Critical value
of some non-convex Hamiltonians
Martino Bardi
Department of Pure and Applied Mathematics
University of Padua, Italy
Nonlinear PDEs Workshop
Lisbon, 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).
Martino Bardi (Università di Padova)
Critical value
Lisbon, June 2011
2 / 23
Classical result
Consider H : RN × RN → R continuous and ZN − periodic
∀k ∈ ZN .
H(x + k , p) = H(x, p),
H is COERCIVE in p if
lim H(x, p) = +∞
|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.
Martino Bardi (Università di Padova)
Critical value
Lisbon, June 2011
3 / 23
Moreover, for δ > 0,
c = − lim δwδ ,
δ→0
χ(x) = lim (wδ (x) − min wδ ),
δ→0+
with
δwδ + 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 the
homogenization of
x
utε + H
, Dx u ε = 0,
ε
and the method was refined and generalized by Evans ’92.
Martino Bardi (Università di Padova)
Critical value
Lisbon, June 2011
4 / 23
Moreover, for δ > 0,
c = − lim δwδ ,
δ→0
χ(x) = lim (wδ (x) − min wδ ),
δ→0+
with
δwδ + 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 the
homogenization of
x
utε + H
, Dx u ε = 0,
ε
and the method was refined and generalized by Evans ’92.
Martino Bardi (Università di Padova)
Critical value
Lisbon, June 2011
4 / 23
Calculus of Variations
If, in addition, p 7→ H(x, p) is convex, we can use Fenchel conjugates
H(x, p) = sup[a · p − L(x, a)] =: L∗ (x, p),
a
L(x, a) = sup[a · p − H(x, p)]
p
and, by uniqueness of viscosity solutions of HJ equations we can write
wδ as the value function of an infinite-horizon discounted problem in
Calculus of Variations:
Z +∞
−δs
wδ (x) = inf
L(x(s), ẋ(s))e
ds : x(0) = x
0
By Abelian-Tauberian type theorems, c = − limδ→0 δwδ (x) must also
be the long-time minimal average action
)
( Z
1 T
c = − lim inf
L(x(s), ẋ(s)) : x(0) = x
T 0
T →+∞
Martino Bardi (Università di Padova)
Critical value
Lisbon, June 2011
5 / 23
Calculus of Variations
If, in addition, p 7→ H(x, p) is convex, we can use Fenchel conjugates
H(x, p) = sup[a · p − L(x, a)] =: L∗ (x, p),
a
L(x, a) = sup[a · p − H(x, p)]
p
and, by uniqueness of viscosity solutions of HJ equations we can write
wδ as the value function of an infinite-horizon discounted problem in
Calculus of Variations:
Z +∞
−δs
L(x(s), ẋ(s))e
ds : x(0) = x
wδ (x) = inf
0
By Abelian-Tauberian type theorems, c = − limδ→0 δwδ (x) must also
be the long-time minimal average action
( Z
)
1 T
c = − lim inf
L(x(s), ẋ(s)) : x(0) = x
T 0
T →+∞
Martino Bardi (Università di Padova)
Critical value
Lisbon, June 2011
5 / 23
Calculus of Variations
If, in addition, p 7→ H(x, p) is convex, we can use Fenchel conjugates
H(x, p) = sup[a · p − L(x, a)] =: L∗ (x, p),
a
L(x, a) = sup[a · p − H(x, p)]
p
and, by uniqueness of viscosity solutions of HJ equations we can write
wδ as the value function of an infinite-horizon discounted problem in
Calculus of Variations:
Z +∞
−δs
L(x(s), ẋ(s))e
ds : x(0) = x
wδ (x) = inf
0
By Abelian-Tauberian type theorems, c = − limδ→0 δwδ (x) must also
be the long-time minimal average action
( Z
)
1 T
c = − lim inf
L(x(s), ẋ(s)) : x(0) = x
T 0
T →+∞
Martino Bardi (Università di Padova)
Critical value
Lisbon, June 2011
5 / 23
If the Lagrangian L is smooth and superlinear
L(x, a)
= +∞,
|a|
|a|→∞
lim
the minimal action is attained at some extremal trajectory x ∗ (s) that
solves the Euler-Lagrange system of ODEs associated to L.
Then the critical value c gives some information on the long time
behavior of the Lagrangian flow.
For general (compact) manifolds this flow was deeply studied by
J. Mather, ’91, ’93, c(P) = α(P), α = Mather’s function (defined in
terms 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 C 1 critical subsolutions.
