Generalized characteristics, singularities, and Lax ...acesar/slides MB60/Cannarsa.pdf ·...

Generalized characteristics, singularities, andLax-Oleinik operators

Piermarco Cannarsa

University of Rome “Tor Vergata”

NONLINEAR PDES: OPTIMAL CONTROL, ASYPTOTIC PROBLEMS,AND MEAN FIELD GAMES

Conference on the occasion of Martino Bardi’s 60th birthdayUniversity of Padova February 25-26, 2016

P. Cannarsa (Rome Tor Vergata) singular dynamics 25/02/2016 1 / 26

Outline

Outline


The euclidean distance function

Ω ⊂ RN bounded open set

dΩ(x) = miny∈∂Ω |x − y | x ∈ Ω

dΩ locally Lipschitz in Ω

dΩ solution of eikonal equation|Du|2 − 1 = 0 a.e. in Ω

u = 0 on ∂Ω

singular setΣ(dΩ) =

x ∈ Ω | proj∂Ω(x) multivalued

6= ∅

also known as medial axis (computer graphics)


'

&

$

%

ΩΣ(dΩ)

x

dΩ(x) -

∇dΩ(x) rrx

rr

6rrr

rr

r

r

rr

@@@@@

@@@

@@

Figure: distance function dΩ and singular set Σ(dΩ)


A problem in homotopy theory

Problem

Ω and Σ(dΩ) same homotopy?

'

&

$

%

ΩΣ(dΩ)

x∇dΩ(x) r6r

rrr

r

r

rr

@@@@@

@@@

@@

Figure: deformation retract techniqueP. Cannarsa (Rome Tor Vergata) singular dynamics 25/02/2016 5 / 26

The deformation retract techniqueF. Wolter (1993): deformation retract technique works if

Ω ⊂ RN and ∂Ω ∈ C2

Ω ⊂ R2 and ∂Ω is piecewise C2

Figure: r(·) discontinuous along edges above A,B,C and D


Homotopy equivalence for general open sets

in general

Theorem (A. Lieutier, Computer-Aided Design 2004)

Ω has the same homotopy type as Σ(dΩ)

proof by Lieutier is self-contained but technical

will show easy corollary of the invariance of Σ(dΩ) under gradient flow of dΩ


Semiconcave solutions of HJB equations

u : Ω→ R semiconcave H = H(x , u, p) continuous

H(x , u(x),Du(x)

)= 0 x ∈ Ω ⊂ RN a.e.

where H(x , u, ·) is convex with strictly convex level sets ∀(x , u) ∈ Ω× Rx0 ∈ Σ(u)

Definition

singularity at x0 propagates

∃δ > 0 and xkk ⊂ Σ(u) \ x0 such that xk → x0 with diam(D+u(xk )

)> δ

observe

for any x ∈ Ω H(x , u(x), p) = 0 ∀p ∈ D∗u(x)

x ∈ Σ(u) ⇐⇒ minp∈D+u(x) H(x , u(x), p)<0


Generalized characteristics (GC)u : Ω→ R semiconcave H(x , u, ·) ∈ C1(Rn)

Definition (Dafermos 1977, Albano – C 2000)

Lipschitz arc ξ(·) : [0, τ)→ Ω generalized characteristic for (u,H)

ξ(t)∈ co DpH(ξ(t), u

(ξ(t)

),D+u

(ξ(t)

))for a.e. t ∈ [0, τ)

Theorem

x0 ∈ Ω and p0 ∈ D+u(x0)

H(x0, u(x0), p0

)= min

p∈D+u(x0)H(x0, u(x0), p

)then ∃ξ : [0, τ ]→ Ω generalized characteristic for (u,H) starting at x0

ξ+(0) = DpH(x0, u(x0), p0

)& lim

t→0+sup

s∈[0,t]|ξ(s)− ξ+(0)| = 0

ref: [Albano – C 2000] , [C – Yu 2009]P. Cannarsa (Rome Tor Vergata) singular dynamics 25/02/2016 9 / 26

