Notes by S. Bolotin
- (Albert Fathi) Let
be a smooth vector field on a compact
manifold
and let
be its flow. There is a
standard way to include the flow
in the flow of a
positive definite Lagrangian system. Take a Riemannian metric on
and let
be the corresponding norm. Define a
Lagrangian
by
Then the zero section
of the tangent bundle is an
invariant manifold for the Lagrangian flow. The Aubry set
corresponding to the zero cohomology class
is contained in
.
Problem: give a characterization of
in terms of the dynamics
of the flow
. In particular, does
contain the
chain recurrent set of the flow
?
- Let
,
be the Hamiltonian corresponding to
. Consider the
Hamilton-Jacobi equation
Question: Under what conditions is
the unique viscosity
solution?
- (John Mather)
Let
be a compact manifold and
a
Lagrangian satisfying the usual hypotheses of the
Mather theory - convexity, superlinearity and completeness. Let
be the Aubry set corresponding to the cohomology
class
. Define a pseudo metric
on
as follows. Modify the Lagrangian by subtracting a closed 1-form
from the cohomology class
and adding a constant
so
that for the new
we have
Then set
and
Define an equivalence relation on
by
iff
. Let
be the corresponding quotient
space.
Question: Is it true that
is totally disconnected? Does
have zero Hausdorff dimension?
- (Albert Fathi) Is it true that for generic
there exists only a one-parameter family
of viscosity solutions of the corresponding Hamilton-Jacobi
equation?
- (Patrick Bernard)
Consider an analytic positive definite Lagrangian
,
, satisfying
Mather's conditions. Let
be Mather's function
.
Question: Is it true that if
is analytic with positive
definite second derivative, then the Lagrangian system is
completely integrable?
- (Gonzalo Contreras) Is it true that under the same
condition on
, each Aubry set
covers the whole
configuration space
?
- (John Mather) Let
be an
analytic twist map. Suppose that
is foliated by
invariant curves
(which are Aubry-Mather sets). Are
these curves
necessarily analytic?
- (Vadim Kaloshin) Consider a billiard
inside a region bounded by a closed convex curve
. Suppose that a neighborhood of
is
foliated by caustics for the billiard. Is it true that then
is an ellipse and hence the caustics are analytic curves?
The billiard in
is described by a twist map
, and
caustics correspond to invariant curves of
. Hence this
questions is a particular case of the previous one.
- (John Mather) Does there exist a
twist map
which has an invariant curve
with irrational rotation number and such that
is not
conjugate to a rotation? By the Denjoy example there exist
maps of a circle with irrational rotation number not conjugate to
a rotation. Can this happen for invariant curves of twist maps?
- (Patrick Bernard) The existence of KAM invariant tori of a
Hamiltonian system on
is equivalent to the
existence of regular solutions of the Hamilton-Jacobi equation
. Hence KAM theory can be regarded as
regularity result for viscosity solutions.
Question: Is it possible to obtain proofs of KAM results using
this connection?
- (Albert Fathi) For a Hamiltonian
on
, which is superlinear and strictly convex in
,
we have a fairly good description both in dynamical terms (Mather
theory) and in PDE terms (viscosity solutions) of the Aubry set.
It would be nice to understand the situation where
is still
superlinear but not necessarily convex:
- Does there exist an Aubry set from the PDE point of
view (for example as a uniqueness set for viscosity solutions)?
- Does there exist an Aubry set from the point of view of dynamics,
i.e. a canonical graph invariant under the Hamiltonian flow?
- Is there a relationship between the two sets if they exist?
- Let
be the infimum of the
's such that the
Hamilton-Jacobi equation
with superlinear
admits a global viscosity subsolution
. Let
be
the set of global viscosity subsolutions of
.
Define a function
on
by
Then for fixed
, the function
is a viscosity
subsolution of
on
, and a viscosity
solution on
. When
is convex in
, the
Aubry set
is the set of
such that
is
a global viscosity solution on the whole
.
Question: If
is not necessarily convex, does there exist
such that
is a viscosity solution on the
whole
?
- (Walter Craig)
Consider a positive definite Lagrangian
. Let
be Mather's
-function.
Problem: Relate the regularity of the Mather set
and the
arithmetic properties of the frequency set
.
- (Walter Craig)
This question concerns the minimax Birkhoff orbits in an area
preserving twist map of an annulus. It is well known that
Question: What is the fate of the minimax orbits for the case when
the Mather set is a Cantor set?
- (Gonzalo Contreras)
Let
be a compact manifold,
a convex superlinear
Lagrangian, and
- lift of
to the
universal cover
of
. Let
be the Mane critical
value for
.
Question: Is it true that
- Let
be a compact manifold,
a convex superlinear
Lagrangian. The only example known of an energy level without
periodic orbits or singularities is
with
a Mane
critical value for the lift of the Lagrangian to
.
Question: Is there an example with other energy levels without
periodic orbits or singularities?
- Let
be a compact manifold,
a convex superlinear
Hamiltonian. The energy level
is of contact
type if there exists a 1-form
on
such that
is the symplectic 2-form on
and
, where
is the Hamiltonian vector field. Let
be the critical value on
and let
Question: Is it true that if
, then for all
the energy level
is not of contact type?
