

Outline

1 Rate-independent evolution and singular perturbation of gradient flows

2 Transversality conditions for the critical set

3 Compactness and variational characterization of the limit evolution

4 A useful tool: graph convergence

Evolution by critical/stable points

I _{H := R}d( Hilbert space),

I E : [0, T] × H → Ris aC2_{time dependent energy with}

H-differential

DE : [0, T] × H → H.

Typical example: time dependent linear perturbation

E(t, x) := E(x) − hf(t), xi, DE(t, x) = DE(x) − f(t).

I u0∈ H. I Critical points: C := (t, x) : DE(t, x) = 0, C(t) := x :DE(t, x) = 0  =“section ofCat timet”. I ρ-critical points: fix someρ > 0 kDE(t, x)k 6 ρ

I Globallyρ-stable points:

E(t, x) 6 E(t, y) + ρky − xk ∀ y ∈ H.

Globallyρ-stable points areρ-critical.

Aim:

Select “reasonable” evolution curvest7→ u(t)starting fromu0such thatu(t)is

critical/stable for every timet∈ [0, T ].

Evolution by critical/stable points

I _{H := R}d( Hilbert space),

I E : [0, T] × H → Ris aC2_{time dependent energy with}

H-differential

DE : [0, T] × H → H.

Typical example: time dependent linear perturbation

E(t, x) := E(x) − hf(t), xi, DE(t, x) = DE(x) − f(t).

I u0∈ H. I Critical points: C := (t, x) : DE(t, x) = 0, C(t) := x :DE(t, x) = 0  =“section ofCat timet”.

I ρ-critical points: fix someρ>0 kDE(t, x)k 6ρ I Globallyρ-stable points:

E(t, x) 6 E(t, y) +ρky − xk ∀ y ∈ H.

Globallyρ-stable points areρ-critical. Aim:

Select “reasonable” evolution curvest7→ u(t)starting fromu0such thatu(t)is

critical/stable for every timet∈ [0, T ].

Evolution by critical/stable points

I _{H := R}d( Hilbert space),

I E : [0, T] × H → Ris aC2_{time dependent energy with}

H-differential

DE : [0, T] × H → H.

Typical example: time dependent linear perturbation

E(t, x) := E(x) − hf(t), xi, DE(t, x) = DE(x) − f(t).

I u0∈ H. I Critical points: C := (t, x) : DE(t, x) = 0, C(t) := x :DE(t, x) = 0  =“section ofCat timet”.

I ρ-critical points: fix someρ>0 kDE(t, x)k 6ρ I Globallyρ-stable points:

E(t, x) 6 E(t, y) +ρky − xk ∀ y ∈ H.

Globallyρ-stable points areρ-critical.

Aim:

Select “reasonable” evolution curvest7→ u(t)starting fromu0such thatu(t)is critical/stable for every timet∈ [0, T ].

Simple examples in 1D

Time Incremental Minimization Scheme

In the case of globalρ-stable evolutions, the main tool to provide existence and to approximate solutions is

The time Incremental Minimization scheme

Fixτ := T/N(for simplicity),tn

τ:= nτ,U0τ= u(0). Recursively chooseUn

τ among the minimizers of

U7→E(tn

τ, U) +ρkU − Un−1τ k

Uτ:=the piecewise constant interpolant of the valuesUnτ. Theorem[Mainik-Mielke ’05]:

There exists a sequencek7→ τ(k) ↓ 0andu : [0, T ] → Hsuch that and

Uτ(k)(t)→ u(t), E(t, Uτ(k)(t)→E(t, u(t)) for everyt∈ [0, T ]

anduis called anEnergetic solution to the Rate Independent Ssystem (R.I.S.)(H, E, ρ).

Time Incremental Minimization Scheme

In the case of globalρ-stable evolutions, the main tool to provide existence and to approximate solutions is

The time Incremental Minimization scheme

Fixτ := T/N(for simplicity),tn

τ:= nτ,U0τ= u(0). Recursively chooseUn

τ among the minimizers of

U7→E(tn

τ, U) +ρkU − Un−1τ k

Uτ:=the piecewise constant interpolant of the valuesUnτ. Theorem[Mainik-Mielke ’05]:

There exists a sequencek7→ τ(k) ↓ 0andu : [0, T ] → Hsuch that and

Uτ(k)(t)→ u(t), E(t, Uτ(k)(t)→E(t, u(t)) for everyt∈ [0, T ] anduis called anEnergetic solution to the Rate Independent Ssystem (R.I.S.)(H, E,ρ).

Energetic solutions

Energetic solution:a curveu : [0, T ] → Hsatisfying for everyt∈ [0, T ]the ρ-stability condition

E(t, u(t)) 6 E(t, v) +ρku(t) − vk for everyv∈ H, (S) and theenergy balance

E(t, u(t)) +ρVar(u, [0, t]) =E(0, u(0)) + Zt 0 P(r, u(r)) dr (E) where P(t, x) = ∂ ∂tE(t, x). [Mielke-Theil-Levitas ’02, Mielke-Theil ’04

Francfort-Marigo ’93/’98, DalMaso-Toader ’02, Francfort-Larsen ’05, DalMaso-Francfort-Toader ’05

Mainik-Mielke ’05, Francfort-Mielke ’06 . . .

Mielke-Roubicek ’15]

The “smooth” finite dimensional case

Energetic solutions provides a variational selection among trajectories satisfying ρsign( ˙u(t)) + DE(t, u(t)) 3 0 in particularkDE(t, u(t))k 6ρ,

and at every jump pointt∈ J(u)theenergetic jump conditions

E(t, (u(t−)) − E(t, u(t+)) =ρku(t+) − u(t+)k

u(t) f(t) − ρ

E0_(u)

u(t−) u(t+)

Energetic solution in the

1-dimensional case for a strictly increasingf.

The “smooth” finite dimensional case

Energetic solutions provides a variational selection among trajectories satisfying ρsign( ˙u(t)) + DE(t, u(t)) 3 0 in particularkDE(t, u(t))k 6ρ,

and at every jump pointt∈ J(u)theenergetic jump conditions

E(t, (u(t−)) − E(t, u(t+)) =ρku(t+) − u(t+)k

u(t) f(t) −ρ

E0_(u)

u(t−) u(t+)

Energetic solution in the

1-dimensional case for a strictly increasingf.

A few technical points

I Compactness w.r.t. space:it follows by the compactness of the sublevels ofEand the estimate

E(t, Uτ(t))6 C

I Compactness w.r.t. time: it follows by the uniform BV estimate (Helly’s

Theorem)

ρ Var(Uτ, [0, T ]) 6 C

I Stability: it follows by the stability of each minimizer and from the

closure of theρ-stable set

(t, x) : E(t, x) 6 E(t, y) + ρky − xk 

.

A few technical points

I Compactness w.r.t. space:it follows by the compactness of the sublevels ofEand the estimate

E(t, Uτ(t))6 C

I Compactness w.r.t. time:it follows by the uniform BV estimate (Helly’s Theorem)

ρVar(Uτ, [0, T ]) 6 C

I Stability: it follows by the stability of each minimizer and from the

closure of theρ-stable set

(t, x) : E(t, x) 6 E(t, y) + ρky − xk 

.

A few technical points

I Compactness w.r.t. space:it follows by the compactness of the sublevels ofEand the estimate

E(t, Uτ(t))6 C

I Compactness w.r.t. time:it follows by the uniform BV estimate (Helly’s Theorem)

ρVar(Uτ, [0, T ]) 6 C

I Stability:it follows by the stability of each minimizer and from the closure of theρ-stable set

(t, x) : E(t, x) 6 E(t, y) +ρky − xk.