Martino Bardi (Università di Padova)
Critical value
Lisbon, June 2011
6 / 23
If the Lagrangian L is smooth and superlinear
L(x, a)
= +∞,
|a|
|a|→∞
lim
the minimal action is attained at some extremal trajectory x ∗ (s) that
solves the Euler-Lagrange system of ODEs associated to L.
Then the critical value c gives some information on the long time
behavior of the Lagrangian flow.
For general (compact) manifolds this flow was deeply studied by
J. Mather, ’91, ’93, c(P) = α(P), α = Mather’s function (defined in
terms 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 C 1 critical subsolutions.
Martino Bardi (Università di Padova)
Critical value
Lisbon, June 2011
6 / 23
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., for
z = (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 supersolutions
holds for the stationary equation, with δ > 0,
(SE)
δwδ + H(z, Dz wδ ) = 0
in RN
and for the evolutive equation
(EE)
ut + H(z, Dz u) = 0
Martino Bardi (Università di Padova)
Critical value
in (0, T ) × RN .
Lisbon, June 2011
7 / 23
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., for
z = (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 supersolutions
holds for the stationary equation, with δ > 0,
(SE)
δwδ + H(z, Dz wδ ) = 0
in RN
and for the evolutive equation
(EE)
ut + H(z, Dz u) = 0
Martino Bardi (Università di Padova)
Critical value
in (0, T ) × RN .
Lisbon, June 2011
7 / 23
A general definition of critical value
Consider the Cell Problem
(CP)
H(z, Dz v ) = 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 satisfies
limt→+∞ 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 a
continuous function of P.
Definition
If (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, to
homogenization and singular perturbations (dimension reduction) of
Bellman-Isaacs PDEs with non-coercive and/or non- convex
Hamiltonians.
Martino Bardi (Università di Padova)
Critical value
Lisbon, June 2011
9 / 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 a
continuous function of P.
Definition
If (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, to
homogenization and singular perturbations (dimension reduction) of
Bellman-Isaacs PDEs with non-coercive and/or non- convex
Hamiltonians.
Martino Bardi (Università di Padova)
Critical value
Lisbon, June 2011
9 / 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 a
continuous function of P.
Definition
If (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, to
homogenization and singular perturbations (dimension reduction) of
Bellman-Isaacs PDEs with non-coercive and/or non- convex
Hamiltonians.
Martino Bardi (Università di Padova)
Critical value
Lisbon, June 2011
9 / 23
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 − max{g(y , b) · q + f (y , b)}
b∈B
B compact, σ, g ∈ Lip, f ∈ C.
Martino Bardi (Università di Padova)
Critical value
Lisbon, June 2011
10 / 23
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 − max{g(y , b) · q + f (y , b)}
b∈B
B compact, σ, g ∈ Lip, f ∈ C.
Martino Bardi (Università di Padova)
Critical value
Lisbon, June 2011
10 / 23
Connection with Lagrangian flows?
Let L : Tn × Tm × Rn × Rm → R continuous, convex in a = ẋ,
concave in b = ẏ , e.g., L(x, y , a, b) = H1∗ (x, y , a) − H2∗ (x, y , b).
Action functional
Z t
L[t, x(·), y (·)] :=
L(x(s), y (s), ẋ(s), ẏ (s)) ds
0
Assume L has a saddle (x ∗ (·), y ∗ (·)) with given conditions at 0 and/or
t, 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 usual
argument 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 of
E-L equations as the minimizers of the action.
Martino Bardi (Università di Padova)
Critical value
Lisbon, June 2011
11 / 23
Connection with Lagrangian flows?
Let L : Tn × Tm × Rn × Rm → R continuous, convex in a = ẋ,
concave in b = ẏ , e.g., L(x, y , a, b) = H1∗ (x, y , a) − H2∗ (x, y , b).
Action functional
Z t
L[t, x(·), y (·)] :=
L(x(s), y (s), ẋ(s), ẏ (s)) ds
0
Assume L has a saddle (x ∗ (·), y ∗ (·)) with given conditions at 0 and/or
t, 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 usual
argument 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 of
E-L equations as the minimizers of the action.
Martino Bardi (Università di Padova)
Critical value
Lisbon, June 2011
11 / 23
Connection with Lagrangian flows?
Let L : Tn × Tm × Rn × Rm → R continuous, convex in a = ẋ,
concave in b = ẏ , e.g., L(x, y , a, b) = H1∗ (x, y , a) − H2∗ (x, y , b).
