Critical value of some non-convex Hamiltonians -...

Critical valueof some non-convex Hamiltonians

Martino Bardi

Department of Pure and Applied MathematicsUniversity of Padua, Italy

Nonlinear PDEs WorkshopLisbon, June 20-24, 2011

Martino Bardi (Università di Padova) Critical value Lisbon, June 2011 1 / 23

Plan

Some classical results: coercive and convex H

A general definition of critical value and some properties

Connections with dynamical systems and differential games

An existence result under a saddle condition

Some results without saddle conditions(joint work with G. Terrone).


Classical result

Consider H : RN × RN → R continuous and ZN− periodic

H(x + k ,p) = H(x ,p), ∀k ∈ ZN .

H is COERCIVE in p if

lim|p|→∞

H(x ,p) = +∞ uniformly in x .

P.-L. Lions - Papanicolaou - Varadhan ’86:

There exists a unique constant c = H(0) such that

H(x ,Dχ) = c, in RN ,

has a ZN−periodic (viscosity) solution χ.


Moreover, for δ > 0,

c = − limδ→0

δwδ , χ(x) = limδ→0+

(wδ(x)−min wδ),

withδwδ + H(x ,Dwδ) = 0,

and ∀P ∈ RN there is c(P) = H(P) such that the cell problem

H(x ,Dχ+ P) = c(P), in RN ,

has a ZN−periodic (viscosity) solution χ(x ; P), called the corrector.

[LPV] prove that H(·) = c(·) is the effective Hamiltonian for thehomogenization of

uεt + H(xε,Dxuε

)= 0,

and the method was refined and generalized by Evans ’92.


Moreover, for δ > 0,

c = − limδ→0

δwδ , χ(x) = limδ→0+

(wδ(x)−min wδ),

withδwδ + H(x ,Dwδ) = 0,

and ∀P ∈ RN there is c(P) = H(P) such that the cell problem

H(x ,Dχ+ P) = c(P), in RN ,

has a ZN−periodic (viscosity) solution χ(x ; P), called the corrector.

[LPV] prove that H(·) = c(·) is the effective Hamiltonian for thehomogenization of

uεt + H(xε,Dxuε

)= 0,

and the method was refined and generalized by Evans ’92.


Calculus of Variations

If, in addition, p 7→ H(x ,p) is convex, we can use Fenchel conjugates

H(x ,p) = supa

[a ·p−L(x ,a)] =: L∗(x ,p), L(x ,a) = supp

[a ·p−H(x ,p)]

and, by uniqueness of viscosity solutions of HJ equations we can writewδ as the value function of an infinite-horizon discounted problem inCalculus of Variations:

wδ(x) = inf{∫ +∞

0L(x(s), x(s))e−δs ds : x(0) = x

}By Abelian-Tauberian type theorems, c = − limδ→0 δwδ(x) must alsobe the long-time minimal average action

c = − limT→+∞

inf

{1T

∫ T

0L(x(s), x(s)) : x(0) = x

}




H(x ,p) = supa


[a ·p−H(x ,p)]


wδ(x) = inf{∫ +∞

0L(x(s), x(s))e−δs ds : x(0) = x


c = − limT→+∞

inf

{1T

∫ T

0L(x(s), x(s)) : x(0) = x

}




H(x ,p) = supa


[a ·p−H(x ,p)]


wδ(x) = inf{∫ +∞

0L(x(s), x(s))e−δs ds : x(0) = x


c = − limT→+∞

inf

{1T

∫ T

0L(x(s), x(s)) : x(0) = x

}


If the Lagrangian L is smooth and superlinear

lim|a|→∞

L(x ,a)

|a|= +∞,

the minimal action is attained at some extremal trajectory x∗(s) thatsolves the Euler-Lagrange system of ODEs associated to L.

Then the critical value c gives some information on the long timebehavior of the Lagrangian flow.