Viscous corrections of the Incremental Minimization Scheme

minimize U7→E(tn τ, U) +ρkU − U n−1 τ k + ε 2τkU − U n−1_k2

and its limit behaviour in three cases: I Visco Energetic solutions:[Minotti-S.]

ε/τ = µ > 0andρ > 0are fixed. (VE) I Balanced Viscosity solutions: [Mielke-Rossi-S.]ρ > 0is fixed andε

satisfies

ε↓ 0, ε/τ↑ +∞, asτ↓ 0. (BV)

I Vanishing Viscosity solutions:ρ = 0and

ε↓ 0, ε/τ↑ +∞ asτ↓ 0. (VV)

The last two methods correspond to the limit behaviour of

ρ sign( ˙u(t)) + ε ˙u(t) + DE(t, u(t)) 3 0

Viscous corrections of the Incremental Minimization Scheme

minimize U7→E(tn τ, U) +ρkU − U n−1 τ k + ε 2τkU − U n−1_k2

and its limit behaviour in three cases: I Visco Energetic solutions:[Minotti-S.]

ε/τ = µ > 0andρ > 0are fixed. (VE) I Balanced Viscosity solutions:[Mielke-Rossi-S.]ρ > 0is fixed andε

satisfies

ε↓ 0, ε/τ↑ +∞, asτ↓ 0. (BV)

I Vanishing Viscosity solutions:ρ = 0and

ε↓ 0, ε/τ↑ +∞ asτ↓ 0. (VV)

The last two methods correspond to the limit behaviour of

ρ sign( ˙u(t)) + ε ˙u(t) + DE(t, u(t)) 3 0

Viscous corrections of the Incremental Minimization Scheme

minimize U7→E(tn τ, U) +ρkU − U n−1 τ k + ε 2τkU − U n−1_k2

and its limit behaviour in three cases: I Visco Energetic solutions:[Minotti-S.]

ε/τ = µ > 0andρ > 0are fixed. (VE) I Balanced Viscosity solutions:[Mielke-Rossi-S.]ρ > 0is fixed andε

satisfies

ε↓ 0, ε/τ↑ +∞, asτ↓ 0. (BV)

I Vanishing Viscosity solutions:ρ = 0and

ε↓ 0, ε/τ↑ +∞ asτ↓ 0. (VV)

The last two methods correspond to the limit behaviour of

ρ sign( ˙u(t)) + ε ˙u(t) + DE(t, u(t)) 3 0

Viscous corrections of the Incremental Minimization Scheme

minimize U7→E(tn τ, U) +ρkU − U n−1 τ k + ε 2τkU − U n−1_k2

and its limit behaviour in three cases: I Visco Energetic solutions:[Minotti-S.]

ε/τ = µ > 0andρ > 0are fixed. (VE) I Balanced Viscosity solutions:[Mielke-Rossi-S.]ρ > 0is fixed andε

satisfies

ε↓ 0, ε/τ↑ +∞, asτ↓ 0. (BV)

I Vanishing Viscosity solutions:ρ = 0and

ε↓ 0, ε/τ↑ +∞ asτ↓ 0. (VV)

The last two methods correspond to the limit behaviour of ρsign( ˙u(t)) +ε ˙u(t)+DE(t, u(t)) 3 0

Singular limit of gradient flows (

ρ = 0

)

Main problem

Study the asymptotic behaviour asε↓ 0of the solutionuε: [0, T ] → Hof the

gradient flow _

εuε0(t) = −DE(t, uε(t))

uε(0) = u0.

Formally, the limitushould be a suitable curve solving

DE(t, u(t)) = 0

In order to avoid a transition layer att =0we will assumeDE(0, u0) =0.

Singular limit of gradient flows (

ρ = 0

)

Main problem

Study the asymptotic behaviour asε↓ 0of the solutionuε: [0, T ] → Hof the

gradient flow _

εuε0(t) = −DE(t, uε(t))

uε(0) = u0.

Formally, the limitushould be a suitable curve solving

DE(t, u(t)) = 0

In order to avoid a transition layer att =0we will assumeDE(0, u0) =0.

Singular limit of gradient flows (

ρ = 0

)

Main problem

Study the asymptotic behaviour asε↓ 0of the solutionuε: [0, T ] → Hof the

gradient flow _

εuε0(t) = −DE(t, uε(t))

uε(0) = u0.

Formally, the limitushould be a suitable curve solving

DE(t, u(t)) = 0

In order to avoid a transition layer att =0we will assumeDE(0, u0) =0.

Basic energy estimate

SetP(t, x) = ∂tE(t, x).Chain rule:

d dtE(t, x(t)) = hDE(t, x), x 0_{(t)i +P(t, x(t)).} εku0 ε(t)k 2₌ 1 εkDE(t, uε(t))k 2_{= −}d dtE(t, x(t)) + P(t, uε(t)), Energy identity E(T, uε(T ))+ ZT 0 ε 2ku 0 ε(t)k 2₊ 1 2εkDE(t, uε(t))k 2_{dt =E(0, u} 0)+ ZT 0 P(t, uε(t))dt If lim |x|→∞E(t, x) = +∞, P(t, x) 6 A + B|E(t, x)| Basic estimates kuε(t)k 6 C, E(t, uε(t))6 C, ZT 0 ku0 ε(t)k 2_dt 6 C/ε, ZT 0 kDE(t, uε(t)k2dt6 Cε. 12

Basic energy estimate

SetP(t, x) = ∂tE(t, x).Chain rule:

d dtE(t, x(t)) = hDE(t, x), x 0_{(t)i +P(t, x(t)).} εku0 ε(t)k 2₌ 1 εkDE(t, uε(t))k 2_{= −}d dtE(t, x(t)) + P(t, uε(t)), Energy identity E(T, uε(T ))+ ZT 0 ε 2ku 0 ε(t)k 2₊ 1 2εkDE(t, uε(t))k 2_{dt =E(0, u} 0)+ ZT 0 P(t, uε(t))dt If lim |x|→∞E(t, x) = +∞, P(t, x) 6 A + B|E(t, x)| Basic estimates kuε(t)k 6 C, E(t, uε(t))6 C, ZT 0 ku0 ε(t)k 2_dt 6 C/ε, ZT 0 kDE(t, uε(t)k2dt6 Cε. 12

Basic energy estimate

SetP(t, x) = ∂tE(t, x).Chain rule:

d dtE(t, x(t)) = hDE(t, x), x 0_{(t)i +P(t, x(t)).} εku0 ε(t)k 2₌ 1 εkDE(t, uε(t))k 2_{= −}d dtE(t, x(t)) + P(t, uε(t)), Energy identity E(T, uε(T ))+ ZT 0 ε 2ku 0 ε(t)k 2₊ 1 2εkDE(t, uε(t))k 2_{dt =E(0, u} 0)+ ZT 0 P(t, uε(t))dt If lim |x|→∞E(t, x) = +∞, P(t, x) 6 A + B|E(t, x)| Basic estimates kuε(t)k 6 C, E(t, uε(t))6 C, ZT 0 ku0 ε(t)k 2_dt 6 C/ε, ZT 0 kDE(t, uε(t)k2dt6 Cε. 12