Action functional
Z t
L[t, x(·), y (·)] :=
L(x(s), y (s), ẋ(s), ẏ (s)) ds
0
Assume L has a saddle (x ∗ (·), y ∗ (·)) with given conditions at 0 and/or
t, 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 usual
argument 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 of
E-L equations as the minimizers of the action.
Martino Bardi (Università di Padova)
Critical value
Lisbon, June 2011
11 / 23
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 by
C.F. Roos "A Mathematical Theory of Competition" motivated by
Economics:
Problem: for given Lagrangians L(1) , L(2) , search (x ∗ (·), y ∗ (·)) s.t.
∗
t
Z
x (·) minimizes
L(1) (x(s), y ∗ (s), ẋ(s), ẏ ∗ (s)) ds
0
y ∗ (·) minimizes
Z
t
L(2) (x ∗ (s), y (s), ẋ ∗ (s), ẏ (s)) ds
0
This is a Nash equilibrium (open loop) for the pair of action functionals,
but it predates of some decades the Theory of Games by Von
Neumann - Morgenstern and Nash himself !
In our saddle problem L = L(1) = −L(2) : a 0 - sum game.
Martino Bardi (Università di Padova)
Critical value
Lisbon, June 2011
12 / 23
Differential Games
Question: can we define a "value function"
v (t, x, y ) := ” inf sup ” or ” sup inf ”{L[t, x(·), y (·)] : x(0) = x, y (0) = y }
x(·) y (·)
y (·) x(·)
such that v solves
vt + H1 (x, y , Dx v ) − H2 (x, y , Dy v ) = 0
?
If so we could try to compute c = − limt→+∞ v (t, x, y )/t.
Nonanticipating strategies
NAS for 1st player, α[·] ∈ Γ, is a map from velocities ẏ to velocities ẋ,
α : L∞ ([0, T ], Rm ) → L∞ ([0, T ], Rn ) s.t. ∀ t ≤ T
ẏ1 (s) = ẏ2 (s) ∀ s ≤ t
=⇒
α[ẏ1 ](s) = α[ẏ2 ](s) ∀ s ≤ t
The definition is symmetric for NAS of 2nd player β ∈ ∆ .
Martino Bardi (Università di Padova)
Critical value
Lisbon, June 2011
13 / 23
Differential Games
Question: can we define a "value function"
v (t, x, y ) := ” inf sup ” or ” sup inf ”{L[t, x(·), y (·)] : x(0) = x, y (0) = y }
x(·) y (·)
y (·) x(·)
such that v solves
vt + H1 (x, y , Dx v ) − H2 (x, y , Dy v ) = 0
?
If so we could try to compute c = − limt→+∞ v (t, x, y )/t.
Nonanticipating strategies
NAS for 1st player, α[·] ∈ Γ, is a map from velocities ẏ to velocities ẋ,
α : L∞ ([0, T ], Rm ) → L∞ ([0, T ], Rn ) s.t. ∀ t ≤ T
ẏ1 (s) = ẏ2 (s) ∀ s ≤ t
=⇒
α[ẏ1 ](s) = α[ẏ2 ](s) ∀ s ≤ t
The definition is symmetric for NAS of 2nd player β ∈ ∆ .
Martino Bardi (Università di Padova)
Critical value
Lisbon, June 2011
13 / 23
Values of the game (Elliott - Kalton ’72)
Lower value:
v (t, x, y ) := inf sup {L[t, x(·), y (·)] : x(0) = x, y (0) = y , ẋ = α[ẏ ]}
α∈Γ y (·)
Upper value:
u(t, x, y ) := sup inf {L[t, x(·), y (·)] : x(0) = x, y (0) = y , ẏ = β[ẋ]}
β∈∆ x(·)
Remark: If there exists a saddle (x ∗ (·), y ∗ (·)) among trajectories with
x(0) = x, y (0) = y , then
L(t, x ∗ (·), y ∗ (·)) = v (t, x, y ) = u(t, x, y ).
Martino Bardi (Università di Padova)
Critical value
Lisbon, June 2011
14 / 23
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, + some
more 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 , Dx v , Dy v ) = 0
v (0, x, y ) = 0
therefore they coincide:
Martino Bardi (Università di Padova)
in (0, +∞) × RN ,
in RN ;
v (t, x, y ) = u(t, x, y )
Critical value
∀ t, x, y .
Lisbon, June 2011
15 / 23
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 Theorem
If
λ1 := min max (L1 (x, y , 0) + c2 (x))
x
y
≤ max min (−L2 (x, y , 0) − c1 (x)) =: λ2
y
x
then λ1 = λ2 =: c and c is the critical value of H .