For general (compact) manifolds this flow was deeply studied byJ. Mather, ’91, ’93, c(P) = α(P), α = Mather’s function (defined interms of invariant measures for the Lagrangian flow);R. Mañé, ’92, ’97, c = min{λ : H(x ,Du) = λ has a subsolution } ;A. Fathi, ’97....., weak KAM Theorem;Namah - Roquejioffre ’99, PDE results related to Fathi’s;Fathi - Siconolfi, ’04, existence of C1 critical subsolutions.


If the Lagrangian L is smooth and superlinear

lim|a|→∞

L(x ,a)

|a|= +∞,

the minimal action is attained at some extremal trajectory x∗(s) thatsolves the Euler-Lagrange system of ODEs associated to L.

Then the critical value c gives some information on the long timebehavior of the Lagrangian flow.

For general (compact) manifolds this flow was deeply studied byJ. Mather, ’91, ’93, c(P) = α(P), α = Mather’s function (defined interms of invariant measures for the Lagrangian flow);R. Mañé, ’92, ’97, c = min{λ : H(x ,Du) = λ has a subsolution } ;A. Fathi, ’97....., weak KAM Theorem;Namah - Roquejioffre ’99, PDE results related to Fathi’s;Fathi - Siconolfi, ’04, existence of C1 critical subsolutions.


Non-coercive non-convex Hamiltonians

I want to define and study a critical value for more general (continuous)Hamiltonians, including convex-concave ones, i.e., forz = (x , y) ∈ Tn × Tm, ξ = (p,q) ∈ Rn × Rm, n + m = N,

H(z, ξ) = H(x , y ,p,q) = H1(x , y ,p)−H2(x , y ,q)

H1 = L∗1 convex in p, H2 = L∗2 convex in q.

MAIN ASSUMPTION:the Comparison Principle among viscosity sub- and supersolutionsholds for the stationary equation, with δ > 0,

(SE) δwδ + H(z,Dzwδ) = 0 in RN

and for the evolutive equation

(EE) ut + H(z,Dzu) = 0 in (0,T )× RN .


Non-coercive non-convex Hamiltonians

I want to define and study a critical value for more general (continuous)Hamiltonians, including convex-concave ones, i.e., forz = (x , y) ∈ Tn × Tm, ξ = (p,q) ∈ Rn × Rm, n + m = N,

H(z, ξ) = H(x , y ,p,q) = H1(x , y ,p)−H2(x , y ,q)

H1 = L∗1 convex in p, H2 = L∗2 convex in q.

MAIN ASSUMPTION:the Comparison Principle among viscosity sub- and supersolutionsholds for the stationary equation, with δ > 0,

(SE) δwδ + H(z,Dzwδ) = 0 in RN

and for the evolutive equation

(EE) ut + H(z,Dzu) = 0 in (0,T )× RN .


A general definition of critical value

Consider the Cell Problem

(CP) H(z,Dzv) = λ in RN , v ZN − periodic

Theorem (O. Alvarez - M.B.: ARMA 2003)Under the Main Assumption

λ1 := inf{λ : (CP) has a subsolution }≤ sup{λ : (CP) has a supersolution } =: λ2.

Moreover, the following are equivalent(i) λ1 = λ2 =: c ;(ii) the solution u of (EE) with u(0, x) = 0 satisfieslimt→+∞ u(t , z)/t = constant, uniformly;(iii) the solution wδ of (SE) satisfies limδ→0 δwδ(x) = constant,uniformly.Martino Bardi (Università di Padova) Critical value Lisbon, June 2011 8 / 23

Finally,if (i), (ii), or (iii) holds, then constant= −c in (ii) and (iii);if H is replaced by H(z, ·+ P) then c(P) = H(P) is acontinuous function of P.

DefinitionIf (i), (ii), or (iii) of the previous Theorem holds then

critical value of H := c .

Remark: There may not be a continuous sol. χ of H(z,Dχ) = c.

This result was applied by O. Alvarez - M.B., Mem. AMS 2010, tohomogenization and singular perturbations (dimension reduction) ofBellman-Isaacs PDEs with non-coercive and/or non- convexHamiltonians.