Basic energy estimate

SetP(t, x) = ∂tE(t, x).Chain rule:

d dtE(t, x(t)) = hDE(t, x), x 0_{(t)i +P(t, x(t)).} εku0 ε(t)k 2₌ 1 εkDE(t, uε(t))k 2_{= −}d dtE(t, x(t)) + P(t, uε(t)), Energy identity E(T, uε(T ))+ ZT 0 ε 2ku 0 ε(t)k 2₊ 1 2εkDE(t, uε(t))k 2_{dt =E(0, u} 0)+ ZT 0 P(t, uε(t))dt If lim |x|→∞E(t, x) = +∞, P(t, x) 6 A + B|E(t, x)| Basic estimates kuε(t)k 6 C, E(t, uε(t))6 C, ZT 0 ku0 ε(t)k 2_dt 6 C/ε, ZT 0 kDE(t, uε(t)k2dt6 Cε. 12

Basic energy estimate

SetP(t, x) = ∂tE(t, x).Chain rule:

d dtE(t, x(t)) = hDE(t, x), x 0_{(t)i +P(t, x(t)).} εku0 ε(t)k 2₌ 1 εkDE(t, uε(t))k 2_{= −}d dtE(t, x(t)) + P(t, uε(t)), Energy identity E(T, uε(T ))+ ZT 0 ε 2ku 0 ε(t)k 2₊ 1 2εkDE(t, uε(t))k 2_{dt =E(0, u} 0)+ ZT 0 P(t, uε(t))dt If lim |x|→∞E(t, x) = +∞, P(t, x) 6 A + B|E(t, x)| Basic estimates kuε(t)k 6 C, E(t, uε(t))6 C, ZT 0 ku0 ε(t)k 2 dt 6 C/ε, ZT 0 kDE(t, uε(t)k2dt 6 Cε. 12

Basic energy estimate

SetP(t, x) = ∂tE(t, x).Chain rule:

d dtE(t, x(t)) = hDE(t, x), x 0_{(t)i +P(t, x(t)).} εku0 ε(t)k 2₌ 1 εkDE(t, uε(t))k 2_{= −}d dtE(t, x(t)) + P(t, uε(t)), Energy identity E(T, uε(T ))+ ZT 0 ε 2ku 0 ε(t)k 2₊ 1 2εkDE(t, uε(t))k 2_{dt =E(0, u} 0)+ ZT 0 P(t, uε(t))dt If lim |x|→∞E(t, x) = +∞, P(t, x) 6 A + B|E(t, x)| Basic estimates kuε(t)k 6 C, E(t, uε(t))6 C, ZT 0 ku0 ε(t)k 2 dt 6 C/ε, ZT 0 kDE(t, uε(t)k2dt 6 Cε. 12

Basic energy estimate

SetP(t, x) = ∂tE(t, x).Chain rule:

d dtE(t, x(t)) = hDE(t, x), x 0_{(t)i +P(t, x(t)).} εku0 ε(t)k 2₌ 1 εkDE(t, uε(t))k 2_{= −}d dtE(t, x(t)) + P(t, uε(t)), Energy identity E(T, uε(T ))+ ZT 0 ε 2ku 0 ε(t)k 2₊ 1 2εkDE(t, uε(t))k 2_{dt =E(0, u} 0)+ ZT 0 P(t, uε(t))dt If lim |x|→∞E(t, x) = +∞, P(t, x) 6 A + B|E(t, x)| Basic estimates kuε(t)k 6 C, E(t, uε(t))6 C, ZT 0 ku0 ε(t)k 2 dt 6 C/ε, ZT 0 kDE(t, uε(t)k2dt 6 Cε. 12

Basic energy estimate

SetP(t, x) = ∂tE(t, x).Chain rule:

d dtE(t, x(t)) = hDE(t, x), x 0_{(t)i +P(t, x(t)).} εku0 ε(t)k 2₌ 1 εkDE(t, uε(t))k 2_{= −}d dtE(t, x(t)) + P(t, uε(t)), Energy identity E(T, uε(T ))+ ZT 0 ε 2ku 0 ε(t)k 2₊ 1 2εkDE(t, uε(t))k 2_{dt =E(0, u} 0)+ ZT 0 P(t, uε(t))dt If lim |x|→∞E(t, x) = +∞, P(t, x) 6 A + B|E(t, x)| Basic estimates kuε(t)k 6 C, E(t, uε(t))6 C, ZT 0 ku0 ε(t)k 2 dt 6 C/ε, ZT 0 kDE(t, uε(t)k2dt 6 Cε. 12

Main difficulties

I Lack of compactness in time: hard to prove BV estimates ([Krejci ’2005]: regulated function, 1-D)

I Characterization of the limit.

Formally, we may expect that a limit curveuprovides an

“evolution by critical points ofE”

satisfying

DE(t, u(t)) = 0, i.e. u(t)∈ C(t) t∈ [0, T ].

WhenEis uniformly convexC(t)contains only the minimizer ofE(t, ·)so that the limit evolution is uniquely characterized.

WhenEis not convex, one may expectjumps and more complex bifurcation

behaviour whenu(t)hits adegenerate critical point of Cd:=

(t, x) ∈ C(t) : D2

xxE(t, x) is not invertible 

.

Main difficulties

I Lack of compactness in time: hard to prove BV estimates ([Krejci ’2005]: regulated function, 1-D)

I Characterization of the limit.

Formally, we may expect that a limit curveuprovides an

“evolution by critical points ofE”

satisfying

DE(t, u(t)) = 0, i.e. u(t)∈ C(t) t∈ [0, T ].

WhenEis uniformly convexC(t)contains only the minimizer ofE(t, ·)so that the limit evolution is uniquely characterized.

WhenEis not convex, one may expectjumps and more complex bifurcation

behaviour whenu(t)hits adegenerate critical point of Cd:=

(t, x) ∈ C(t) : D2

xxE(t, x) is not invertible 

.

Main difficulties

I Lack of compactness in time: hard to prove BV estimates ([Krejci ’2005]: regulated function, 1-D)

I Characterization of the limit.

Formally, we may expect that a limit curveuprovides an “evolution by critical points ofE” satisfying

DE(t, u(t)) = 0, i.e. u(t)∈ C(t) t∈ [0, T ].

WhenEis uniformly convexC(t)contains only the minimizer ofE(t, ·)so that the limit evolution is uniquely characterized.

WhenEis not convex, one may expectjumps and more complex bifurcation

behaviour whenu(t)hits adegenerate critical point of Cd:=

(t, x) ∈ C(t) : D2

xxE(t, x) is not invertible 

.

Main difficulties

I Lack of compactness in time: hard to prove BV estimates ([Krejci ’2005]: regulated function, 1-D)

I Characterization of the limit.

Formally, we may expect that a limit curveuprovides an “evolution by critical points ofE” satisfying

DE(t, u(t)) = 0, i.e. u(t)∈ C(t) t∈ [0, T ].

WhenEis uniformly convexC(t)contains only the minimizer ofE(t, ·)so that the limit evolution is uniquely characterized.

WhenEis not convex, one may expectjumps and more complex bifurcation

behaviour whenu(t)hits adegenerate critical point of Cd:=

(t, x) ∈ C(t) : D2

xxE(t, x) is not invertible 

.

Main difficulties

I Lack of compactness in time: hard to prove BV estimates ([Krejci ’2005]: regulated function, 1-D)

I Characterization of the limit.

Formally, we may expect that a limit curveuprovides an “evolution by critical points ofE” satisfying

DE(t, u(t)) = 0, i.e. u(t)∈ C(t) t∈ [0, T ].

WhenEis uniformly convexC(t)contains only the minimizer ofE(t, ·)so that the limit evolution is uniquely characterized.