Propagation of singularities along GC

generalized characteristics

ξ(t)∈ co DpH(ξ(t), u

(ξ(t)

),D+u

(ξ(t)

))for a.e. t ∈ [0, τ)

may well be constant however:

Theorem (Albano – C 2000, Yu 2006, C – Yu 2009)

u : Ω→ R semiconcave solution H(x , u,Du) = 0 a.e. in Ω

x0 ∈ Σ(u) such that 0 /∈ DpH(x0, u(x0),D+u(x0)

)then ∃ξ : [0, τ)→ Ω generalized characteristic for (u,H) with

ξ(0) = x0

ξ+(0) 6= 0

ξ(t) ∈ Σ(u) for all t ∈ [0, τ)


References

C – Soner (1987, 1989)

Ambrosio – C – Soner (1993)

Albano – C (1999, 2000, 2002)

Albano (2002)

C – Sinestrari (2004)

Bogaevsky (2006)

Yu (2006, 2007)

C – Yu (2009)

Stromberg (2013)

C – Cheng – Zhang (2014)

Khanin – Sobolevski (2014)

Stromberg – Ahmadzdeh (2014)


A class of equations in weak KAM theory

TN = N−dimensional flat torusV (·) smooth TN−periodic function〈A(·)·, ·〉 is a smooth Tn-periodic Riemannian metric on Rn

Theorem

For each c ∈ RN there is a unique real number α(c) such that the (cell) problem

H(x ,Du(x) + c) = α(c) (x ∈ TN)

with H(x , p) = 12

⟨A(x)p, p

⟩+ V (x) has a TN−periodic (weak KAM) solution uc .

Moreover,

uc is semiconcave in Rn

α(·) is convex and minRN α = α(0) = maxTn V (Mane’s critical value)

Lions, Papanicolau, and Varadhan (circa 1988)Evans (1992)Fathi (2007)


A local propagation result

Theorem (C – Cheng – Zhang, 2014)

Fix c ∈ RN and let uc be a weak KAM solution of

H(x ,Du(x) + c) = α(c) (x ∈ TN)

whereH(x , p) =

12⟨A(x)p, p

⟩+ V (x)

Assume

the energy condition α(c) > maxTn V

x0 ∈ Σ(uc)

Then there exists ξ : [0, τ)→ Σ(uc) Lipschitz such that ξ(0) = x0 and ξ+(0) 6= 0

Ωx0


Replicating structure of the singular set

Theorem (C – Yu, 2009)

Fix c ∈ RN and let uc be a weak KAM solution of

H(x ,Du(x) + c) = α(c) (x ∈ TN)

whereH(x , p) =

12⟨A(x)p, p

⟩+ V (x)

Assume

the energy condition α(c) > maxTn V

x0 ∈ Ω ⊂ RN bounded with ∂Ω ∼ SN−1

Thenx0 ∈ Σ(uc) =⇒ ∂Ω ∩ Σ(uc) 6= ∅

Ωx0


An invariance problem

u : Ω→ R semiconcave H(x , u,Du) = 0

Problem

is Σ(u) invariant (viable) for the generalized characteristic flow ?

equivalently:

given x0 ∈ Σ(u)

does there exist a generalized characteristic ξξ ∈ co DpH

(ξ, u(ξ),D+u(ξ)

)ξ(0) = x0

such that ξ(t) ∈ Σ(u) for all t ≥ 0?

In general, the answer is negative but. . .


Invariance of Σ(dΩ) for generalized gradient flow

back to the distance function dΩ(x) = miny∈∂Ω |x − y | x ∈ Ω

1 ∀x0 ∈ Ω there exists a unique ξ : [0,∞)→ Ω such thatξ(t) ∈ D+dΩ

(ξ(t)

)t ∈ [0,∞) a.e.