Sufficient conditions for the Main Assumption (Comparison Principle),see

Barles, book 1994,

M.B. - I. Capuzzo Dolcetta, book 1997,

papers by Ishii, Crandall - Ishii - Lions ’86 - ’88.

papers by Da Lio -Ley 2006 -...

Examples:

H = H0(p,q)− f (x , y) , H ∈ C(RN), f ∈ C(TN).

H(x , y ,p,q) = |σ(x)p|2 −maxb∈B{g(y ,b) · q + f (y ,b)}

B compact, σ, g ∈ Lip, f ∈ C.


Sufficient conditions for the Main Assumption (Comparison Principle),see

Barles, book 1994,

M.B. - I. Capuzzo Dolcetta, book 1997,

papers by Ishii, Crandall - Ishii - Lions ’86 - ’88.

papers by Da Lio -Ley 2006 -...

Examples:

H = H0(p,q)− f (x , y) , H ∈ C(RN), f ∈ C(TN).

H(x , y ,p,q) = |σ(x)p|2 −maxb∈B{g(y ,b) · q + f (y ,b)}

B compact, σ, g ∈ Lip, f ∈ C.


Connection with Lagrangian flows?

Let L : Tn × Tm × Rn × Rm → R continuous, convex in a = x ,concave in b = y , e.g., L(x , y ,a,b) = H∗1(x , y ,a)− H∗2(x , y ,b).Action functional

L[t , x(·), y(·)] :=

∫ t

0L(x(s), y(s), x(s), y(s)) ds

Assume L has a saddle (x∗(·), y∗(·)) with given conditions at 0 and/ort , i.e.,

L[t , x∗(·), y(·)] ≤ L[t , x∗(·), y∗(·)] ≤ L[t , x(·), y∗(·)]

Plugging y(s) = y∗(s) + εφ(s) in the first inequality, the usualargument of C.o.V. gives m Euler-Lagrange equations,x(s) = x∗(s) + εψ(s) in the second inequality gives n E-L equations,so the saddle trajectory (x∗(·), y∗(·)) solves the same N-d system ofE-L equations as the minimizers of the action.




L[t , x(·), y(·)] :=

∫ t

0L(x(s), y(s), x(s), y(s)) ds


L[t , x∗(·), y(·)] ≤ L[t , x∗(·), y∗(·)] ≤ L[t , x(·), y∗(·)]





L[t , x(·), y(·)] :=

∫ t

0L(x(s), y(s), x(s), y(s)) ds


L[t , x∗(·), y(·)] ≤ L[t , x∗(·), y∗(·)] ≤ L[t , x(·), y∗(·)]



Questions: does a saddle trajectory exist? how regular is it?Not much seems to be known!But a more general problem was posed and studied in 1925-1930 byC.F. Roos "A Mathematical Theory of Competition" motivated byEconomics:Problem: for given Lagrangians L(1),L(2), search (x∗(·), y∗(·)) s.t.

x∗(·) minimizes∫ t

0L(1)(x(s), y∗(s), x(s), y∗(s)) ds

y∗(·) minimizes∫ t

0L(2)(x∗(s), y(s), x∗(s), y(s)) ds

This is a Nash equilibrium (open loop) for the pair of action functionals,but it predates of some decades the Theory of Games by VonNeumann - Morgenstern and Nash himself !In our saddle problem L = L(1) = −L(2) : a 0 - sum game.


Differential Games

Question: can we define a "value function"

v(t , x , y) := ” infx(·)

supy(·)

” or ” supy(·)

infx(·)

”{L[t , x(·), y(·)] : x(0) = x , y(0) = y}

such that v solves

vt + H1(x , y ,Dxv)− H2(x , y ,Dyv) = 0 ?

If so we could try to compute c = − limt→+∞ v(t , x , y)/t .

Nonanticipating strategiesNAS for 1st player, α[·] ∈ Γ, is a map from velocities y to velocities x ,α : L∞([0,T ],Rm)→ L∞([0,T ],Rn) s.t. ∀ t ≤ T

y1(s) = y2(s) ∀ s ≤ t =⇒ α[y1](s) = α[y2](s) ∀ s ≤ t

The definition is symmetric for NAS of 2nd player β ∈ ∆ .