WhenEis not convex, one may expectjumpsand more complex bifurcation behaviour whenu(t)hits adegenerate critical pointof

Cd:=

(t, x) ∈ C(t) : D2

xxE(t, x) is not invertible

 .

The critical set (1D)

Jumps between curves

Outline

1 Rate-independent evolution and singular perturbation of gradient flows

2 Transversality conditions for the critical set

3 Compactness and variational characterization of the limit evolution

4 A useful tool: graph convergence

Strong transversality [Zanini, ’08]

Convergence can be proved by assuming suitablystrong transversality conditionson the set of degenerate critical pointsCd.

I Cdis finite;

I any candidate limit evolution can reachCdonly at points where D2_{E(t, x) > 0} I if(t, x) ∈ Cdthen     

Ker D2_{E(t, x)
= span v,} D3E(t, x)[v, v, v] 6= 0

∂tDE(t, x)[v] 6= 0

In1Dthe above conditions mean thatG(t, x) := ∂xE(t, x)satisfies

whenever G(t, x) = 0, ∂xG(t, x) = 0 then∂tG(t, x) 6= 0, ∂2xxG(t, x) 6= 0.

Strong transversality implies thatC(t)is discrete for everyt∈ [0, T ].

[Agostiniani-Rossi, Scilla-Solombrino].

It is a “generic” condition, in the following sense: ifEisC4_{, for aG}

δdense set of g_{∈ H}andQ∈L (H, H)the perturbed energy

Eg,Q(t, x) :=E(t, x) − hg, xi − hQx, xi

satisfies the strong transversality conditions.

Strong transversality [Zanini, ’08]

Convergence can be proved by assuming suitablystrong transversality conditionson the set of degenerate critical pointsCd.

I Cdis finite;

I any candidate limit evolution can reachCdonly at points where D2_{E(t, x) > 0} I if(t, x) ∈ Cdthen     

Ker D2_{E(t, x)
= span v,} D3E(t, x)[v, v, v] 6= 0

∂tDE(t, x)[v] 6= 0

In1Dthe above conditions mean thatG(t, x) := ∂xE(t, x)satisfies

whenever G(t, x) = 0, ∂xG(t, x) = 0 then∂tG(t, x) 6= 0, ∂2xxG(t, x) 6= 0.

Strong transversality implies thatC(t)is discrete for everyt∈ [0, T ].

[Agostiniani-Rossi, Scilla-Solombrino].

It is a “generic” condition, in the following sense: ifEisC4_{, for aG}

δdense set of g_{∈ H}andQ∈L (H, H)the perturbed energy

Eg,Q(t, x) :=E(t, x) − hg, xi − hQx, xi

satisfies the strong transversality conditions.

Strong transversality [Zanini, ’08]

Convergence can be proved by assuming suitablystrong transversality conditionson the set of degenerate critical pointsCd.

I Cdis finite;

I any candidate limit evolution can reachCdonly at points where

D2_{E(t, x) > 0} I if(t, x) ∈ Cdthen     

Ker D2_{E(t, x)
= span v,} D3E(t, x)[v, v, v] 6= 0

∂tDE(t, x)[v] 6= 0

In1Dthe above conditions mean thatG(t, x) := ∂xE(t, x)satisfies

whenever G(t, x) = 0, ∂xG(t, x) = 0 then∂tG(t, x) 6= 0, ∂2xxG(t, x) 6= 0.

Strong transversality implies thatC(t)is discrete for everyt∈ [0, T ].

[Agostiniani-Rossi, Scilla-Solombrino].

It is a “generic” condition, in the following sense: ifEisC4_{, for aG}

δdense set of g_{∈ H}andQ∈L (H, H)the perturbed energy

Eg,Q(t, x) :=E(t, x) − hg, xi − hQx, xi

satisfies the strong transversality conditions.

Strong transversality [Zanini, ’08]

Convergence can be proved by assuming suitablystrong transversality conditionson the set of degenerate critical pointsCd.

I Cdis finite;

I any candidate limit evolution can reachCdonly at points where

D2_{E(t, x) > 0} I if(t, x) ∈ Cdthen     

Ker D2_{E(t, x)
= span v,}

D3E(t, x)[v, v, v] 6= 0 ∂tDE(t, x)[v] 6= 0

In1Dthe above conditions mean thatG(t, x) := ∂xE(t, x)satisfies

whenever G(t, x) = 0, ∂xG(t, x) = 0 then∂tG(t, x) 6= 0, ∂2xxG(t, x) 6= 0.

Strong transversality implies thatC(t)is discrete for everyt∈ [0, T ].

[Agostiniani-Rossi, Scilla-Solombrino].

It is a “generic” condition, in the following sense: ifEisC4_{, for aG}

δdense set of g_{∈ H}andQ∈L (H, H)the perturbed energy

Eg,Q(t, x) :=E(t, x) − hg, xi − hQx, xi

satisfies the strong transversality conditions.

Strong transversality [Zanini, ’08]

Convergence can be proved by assuming suitablystrong transversality conditionson the set of degenerate critical pointsCd.

I Cdis finite;

I any candidate limit evolution can reachCdonly at points where

D2_{E(t, x) > 0} I if(t, x) ∈ Cdthen     

Ker D2_{E(t, x)
= span v,}

D3E(t, x)[v, v, v] 6= 0 ∂tDE(t, x)[v] 6= 0

In1Dthe above conditions mean thatG(t, x) := ∂xE(t, x)satisfies

whenever G(t, x) = 0, ∂xG(t, x) = 0 then∂tG(t, x) 6= 0, ∂2xxG(t, x) 6= 0. Strong transversality implies thatC(t)is discrete for everyt∈ [0, T ].

[Agostiniani-Rossi, Scilla-Solombrino].

It is a “generic” condition, in the following sense: ifEisC4_{, for aG}

δdense set of g_{∈ H}andQ∈L (H, H)the perturbed energy

Eg,Q(t, x) :=E(t, x) − hg, xi − hQx, xi

satisfies the strong transversality conditions.

Strong transversality [Zanini, ’08]

Convergence can be proved by assuming suitablystrong transversality conditionson the set of degenerate critical pointsCd.

I Cdis finite;

I any candidate limit evolution can reachCdonly at points where

D2_{E(t, x) > 0} I if(t, x) ∈ Cdthen     

Ker D2_{E(t, x)
= span v,}

D3E(t, x)[v, v, v] 6= 0 ∂tDE(t, x)[v] 6= 0

In1Dthe above conditions mean thatG(t, x) := ∂xE(t, x)satisfies

whenever G(t, x) = 0, ∂xG(t, x) = 0 then∂tG(t, x) 6= 0, ∂2xxG(t, x) 6= 0. Strong transversality implies thatC(t)is discrete for everyt∈ [0, T ].

[Agostiniani-Rossi, Scilla-Solombrino].

It is a “generic” condition, in the following sense: ifEisC4_{, for aG}

δdense set of

g_{∈ H}andQ∈L (H, H)the perturbed energy

Eg,Q(t, x) :=E(t, x) − hg, xi − hQx, xi satisfies the strong transversality conditions.

Transversality of the critical set

Simpler Generic conditions

Eg(t, x) :=E(t, x) − hg, xi, DEg(t, x) = DE(t, x) − g.

A generic decomposition ofC[Sard, Quinn, Hirsch, Simon; Saut-Temam]

The setOofg∈ Hsuch that the total differential

dDEg(t, x) ∈L (R × H; H) is surjective for every(t, x) ∈ C is open, dense, and its complement isLd_-negligible.