ξ(0) = x0

2 |ξ(t)| = min|p| : p ∈ D+dΩ

(ξ(t)

) 3 ξ(t) ∈ Σ(dΩ) ⇐⇒ |ξ(t)| < 14 x0 ∈ Σ(dΩ) =⇒ ∃τ = τ(x0) > 0 : ξ(t) ∈ Σ(dΩ) , ∀t ∈ [0, τ)

Theorem (Albano – C – Khai T. Nguyen – Sinestrari, Math. Ann. 2013)

If ξ(t0) ∈ Σ(dΩ) for some t0 ≥ 0, then ξ(t) ∈ Σ(dΩ) for all t ∈ [t0,+∞)


Application to homotopy equivalence

want to construct H : Ω× [0, 1]→ Ω continuous with

1 H(x ,0) = x ∀x ∈ Ω2 H(x ,1) ∈ Σ(dΩ) ∀x ∈ Ω3 H(x , t) ∈ Σ(dΩ) ∀(x , t) ∈ Σ(dΩ)× [0,1]

ξ(t , x) generalized gradient flowξ(t) ∈ D+dΩ

(ξ(t)

)t ∈ [0,∞) a.e.

ξ(0) = x

locally Lipschitz in [0,∞)× Ω

easy ∃T > 0 : ξ(T , x) ∈ Σ(dΩ) ∀x ∈ Ω

take H(x , t) = ξ(tT , x)


Tonelli Hamiltonians

H(x ,Du(x)) = 0 (x ∈ Rn)

with H = H(x , p) ∈ C2(Rn × Rn) satisfying

(L1) Uniform convexity: ∃ν : [0,+∞)→ (0,+∞) nonincreasing such that

Hpp(x , p) > ν(|p|)I ∀(x , p) ∈ Rn × Rn

(L2) Superlinearity: ∃θ : [0,+∞)→ [0,+∞) nondecreasing with limr→∞θ(r)

r = +∞and c0 > 0 such that

H(x , p) > θ(|p|)− c0 ∀(x , p) ∈ Rn × Rn

(L3) Uniform regularity: ∃K : [0,+∞)→ [0,+∞) nondecreasing such that

|DαH(x , p)| 6 K (|p|) ∀(x , p) ∈ Rn × Rn

for any multindex |α| 6 2


Global weak KAM solutionsLet H(x , p) be a Tonelli Hamiltonian.

Proposition (Fathi-Maderna, 2007)

There exists a constant c(H) ∈ R such that the Hamilton-Jacobi equation

H(x ,Du(x)) = c (x ∈ Rn)

admits a viscosity solution u : Rn → R for c = c(H) and does not admit any suchsolution for c < c(H). Moreover, u is Lipschitz continuous and semiconcave on Rn.

Indeed, u = T−t u + c(H)t where T−t is the Lax-Oleinik operator

T−t u(x) = infy∈Rnu(y) + At (y , x)

and At (x , y) is the fundamental solution

At (x , y) = minξ∈W 1,1([0,t];Rn)

∫ t

0L(ξ(s), ξ(s))ds : ξ(0) = x , ξ(t) = y

associated with the Lagrangian L(x , v) = maxp∈Rn

〈p, v〉 − H((x , p)


Global propagation of singularitiesLet u : Rn → R be a Lipschitz semiconcave solution of

H(x ,Du(x)) = 0

and consider Lax-Oleinik operator T +t u(x) = supy∈Rn

u(y)− At (x , y)︸︷︷︸

φxt (y)

Theorem (C – Cheng)

∃t0 ∈ (0, 1] such that ∀(t , x) ∈ (0, t0]× Rn ∃!yt,x maximum point of φxt and the curve

y(t) :=

x if t = 0yt,x if t ∈ (0, t0]

satisfies

y(t) is Lipschitz and ∃λ0 > 0 such that |y(t)− x | 6 λ0t

if x ∈ Σ(u), then y(t) ∈ Σ(u) for all t ∈ (0, t0]

y(τ) ∈ co Hp(y(τ),D+u(y(τ))) τ ∈ [0, t0] a.e.

y+(0) = Hp(x , p0) where p0 is such that H(x , p) > H(x , p0), ∀p ∈ D+u(x)


Semiconcavity of the fundamental solution

Proposition

Suppose L is a Tonelli Lagrangian. Then for any λ > 0 there exists a constant Cλ > 0such that for any x ∈ Rn, t ∈ (0, 2/3), y ∈ B(x , λt), and (h, z) ∈ R× Rn satisfying|h| < t/2 and |z| < λt we have

At+h(x , y + z) + At−h(x , y − z)− 2At (x , y) 6Cλt(|h|2 + |z|2

).