Differential Games

Question: can we define a "value function"

v(t , x , y) := ” infx(·)

supy(·)

” or ” supy(·)

infx(·)

”{L[t , x(·), y(·)] : x(0) = x , y(0) = y}

such that v solves

vt + H1(x , y ,Dxv)− H2(x , y ,Dyv) = 0 ?

If so we could try to compute c = − limt→+∞ v(t , x , y)/t .

Nonanticipating strategiesNAS for 1st player, α[·] ∈ Γ, is a map from velocities y to velocities x ,α : L∞([0,T ],Rm)→ L∞([0,T ],Rn) s.t. ∀ t ≤ T

y1(s) = y2(s) ∀ s ≤ t =⇒ α[y1](s) = α[y2](s) ∀ s ≤ t

The definition is symmetric for NAS of 2nd player β ∈ ∆ .


Values of the game (Elliott - Kalton ’72)Lower value:

v(t , x , y) := infα∈Γ

supy(·){L[t , x(·), y(·)] : x(0) = x , y(0) = y , x = α[y ]}

Upper value:

u(t , x , y) := supβ∈∆

infx(·){L[t , x(·), y(·)] : x(0) = x , y(0) = y , y = β[x ]}

Remark: If there exists a saddle (x∗(·), y∗(·)) among trajectories withx(0) = x , y(0) = y , then

L(t , x∗(·), y∗(·)) = v(t , x , y) = u(t , x , y).


Game representation of solutions

Assumptions: H and L convex-concave

H(x , y ,p,q) = H1(x , y ,p)− H2(x , y ,q), H1 = L∗1, H2 = L∗2

L(x , y ,a,b) = L1(x , y ,a)− L2(x , y ,b),

the evolutive H-J eq. (EE) satisfy the Comparison Principle, + somemore technical conditions.Example: H = |p|η − γ|q|ν − f (x , y) , η, ν ≥ 1, γ > 0, f ∈ C(Tn+m).

Theorem (Evans-Souganidis ’84, Ishii ’88)The lower and the upper value, v and u, satisfy the Cauchy problem

vt + H(x , y ,Dxv ,Dyv) = 0 in (0,+∞)× RN ,

v(0, x , y) = 0 in RN ;

therefore they coincide: v(t , x , y) = u(t , x , y) ∀ t , x , y .


An existence result for the critical value

Assume in addition:

H1(x , y ,p) coercive in p for each fixed y , with critical value c1(y),

H2(x , y ,q) coercive in q for each fixed x , with critical value c2(x).

Main TheoremIf

λ1 := minx

maxy

(L1(x , y ,0) + c2(x))

≤ maxy

minx

(−L2(x , y ,0)− c1(x)) =: λ2

then λ1 = λ2 =: c and c is the critical value of H .


Idea of proof

By the equality v = u and the def. of critical value, if

lim supT→+∞

1T

v(T , x , y) ≤ const . ≤ lim infT→+∞

1T

u(T , x , y)

then c := const . is the critical value of H .

Building suitable nonanticipating strategies I prove that

lim supT→+∞

1T

v(T , x , y) ≤ λ1

andλ2 ≤ lim inf

T→+∞

1T

u(T , x , y).

Then λ2 ≤ λ1 and by the assumption they coincide.


Examples

Assume

mina

L1(x , y ,a) = L1(x , y ,0), minb

L2(x , y ,b) = L2(x , y ,0) ∀ x , y ,

i.e., the null velocity is optimal for both players. Then

c1(y) = −minx

L1(x , y ,0), c2(x) = −miny

L2(x , y ,0)

and the assumption of the theorem becomes

minx

maxy

(L1(x , y ,0) + max

y(−L2(x , y ,0))

)≤ max

ymin

x

(−L2(x , y ,0) + min

xL1(x , y ,0)

).

If L1(x , y ,0) is independent of y and L2(x , y ,0) is independent of x ,the condition becomes

minx

maxy

L(x , y ,0,0) ≤ maxy

minx

L(x , y ,0,0).