Ifg∈O,Ccan be decomposed as the disjoint union of at most countableC1

curves. In particular

Cis countably(H1_{, 1)-rectifiable.} I E(t, ·)is constant on every connected component ofC(t).

I There is an at most countable set of timesN⊂ [0, T ]such thatC(t)is not totally disconnected:

- ift∈ [0, T ]\ Nthen every connected component ofC(t)is reduced to a point,

- ift∈ NthenC(t)has “larger” connected components.

Simpler Generic conditions

Eg(t, x) :=E(t, x) − hg, xi, DEg(t, x) = DE(t, x) − g.

A generic decomposition ofC[Sard, Quinn, Hirsch, Simon; Saut-Temam]

The setOofg∈ Hsuch that the total differential

dDEg(t, x) ∈L (R × H; H) is surjective for every(t, x) ∈ C is open, dense, and its complement isLd_-negligible.

Ifg∈O,Ccan be decomposed as the disjoint union of at most countableC1 curves. In particular

Cis countably(H1_{, 1)-rectifiable.} I E(t, ·)is constant on every connected component ofC(t).

I There is an at most countable set of timesN⊂ [0, T ]such thatC(t)is not totally disconnected:

- ift∈ [0, T ]\ Nthen every connected component ofC(t)is reduced to a point,

- ift∈ NthenC(t)has “larger” connected components.

Simpler Generic conditions

Eg(t, x) :=E(t, x) − hg, xi, DEg(t, x) = DE(t, x) − g.

A generic decomposition ofC[Sard, Quinn, Hirsch, Simon; Saut-Temam]

The setOofg∈ Hsuch that the total differential

dDEg(t, x) ∈L (R × H; H) is surjective for every(t, x) ∈ C is open, dense, and its complement isLd_-negligible.

Ifg∈O,Ccan be decomposed as the disjoint union of at most countableC1 curves. In particular

Cis countably(H1_{, 1)-rectifiable.} I E(t, ·)is constant on every connected component ofC(t).

I There is an at most countable set of timesN⊂ [0, T ]such thatC(t)is not totally disconnected:

- ift∈ [0, T ]\ Nthen every connected component ofC(t)is reduced to a point,

- ift∈ NthenC(t)has “larger” connected components.

Simpler Generic conditions

Eg(t, x) :=E(t, x) − hg, xi, DEg(t, x) = DE(t, x) − g.

A generic decomposition ofC[Sard, Quinn, Hirsch, Simon; Saut-Temam]

The setOofg∈ Hsuch that the total differential

dDEg(t, x) ∈L (R × H; H) is surjective for every(t, x) ∈ C is open, dense, and its complement isLd_-negligible.

Ifg∈O,Ccan be decomposed as the disjoint union of at most countableC1 curves. In particular

Cis countably(H1_{, 1)-rectifiable.} I E(t, ·)is constant on every connected component ofC(t).

I There is an at most countable set of timesN⊂ [0, T ]such thatC(t)is not totally disconnected:

- ift∈ [0, T ]\ Nthen every connected component ofC(t)is reduced to a point,

- ift∈ NthenC(t)has “larger” connected components.

A simple example in infinite dimension

LetΩbe a bounded connected open set ofR3,H := L2(Ω), D(E) := H2_{(Ω)∩ H}1 0(Ω). Consider E(u) := Z Ω ₁ 2|∇u(x)| 2 + W(u(x))  dx, f∈ C2 ([0, T ]; L2(Ω)), E(t, u) = E(u) − Z Ω f(t, x)u(x) dx, ifu∈ D(E).

−∆u(t, x) + W0(u(t, x)) − g(x) = f(t, x) inΩ, u(t, ·) = 0on∂Ω.

For a denseGδ-subsetOinL2_{(Ω)the energyEg,g∈O, satisfies the} transversality condition and the critical set is countably(H1_{, 1)-rectifiable.}

One can also consider genericity w.r.t.Ωor w.r.t. the coefficients of the elliptic operator.

A simple example in infinite dimension

LetΩbe a bounded connected open set ofR3,H := L2(Ω), D(E) := H2_{(Ω)∩ H}1 0(Ω). Consider E(u) := Z Ω ₁ 2|∇u(x)| 2 + W(u(x))  dx, f∈ C2 ([0, T ]; L2(Ω)), E(t, u) = E(u) − Z Ω f(t, x)u(x) dx, ifu∈ D(E).

−∆u(t, x) + W0(u(t, x)) − g(x) = f(t, x) inΩ, u(t, ·) = 0on∂Ω.

For a denseGδ-subsetOinL2_{(Ω)the energyEg,g∈O, satisfies the} transversality condition and the critical set is countably(H1_{, 1)-rectifiable.} One can also consider genericity w.r.t.Ωor w.r.t. the coefficients of the elliptic operator.

Outline

1 Rate-independent evolution and singular perturbation of gradient flows

2 Transversality conditions for the critical set

3 Compactness and variational characterization of the limit evolution

4 A useful tool: graph convergence

The main compactness result

Theorem (Agostiniani-Rossi-S.)

Suppose thatCis countably(H1_{, 1)-rectifiable (it is sufficient thatH}1_isσ-finite onC)

Then there exists:

I a subsequencen7→ ε(n) ↓ 0

I a limit curveu : [0, T ] → Hsuch that

lim

n→_∞uε(n)(t) = u(t) for everyt∈ [0, T ].

Properties of limit solutions

Letu : [0, T ] → Hbe a limit solution arising from the previous compactness result.

For the sake of simplicity, we suppose thatC(t)is totally disconnected for every

t∈ [0, T ], i.e.N =∅.

I uis regulated, i.e. for everyt∈ [0, T ] ∃ lims↑tu(s) = u(t−),

∃ lims↓tu(s) = u(t+),

J(u) :={t ∈ [0, T] : u(t) 6= u(t±)}is at most countable.

I u(t)∈ C(t)for everyt∈ [0, T ]\ J(u); ift∈ J(u)thenu(t−), u(t+) ∈ C(t).

I The mapt7→E(t, u(t))hasbounded variation,

its jump set coincides with the jump set ofu

for every0 6 s < t 6 Ttheenergy balance holds: E(t, u(t−))+ X

r∈J(u)∩(s,t)

c(r; u(r−), u(r+)) =E(s, u(s+))+ Zt

s

P(r, u(r)) dr

Properties of limit solutions

Letu : [0, T ] → Hbe a limit solution arising from the previous compactness result.

For the sake of simplicity, we suppose thatC(t)is totally disconnected for every

t∈ [0, T ], i.e.N =∅.

I uis regulated, i.e. for everyt∈ [0, T ] ∃ lims↑tu(s) = u(t−),

∃ lims↓tu(s) = u(t+),

J(u) :={t ∈ [0, T] : u(t) 6= u(t±)}is at most countable.

I u(t)∈ C(t)for everyt∈ [0, T ]\ J(u); ift∈ J(u)thenu(t−), u(t+) ∈ C(t).

I The mapt7→E(t, u(t))hasbounded variation,

its jump set coincides with the jump set ofu

for every0 6 s < t 6 Ttheenergy balance holds: E(t, u(t−))+ X

r∈J(u)∩(s,t)

c(r; u(r−), u(r+)) =E(s, u(s+))+ Zt

s

P(r, u(r)) dr

Properties of limit solutions

Letu : [0, T ] → Hbe a limit solution arising from the previous compactness result.

For the sake of simplicity, we suppose thatC(t)is totally disconnected for every

t∈ [0, T ], i.e.N =∅.