Martino Bardi (Università di Padova)
Critical value
Lisbon, June 2011
16 / 23
Idea of proof
By the equality v = u and the def. of critical value, if
lim sup
T →+∞
1
1
v (T , x, y ) ≤ const. ≤ lim inf u(T , x, y )
T
T →+∞ T
then c := const. is the critical value of H .
Building suitable nonanticipating strategies I prove that
lim sup
T →+∞
1
v (T , x, y ) ≤ λ1
T
and
λ2 ≤ lim inf
T →+∞
1
u(T , x, y ).
T
Then λ2 ≤ λ1 and by the assumption they coincide.
Martino Bardi (Università di Padova)
Critical value
Lisbon, June 2011
17 / 23
Examples
Assume
min L1 (x, y , a) = L1 (x, y ,0),
a
min L2 (x, y , b) = L2 (x, y ,0)
b
∀ x, y ,
i.e., the null velocity is optimal for both players. Then
c1 (y ) = − min L1 (x, y , 0),
x
c2 (x) = − min L2 (x, y , 0)
y
and the assumption of the theorem becomes
min max L1 (x, y , 0) + max(−L2 (x, y , 0))
x
y
y
≤ max min −L2 (x, y , 0) + min L1 (x, y , 0) .
y
x
x
If L1 (x, y , 0) is independent of y and L2 (x, y , 0) is independent of x,
the condition becomes
min max L(x, y , 0, 0) ≤ max min L(x, y , 0, 0).
x
y
y
Martino Bardi (Università di Padova)
Critical value
x
Lisbon, June 2011
18 / 23
Further examples
The critical value of H = |Dx u|η − γ|Dy u|ν − f (x, y ) , η, ν ≥ 1,
γ > 0 is
c = min max f (x, y ) = max min f (x, y ),
x
y
y
x
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) = min max f (x, y ) = max min f (x, y )
x
y
y
x
R1
for (P, Q) in a neighborhood of 0, if 0 f (x, y ) dx > minx f (x, y ),
R1
0 f (x, y ) dy < maxy f (x, y ).
But the method does not give the existence of H(P, Q) for all
(P, Q).
Martino Bardi (Università di Padova)
Critical value
Lisbon, June 2011
19 / 23
Further examples
The critical value of H = |Dx u|η − γ|Dy u|ν − f (x, y ) , η, ν ≥ 1,
γ > 0 is
c = min max f (x, y ) = max min f (x, y ),
x
y
y
x
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) = min max f (x, y ) = max min f (x, y )
x
y
y
x
R1
for (P, Q) in a neighborhood of 0, if 0 f (x, y ) dx > minx f (x, y ),
R1
0 f (x, y ) dy < maxy f (x, y ).
But the method does not give the existence of H(P, Q) for all
(P, Q).
Martino Bardi (Università di Padova)
Critical value
Lisbon, June 2011
19 / 23
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
for all g ∈ C 1 (T) ,
u(t, x, y ) = tg (x − y )
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:
min max g (x − y ) = max g(r ) > min g(r ) = max min g (x − y ) .
x
y
r
r
y
x
But by a different method we can prove the existence of critical value
for many cases of such f without saddle.
Martino Bardi (Università di Padova)
Critical value
Lisbon, June 2011
20 / 23
Coercivity on subspaces
Let V be M–dimensional subspace of RN , zV := ΠV (z) projection.
Consider Hamiltonians of the form H = H (zV , Dz u)
and assume H(θ, p) ZM -periodic in θ and coercive in V : ∀ p̄ ∈ RN
lim
|p|→+∞, p∈V
H(θ, p + p̄) = +∞,
uniformly in θ ∈ V .
Theorem (with G. Terrone, 2011)
Under these conditions H (zV , · + p̄) has critical value c(p̄) = H(p̄)
∀ p̄ ∈ RN .
Proof: by PDE methods, as in [LPV].
This theorem applies to homogenization on subspaces: see Terrone’s
talk tomorrow.
Martino Bardi (Università di Padova)
Critical value
Lisbon, June 2011
21 / 23
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 other
restrictions on the dependence of H on x, y ,
but the results found so far leave many open cases.
Martino Bardi (Università di Padova)
Critical value
Lisbon, June 2011
22 / 23
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 other
restrictions on the dependence of H on x, y ,
but the results found so far leave many open cases.
Martino Bardi (Università di Padova)
Critical value
Lisbon, June 2011
22 / 23
Thanks for your attention !
Martino Bardi (Università di Padova)
Critical value
Lisbon, June 2011
23 / 23