Further examples

The critical value of H = |Dxu|η − γ|Dyu|ν − f (x , y) , η, ν ≥ 1,γ > 0 is

c = minx

maxy

f (x , y) = maxy

minx

f (x , y),

provided that minx maxy f ≤ maxy minx f .

For x , y ∈ R , η = ν = 2,

H(x , y , ·+ P, ·+ Q) = |ux +P|2 − γ|uy +Q|2 − f (x , y)

has critical value

c(P,Q) = H(P,Q) = minx

maxy

f (x , y) = maxy

minx

f (x , y)

for (P,Q) in a neighborhood of 0, if∫ 1

0 f (x , y) dx > minx f (x , y),∫ 10 f (x , y) dy < maxy f (x , y).

But the method does not give the existence of H(P,Q) for all(P,Q).


Further examples

The critical value of H = |Dxu|η − γ|Dyu|ν − f (x , y) , η, ν ≥ 1,γ > 0 is

c = minx

maxy

f (x , y) = maxy

minx

f (x , y),

provided that minx maxy f ≤ maxy minx f .

For x , y ∈ R , η = ν = 2,

H(x , y , ·+ P, ·+ Q) = |ux +P|2 − γ|uy +Q|2 − f (x , y)

has critical value

c(P,Q) = H(P,Q) = minx

maxy

f (x , y) = maxy

minx

f (x , y)

for (P,Q) in a neighborhood of 0, if∫ 1

0 f (x , y) dx > minx f (x , y),∫ 10 f (x , y) dy < maxy f (x , y).

But the method does not give the existence of H(P,Q) for all(P,Q).


And without saddle conditions?

Example 1

ut + |ux |η − |uy |η = g (x − y) x , y ∈ R

u(0, x , y) = 0

is solved by u(t , x , y) = tg (x − y) for all g ∈ C1(T) ,

so limt→+∞ u(t , x , y)/t is NOT a constant if g is not constant.

Note that f (x , y) = g(x − y) has no saddle:

minx

maxy

g (x − y) = maxr

g(r)> minr

g(r) = maxy

minx

g (x − y) .

But by a different method we can prove the existence of critical valuefor many cases of such f without saddle.


Coercivity on subspaces

Let V be M–dimensional subspace of RN , zV := ΠV (z) projection.

Consider Hamiltonians of the form H = H (zV ,Dzu)

and assume H(θ,p) ZM -periodic in θ and coercive in V : ∀ p ∈ RN

lim|p|→+∞, p∈V

H(θ,p + p) = +∞, uniformly in θ ∈ V .

Theorem (with G. Terrone, 2011)

Under these conditions H (zV , ·+ p) has critical value c(p) = H(p)∀ p ∈ RN .

Proof: by PDE methods, as in [LPV].

This theorem applies to homogenization on subspaces: see Terrone’stalk tomorrow.


Example 2

H = |ux + P|η − γ|uy + Q|η = g (x − y) x , y ∈ RN/2, η ≥ 1,

has critical value ∀g ZN/2-periodic ∀P ∈ Rn,Q ∈ Rm ⇐⇒ γ 6= 1

γ = 1 by Example 1;

γ 6= 1 fits in the preceding Theorem with

V = {(q,−q) : q ∈ RN/2}.

Conclusion: the saddle-type condition can be replaced by otherrestrictions on the dependence of H on x , y ,but the results found so far leave many open cases.


Example 2

H = |ux + P|η − γ|uy + Q|η = g (x − y) x , y ∈ RN/2, η ≥ 1,

has critical value ∀g ZN/2-periodic ∀P ∈ Rn,Q ∈ Rm ⇐⇒ γ 6= 1

γ = 1 by Example 1;

γ 6= 1 fits in the preceding Theorem with

V = {(q,−q) : q ∈ RN/2}.

Conclusion: the saddle-type condition can be replaced by otherrestrictions on the dependence of H on x , y ,but the results found so far leave many open cases.


Thanks for your attention !


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