I uis regulated, i.e. for everyt∈ [0, T ] ∃ lims↑tu(s) = u(t−),

∃ lims↓tu(s) = u(t+),

J(u) :={t ∈ [0, T] : u(t) 6= u(t±)}is at most countable.

I u(t)∈ C(t)for everyt∈ [0, T ]\ J(u); ift∈ J(u)thenu(t−), u(t+) ∈ C(t). I The mapt7→E(t, u(t))hasbounded variation,

its jump set coincides with the jump set ofu

for every0 6 s < t 6 Ttheenergy balance holds: E(t, u(t−))+ X

r∈J(u)∩(s,t)

c(r; u(r−), u(r+)) =E(s, u(s+))+ Zt

s

P(r, u(r)) dr

Properties of limit solutions

Letu : [0, T ] → Hbe a limit solution arising from the previous compactness result.

For the sake of simplicity, we suppose thatC(t)is totally disconnected for every

t∈ [0, T ], i.e.N =∅.

I uis regulated, i.e. for everyt∈ [0, T ] ∃ lims↑tu(s) = u(t−),

∃ lims↓tu(s) = u(t+),

J(u) :={t ∈ [0, T] : u(t) 6= u(t±)}is at most countable.

I u(t)∈ C(t)for everyt∈ [0, T ]\ J(u); ift∈ J(u)thenu(t−), u(t+) ∈ C(t). I The mapt7→E(t, u(t))hasbounded variation,

its jump set coincides with the jump set ofu

for every0 6 s < t 6 Ttheenergy balanceholds:

E(t, u(t−))+ X

r∈J(u)∩(s,t)

c(r; u(r−), u(r+)) =E(s, u(s+))+ Zt

s

P(r, u(r)) dr

The transition cost

c

At every jumpt∈ J(u), the energy dissipation corresponds to theoptimal transition cost:

E(t, u(t−)) − E(t, u(t+)) = c(t; u(t−), u(t+)) For everyt∈ [0, T ]the transition costc(t; u−, u+)between two points u−, u+∈ His given by the “Finsler” metric induced bykDEk:

c(t; u−, u+) :=inf Z Ω(ϑ) kDE(t, ϑ(τ))k kϑ0_{(τ))k dτ :} ϑ_{∈ C([0, 1]; H), ϑ(0) = u}−, ϑ(1) = u+ Ω(ϑ) :={τ : ϑ(τ) 6∈ C(t)}

ϑ∈ Liploc(Ω), E(t, ϑ(·)) ∈ Lip([0, 1]) 

We always have

E(t, u−) −E(t, u+)6 c(t; u−, u+)

The transition cost

c

At every jumpt∈ J(u), the energy dissipation corresponds to theoptimal transition cost:

E(t, u(t−)) − E(t, u(t+)) = c(t; u(t−), u(t+))

For everyt∈ [0, T ]the transition costc(t; u−, u+)between two points

u−, u+∈ His given by the “Finsler” metric induced bykDEk:

c(t; u−, u+) :=inf Z Ω(ϑ) kDE(t, ϑ(τ))k kϑ0_{(τ))k dτ :} ϑ_{∈ C([0, 1]; H), ϑ(0) = u}−, ϑ(1) = u+ Ω(ϑ) :={τ : ϑ(τ) 6∈ C(t)}

ϑ∈ Liploc(Ω), E(t, ϑ(·)) ∈ Lip([0, 1]) 

We always have

E(t, u−) −E(t, u+)6 c(t; u−, u+)

The transition cost

c

At every jumpt∈ J(u), the energy dissipation corresponds to theoptimal transition cost:

E(t, u(t−)) − E(t, u(t+)) = c(t; u(t−), u(t+))

For everyt∈ [0, T ]the transition costc(t; u−, u+)between two points

u−, u+∈ His given by the “Finsler” metric induced bykDEk:

c(t; u−, u+) :=inf Z Ω(ϑ) kDE(t, ϑ(τ))k kϑ0_{(τ))k dτ :} ϑ_{∈ C([0, 1]; H), ϑ(0) = u}−, ϑ(1) = u+ Ω(ϑ) :={τ : ϑ(τ) 6∈ C(t)}

ϑ∈ Liploc(Ω), E(t, ϑ(·)) ∈ Lip([0, 1]) 

We always have

E(t, u−) −E(t, u+)6 c(t; u−, u+)

The transition cost

c

At every jumpt∈ J(u), the energy dissipation corresponds to theoptimal transition cost:

E(t, u(t−)) − E(t, u(t+)) = c(t; u(t−), u(t+))

For everyt∈ [0, T ]the transition costc(t; u−, u+)between two points

u−, u+∈ His given by the “Finsler” metric induced bykDEk:

c(t; u−, u+) :=inf Z Ω(ϑ) kDE(t, ϑ(τ))k kϑ0_{(τ))k dτ :} ϑ_{∈ C([0, 1]; H), ϑ(0) = u}−, ϑ(1) = u+ Ω(ϑ) :={τ : ϑ(τ) 6∈ C(t)}

ϑ∈ Liploc(Ω), E(t, ϑ(·)) ∈ Lip([0, 1]) 

We always have

E(t, u−) −E(t, u+)6 c(t; u−, u+)

The transition cost

c

At every jumpt∈ J(u), the energy dissipation corresponds to theoptimal transition cost:

E(t, u(t−)) − E(t, u(t+)) = c(t; u(t−), u(t+))

For everyt∈ [0, T ]the transition costc(t; u−, u+)between two points

u−, u+∈ His given by the “Finsler” metric induced bykDEk:

c(t; u−, u+) :=inf Z Ω(ϑ) kDE(t, ϑ(τ))k kϑ0_{(τ))k dτ :} ϑ_{∈ C([0, 1]; H), ϑ(0) = u}−, ϑ(1) = u+ Ω(ϑ) :={τ : ϑ(τ) 6∈ C(t)}

ϑ∈ Liploc(Ω), E(t, ϑ(·)) ∈ Lip([0, 1])

 We always have

E(t, u−) −E(t, u+)6 c(t; u−, u+)

The transition cost

c

At every jumpt∈ J(u), the energy dissipation corresponds to theoptimal transition cost:

E(t, u(t−)) − E(t, u(t+)) = c(t; u(t−), u(t+))

For everyt∈ [0, T ]the transition costc(t; u−, u+)between two points

u−, u+∈ His given by the “Finsler” metric induced bykDEk:

c(t; u−, u+) :=inf Z Ω(ϑ) kDE(t, ϑ(τ))k kϑ0_{(τ))k dτ :} ϑ_{∈ C([0, 1]; H), ϑ(0) = u}−, ϑ(1) = u+ Ω(ϑ) :={τ : ϑ(τ) 6∈ C(t)}

ϑ∈ Liploc(Ω), E(t, ϑ(·)) ∈ Lip([0, 1])

We always have

E(t, u−) −E(t, u+)6 c(t; u−, u+)

Transitions

Structure of limit solutions

E(t, u(t−)) + X

r∈J(u)∩(s,t)

c(r; u(r−), u(r+)) =E(s, u(s+)) + Zt

s

P(r, u(r)) dr

has two important consequences:

I At every jump pointt∈ J(u)there exists anoptimal transition ϑ∈ C([0, 1]; H)connectingu(t−)tou(t+)such that

E(t, u(t−)) − E(t, u(t+)) = c(t; u(t−), u(t+))

I in each connected component ofΩ(ϑ)⊂ [0, 1]whereϑ6∈ C(t)there is an increasing change of variableτ = τ(r), r ∈ (a, b), such that the

reparametrized transitionθ(r) := ϑ(τ(r))satisfiesthe gradient flow equation

d

drθ(r) = −DE(t, θ(r))

at the “frozen time”t.