Consequently, (t , y) 7→ At (x , y) is semiconcave in (0, 1)× Rn uniformly in x


Convexity of the fundamental solution for small time

Let L be a Tonelli Lagrangian

Proposition

∀λ > 0 there exists tλ > 0 such that ∀x ∈ Rn the function (t , y) 7→ At (x , y) is:

semiconvex on the cone

Sλ(x , tλ) :=

(t , y) ∈ R× Rn : 0 < t < tλ, |y − x | < λt,

and there exists a constant Cλ > 0 such that for all (t , y) ∈ Sλ(x , tλ), allh ∈ [0, t/2), and all z ∈ B(0, λt) we have that

At+h(x , y + z) + At−h(x , y − z)− 2At (x , y) > −Cλt

(h2 + |z|2)

uniformly convex on B(x , λt) for all 0 < t < tλ and there exists a constantC′λ > 0 such that ∀y ∈ B(x , λt) and z ∈ B(0, λt) we have that

At (x , y + z) + At (x , y − z)− 2At (x , y) >C′λt|z|2


Time dependent Hamilton-Jacobi equations

ut (t , x) + H

(x ,∇u(t , x)

)= 0 (t , x) ∈ R+ × Rn

u(0, x) = u0(x) x ∈ Rn

where

H : Rn × Rn → R is a C2 smooth function such that

(a) lim|p|→∞

infx∈Rn

H(x , p)

|p| = +∞

(b) D2pH(x , p) > 0, ∀(x , p) ∈ Rn × Rn

u0 : Rn → R is a Lipschitz function


Invariance of the singular set of u

The invariance problem for the genuine singular set of a solution tout (t , x) + H

(x ,∇u(t , x)

)= 0 (t , x) ∈ (0,T )× Rn

u(0, ·) = u0 Lipschitz

is widely open: just three results are available

Dafermos (Indiana Univ. Math. J. 1977) n = 1

Albano – C (Equadiff 99, World Scientific 2000)ut (t , x) + H

(∇u(t , x)

)= 0 (t , x) ∈ (0,∞)× Rn

u(0, ·) = u0 concave

C – Mazzola – Sinestrari (DCDS 2015)ut (t , x) + 1

2 〈A∇u,∇u〉 = 0 (t , x) ∈ (0,∞)× Rn

u(0, ·) = u0 Lipschitz

with A ∈ Rn×n > 0


Time dependent eikonal equations

Theorem (C – Mazzola – Sinestrari)

Let Ω ⊂ Rn be an open set and let u be the viscosity solution ofut (t , x) + 1

2 〈A∇u,∇u〉 = 0 (t , x) ∈ (0,∞)× Ω =: Qu(t , x) = φ(t , x) (t , x) ∈ ∂Q

with A ∈ Rn×n > 0, φ : Q → R Lipschitz such that

φ(t , x)− φ(s, y) ≤ 〈A−1(x − y)x − y〉

2(t − s)∀(t , x), (s, y) ∈ ∂Q, t > s ≥ 0

If (t0, x0) ∈ Σ(u), then ∃T ∈ (t0,∞] such that the solution ofx ′(t) ∈ A∇+u(t , x(t)) a.e. t ∈ (t0,T )

x(t0) = x0

satisfies(t , x(t)

)∈ Σ(u) for all t ∈ [t0,T )

T <∞ =⇒ limt↑T x(t) ∈ ∂ΩP. Cannarsa (Rome Tor Vergata) singular dynamics 25/02/2016 25 / 26

Happy Birthday Martino!


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