Structure of limit solutions

E(t, u(t−)) + X

r∈J(u)∩(s,t)

c(r; u(r−), u(r+)) =E(s, u(s+)) + Zt

s

P(r, u(r)) dr

has two important consequences:

I At every jump pointt∈ J(u)there exists anoptimal transition

ϑ∈ C([0, 1]; H)connectingu(t−)tou(t+)such that

E(t, u(t−)) − E(t, u(t+)) = c(t; u(t−), u(t+))

I in each connected component ofΩ(ϑ)⊂ [0, 1]whereϑ6∈ C(t)there is an increasing change of variableτ = τ(r), r ∈ (a, b), such that the

reparametrized transitionθ(r) := ϑ(τ(r))satisfiesthe gradient flow equation

d

drθ(r) = −DE(t, θ(r))

at the “frozen time”t.

Structure of limit solutions

E(t, u(t−)) + X

r∈J(u)∩(s,t)

c(r; u(r−), u(r+)) =E(s, u(s+)) + Zt

s

P(r, u(r)) dr

has two important consequences:

I At every jump pointt∈ J(u)there exists anoptimal transition

ϑ∈ C([0, 1]; H)connectingu(t−)tou(t+)such that

E(t, u(t−)) − E(t, u(t+)) = c(t; u(t−), u(t+))

I in each connected component ofΩ(ϑ)⊂ [0, 1]whereϑ6∈ C(t)there is an increasing change of variableτ = τ(r), r ∈ (a, b), such that the

reparametrized transitionθ(r) := ϑ(τ(r))satisfiesthe gradient flow equation

d

drθ(r) = −DE(t, θ(r))

at the “frozen time”t.

Jumps in the smooth

1

-dimensional case: double-well potential

u(t) f(t) _E0_(u)

u(t−) u(t+)

Limit solution in the1-dimensional case for a strictly increasingf. The blue line represents the graph of the jump transitionϑ.

Outline

1 Rate-independent evolution and singular perturbation of gradient flows

2 Transversality conditions for the critical set

3 Compactness and variational characterization of the limit evolution

4 A useful tool: graph convergence

Main idea

Instead of studying the convergence ofuεwe consider thelimit of their graphs:

Gε:=

(t, uε(t)) : t∈ [0, T ]

⊂ [0, T ] × H ,

a family of compact sets.

We can useHausdorff-Kuratovski convergence:

Lsn→∞Kn:= y :∃ yn(k)∈ Kn(k), yn(k)→ y ask↑∞  Lin→∞Kn:= y :∃ yn∈ Kn: yn→ y asn↑∞  Kn K → K ⇔ K = Lsn→∞Kn=Lin→∞Kn Equivalently, lim

n→∞DH(Kn, K) = 0, DHis the Hausdorff distance.

For everyε >0there existsn¯∈ Nsuch that for everyn> ¯n Knis contained in theε-neighborhood ofK,

Kis contained in theε-neighborhood ofKn,

Main idea

Instead of studying the convergence ofuεwe consider thelimit of their graphs:

Gε:=

(t, uε(t)) : t∈ [0, T ]

⊂ [0, T ] × H ,

a family of compact sets.

We can useHausdorff-Kuratovski convergence:

Lsn→∞Kn:= y :∃ yn(k)∈ Kn(k), yn(k)→ y ask↑∞  Lin→∞Kn:= y :∃ yn∈ Kn: yn→ y asn↑∞  Kn K → K ⇔ K = Lsn→∞Kn=Lin→∞Kn Equivalently, lim

n→∞DH(Kn, K) = 0, DHis the Hausdorff distance.

For everyε >0there existsn¯∈ Nsuch that for everyn> ¯n Knis contained in theε-neighborhood ofK,

Kis contained in theε-neighborhood ofKn,

Main idea

Instead of studying the convergence ofuεwe consider thelimit of their graphs:

Gε:=

(t, uε(t)) : t∈ [0, T ]

⊂ [0, T ] × H ,

a family of compact sets.

We can useHausdorff-Kuratovski convergence:

Lsn→∞Kn:= y :∃ yn(k)∈ Kn(k), yn(k)→ y ask↑∞  Lin→∞Kn:= y :∃ yn∈ Kn: yn→ y asn↑∞  Kn K → K ⇔ K = Lsn→∞Kn=Lin→∞Kn Equivalently, lim

n→∞DH(Kn, K) = 0, DHis the Hausdorff distance.

For everyε >0there existsn¯∈ Nsuch that for everyn> ¯n Knis contained in theε-neighborhood ofK,

Kis contained in theε-neighborhood ofKn,

Main idea

Instead of studying the convergence ofuεwe consider thelimit of their graphs:

Gε:=

(t, uε(t)) : t∈ [0, T ]

⊂ [0, T ] × H ,

a family of compact sets.

We can useHausdorff-Kuratovski convergence:

Lsn→∞Kn:= y :∃ yn(k)∈ Kn(k), yn(k)→ y ask↑∞  Lin→∞Kn:= y :∃ yn∈ Kn: yn→ y asn↑∞  Kn K → K ⇔ K = Lsn→∞Kn=Lin→∞Kn Equivalently, lim

n→∞DH(Kn, K) = 0, DHis the Hausdorff distance.

For everyε >0there existsn¯∈ Nsuch that for everyn> ¯n Knis contained in theε-neighborhood ofK,

Kis contained in theε-neighborhood ofKn,

Main idea

Instead of studying the convergence ofuεwe consider thelimit of their graphs:

Gε:=

(t, uε(t)) : t∈ [0, T ]

⊂ [0, T ] × H ,

a family of compact sets.

We can useHausdorff-Kuratovski convergence:

Lsn→∞Kn:= y :∃ yn(k)∈ Kn(k), yn(k)→ y ask↑∞  Lin→∞Kn:= y :∃ yn∈ Kn: yn→ y asn↑∞  Kn K → K ⇔ K = Lsn→∞Kn=Lin→∞Kn Equivalently, lim

n→∞DH(Kn, K) = 0, DHis the Hausdorff distance.

For everyε >0there existsn¯∈ Nsuch that for everyn> ¯n Knis contained in theε-neighborhood ofK,

Kis contained in theε-neighborhood ofKn,

Main idea

Instead of studying the convergence ofuεwe consider thelimit of their graphs:

Gε:=

(t, uε(t)) : t∈ [0, T ]

⊂ [0, T ] × H ,

a family of compact sets.

We can useHausdorff-Kuratovski convergence:

Lsn→∞Kn:= y :∃ yn(k)∈ Kn(k), yn(k)→ y ask↑∞  Lin→∞Kn:= y :∃ yn∈ Kn: yn→ y asn↑∞  Kn K → K ⇔ K = Lsn→∞Kn=Lin→∞Kn Equivalently, lim

n→∞DH(Kn, K) = 0, DHis the Hausdorff distance. For everyε >0there existsn¯∈ Nsuch that for everyn> ¯n

Knis contained in theε-neighborhood ofK,

Kis contained in theε-neighborhood ofKn,

A general compactness property

Blashke compactness theorem

IfGεare contained in a common compact set, there exists a subsequence

n7→ ε(n) ↓ 0and a limit setGsuch thatGε(n) K

→ G.

If moreover the setsGεare connected then also the limitGis connected.

A general compactness property

Blashke compactness theorem

IfGεare contained in a common compact set, there exists a subsequence

n7→ ε(n) ↓ 0and a limit setGsuch thatGε(n) K

→ G.

If moreover the setsGεare connected then also the limitGis connected.

Examples

Examples

Examples

Examples

Examples

Examples

Examples

Examples

Properties of the limit graph

Blashke compactness theorem

There exists a subsequencen7→ ε(n) ↓ 0and a limit setGsuch that

Gε(n)=Graph(uε(n) K

→ G. I Gis compact;

I π1_{(G) = [0, T ],π}1_{(t, x) := t;}

I Gis connected and all its sectionsG(t) := G∩{t} × Hare connected. Two alternatives:

I G(t)⊂ C(t): thenG(t) ={u(t)}is a singleton;

I G(t)6⊂ C(t): thent∈ J(u)andG(t)contains an optimal transitionϑ

connectingu(t−)withu(t+)along which

E(t, u(t−)) − E(t, u(t+)) > c(t; u(t−), u(t+))

Properties of the limit graph

Blashke compactness theorem

There exists a subsequencen7→ ε(n) ↓ 0and a limit setGsuch that

Gε(n)=Graph(uε(n) K

→ G. I Gis compact;

I π1_{(G) = [0, T ],π}1_{(t, x) := t;}

I Gis connected and all its sectionsG(t) := G∩{t} × Hare connected. Two alternatives:

I G(t)⊂ C(t): thenG(t) ={u(t)}is a singleton;

I G(t)6⊂ C(t): thent∈ J(u)andG(t)contains an optimal transitionϑ

connectingu(t−)withu(t+)along which

E(t, u(t−)) − E(t, u(t+)) > c(t; u(t−), u(t+))

Properties of the limit graph

Blashke compactness theorem

There exists a subsequencen7→ ε(n) ↓ 0and a limit setGsuch that

Gε(n)=Graph(uε(n) K

→ G. I Gis compact;

I π1_{(G) = [0, T ],π}1_{(t, x) := t;}

I Gis connected and all its sectionsG(t) := G∩{t} × Hare connected. Two alternatives:

I G(t)⊂ C(t): thenG(t) ={u(t)}is a singleton;

I G(t)6⊂ C(t): thent∈ J(u)andG(t)contains an optimal transitionϑ

connectingu(t−)withu(t+)along which

E(t, u(t−)) − E(t, u(t+)) > c(t; u(t−), u(t+))

Properties of the limit graph

Blashke compactness theorem

There exists a subsequencen7→ ε(n) ↓ 0and a limit setGsuch that

Gε(n)=Graph(uε(n) K

→ G. I Gis compact;

I π1_{(G) = [0, T ],π}1_{(t, x) := t;}

I Gis connected and all its sectionsG(t) := G∩{t} × Hare connected.

Two alternatives:

I G(t)⊂ C(t): thenG(t) ={u(t)}is a singleton;

I G(t)6⊂ C(t): thent∈ J(u)andG(t)contains an optimal transitionϑ

connectingu(t−)withu(t+)along which

E(t, u(t−)) − E(t, u(t+)) > c(t; u(t−), u(t+))

Properties of the limit graph

Blashke compactness theorem

There exists a subsequencen7→ ε(n) ↓ 0and a limit setGsuch that

Gε(n)=Graph(uε(n) K

→ G. I Gis compact;

I π1_{(G) = [0, T ],π}1_{(t, x) := t;}

I Gis connected and all its sectionsG(t) := G∩{t} × Hare connected. Two alternatives:

I G(t)⊂ C(t): thenG(t) ={u(t)}is a singleton;

I G(t)6⊂ C(t): thent∈ J(u)andG(t)contains an optimal transitionϑ

connectingu(t−)withu(t+)along which

E(t, u(t−)) − E(t, u(t+)) > c(t; u(t−), u(t+))

Properties of the limit graph

Blashke compactness theorem

There exists a subsequencen7→ ε(n) ↓ 0and a limit setGsuch that

Gε(n)=Graph(uε(n) K

→ G. I Gis compact;

I π1_{(G) = [0, T ],π}1_{(t, x) := t;}

I Gis connected and all its sectionsG(t) := G∩{t} × Hare connected. Two alternatives:

I G(t)⊂ C(t): thenG(t) ={u(t)}is a singleton;

I G(t)6⊂ C(t): thent∈ J(u)andG(t)contains an optimal transitionϑ

connectingu(t−)withu(t+)along which

E(t, u(t−)) − E(t, u(t+)) > c(t; u(t−), u(t+))

Extensions

I The assumption thatC(t)is totally disconnected for everyt∈ [0, T ]can be removed, by working with possibly discontinuous functions at a countable set of timesN⊂ [0, T ]. For eacht∈ Nwe can only say thatu(s)

approaches a connected component ofC(t)ass→ t. I _Hmay have infinite dimensions,Ehas to be

λ-convex with compact sublevels.

I DE ∂E, the Fr ´echet subdifferential;C =(t, x) : ∂E(t, x) 3 0 ; I The assumption thatCis(H1_{, 1)-countably rectifiable is sufficient also in}

infinite dimension;

I A finer description is possible whenEis subanalytic.

Extensions

I The assumption thatC(t)is totally disconnected for everyt∈ [0, T ]can be removed, by working with possibly discontinuous functions at a countable set of timesN⊂ [0, T ]. For eacht∈ Nwe can only say thatu(s)

approaches a connected component ofC(t)ass→ t. I _Hmay have infinite dimensions,Ehas to be

λ-convex with compact sublevels.

I DE ∂E, the Fr ´echet subdifferential;C =(t, x) : ∂E(t, x) 3 0 ; I The assumption thatCis(H1_{, 1)-countably rectifiable is sufficient also in}

infinite dimension;

I A finer description is possible whenEis subanalytic.

Extensions

I The assumption thatC(t)is totally disconnected for everyt∈ [0, T ]can be removed, by working with possibly discontinuous functions at a countable set of timesN⊂ [0, T ]. For eacht∈ Nwe can only say thatu(s)

approaches a connected component ofC(t)ass→ t. I _Hmay have infinite dimensions,Ehas to be

λ-convex with compact sublevels.

I DE ∂E, the Fr ´echet subdifferential;C =(t, x) : ∂E(t, x) 3 0 ;

I The assumption thatCis(H1_{, 1)-countably rectifiable is sufficient also in}

infinite dimension;

I A finer description is possible whenEis subanalytic.

Extensions

I The assumption thatC(t)is totally disconnected for everyt∈ [0, T ]can be removed, by working with possibly discontinuous functions at a countable set of timesN⊂ [0, T ]. For eacht∈ Nwe can only say thatu(s)

approaches a connected component ofC(t)ass→ t. I _Hmay have infinite dimensions,Ehas to be

λ-convex with compact sublevels.

I DE ∂E, the Fr ´echet subdifferential;C =(t, x) : ∂E(t, x) 3 0 ;

I The assumption thatCis(H1_{, 1)-countably rectifiable is sufficient also in} infinite dimension;

I A finer description is possible whenEis subanalytic.

Extensions

I The assumption thatC(t)is totally disconnected for everyt∈ [0, T ]can be removed, by working with possibly discontinuous functions at a countable set of timesN⊂ [0, T ]. For eacht∈ Nwe can only say thatu(s)

approaches a connected component ofC(t)ass→ t. I _Hmay have infinite dimensions,Ehas to be

λ-convex with compact sublevels.

I DE ∂E, the Fr ´echet subdifferential;C =(t, x) : ∂E(t, x) 3 0 ;

I The assumption thatCis(H1_{, 1)-countably rectifiable is sufficient also in} infinite dimension;

I A finer description is possible whenEis subanalytic.