- Let
be a non-compact manifold and
a convex
superlinear Lagrangian satisfying appropriate completeness
conditions at infinity. There is a compactification of
by
adding an "extended Aubry set" whose points correspond to
"Busemann viscosity solutions" of the Hamilton-Jacobi equation
(Calc. Var. 13 (2001), 427-458).
Problem: Understand the geometry of this compactification.
- Let
be an autonomous convex superlinear
Lagrangian and
the corresponding Hamiltonian. Let
be the Mane critical value.
Question: Is it true that for any
the set
has finite symplectic capacity?
- Let
be an autonomous convex superlinear
Hamiltonian. Suppose that the Aubry set
satisfies
and there exists a unique (up to a constant)
viscosity solution of the Hamilton-Jacobi equation
. For
, let
be the Aubry set
corresponding to the Hamiltonian
. Let
be
a viscosity solution of
.
Question: is it true that
- Mane proved that for a generic positive definite Lagrangian
and generic
, the minimizing
measure in the Mather set
is unique.
Problem: For generic
and
is the unique minimizing measure
in
supported in a periodic orbit?
Conjecture (Jeff Xia): For a generic
and all
, the number of ergodic invariant measures in
is
finite.
- (Kostia Khanin)
Consider a convex superlinear positive definite Lagrangian on
with white noise perturbation:
where
are independent Brownian motions. Suppose that the
map
is an embedding. Then with
probability 1 there exists a unique global minimizer
.
Conjecture: With probability 1, the minimizer
is a
hyperbolic trajectory of the Lagrangian flow.
This is proved (E-Khanin-Mazel-Sinai) for
.
- (Arnold diffusion)
Consider a
,
, Hamiltonian
of the form
Suppose that
is positive definite and superlinear.
Conjecture: Suppose that
. Then for given open sets
and typical in the
topology
,
there exists
such that for any
,
there exists a trajectory
of the Hamiltonian system
such that
and
.
This conjecture seems very hard in the
category, when
the change of the action is exponentially slow in
by the
Nekhoroshev theorem. The case of finite
should be easier.
A precise definition of "typical" needs to be established. Mather
proved this conjecture for
and for a cusp residual set of
perturbations
in the
topology. This doesn't
prove the above conjecture: for given
the set of admissible
does not cover an interval
.
Question: Does Mather's theorem holds for an
in a cusp
residual set of trigonometric polynomials of high order
?
Exponentially small perturbations of coefficients are allowed, so
the transcendental problem of exponentially small separatrice
splitting is avoided.
- Consider an apriori unstable Hamiltonian of the form
on
, where
is convex and
superlinear, the Hamiltonian
has a separatrix loop,
and
is a generic perturbation. The large gap problem of
Arnold diffusion was overpassed recently for such systems by
variational methods of Mather (Xia), by geometrical methods using
secondary KAM-tori (de la Llave, Delshams, Seara), and by the
method of separatrix map (Treschev).
Problem: Understand the relation between the variational method
and the method of separatrix map. They seem similar in spirit.
- (Paul Rabinowitz)
Is there a reasonable PDE analogue of Arnold's diffusion?
It seems that a version of finite dimensional transition chains is
not the right mechanism for PDE. There are results of Kuksin which
prove "diffusion" for PDE with respect to high order Sobolev norms.
The same problem for infinite lattices like Fermi-Pasta-Ulam
system.
- (Victor Bangert)
Let
,
be a smooth
multidimensional Lagrangian satisfying the usual convexity
assumptions in
. For example,
where the potential
is periodic in all
variables.
A function
is called minimal if it minimizes
the action functional
for all variations with compact support. The set of minimizers
``without self-intersections'' is very well understood. Here
is said to be ``without selfintersections''
if the projection to
of
is a hypersurface without
self-intersections. In particular, for every minimizer
without
self-intersections there exists a ``rotation vector''
such that
is bounded by a constant that only depends on
. Conversely, for
every
there exists a minimizer
without
self-intersections such that
is bounded.
Question: Is there an analytical condition implying that the
minimizers satisfying this condition have no self-intersections.
More concrete question: Suppose
is minimal
and
is bounded. Is it true that the graph of
in
has no self-intersections. For partial results see V.
Bangert, Ann. Inst. Henri Poincaré - Analyse non linéaire
6(1989), 95-138, in particular Sect. 8.
- (Victor Bangert)
Let
be a compact Riemannian manifold and let
be the
set of closed
-currents on
:
Every
-current
defines a homology class
and a mass
Every homology class
has a representative with
minimal mass.
For
the supports of minimal currents consist of minimal
geodesics. For
any minimizer
is given by a measured
lamination by minimizing hypersurfaces (possibly with
singularities).
Problem: What can one say about minimal currents for
.
- (Franz Auer) Conjecture: If
is a minimizing
closed current, then the Hausdorff dimension of the corresponding
singularity set is at most
.
- Define a stable norm on
by
This norm is an analog of Mather's
-function).
Problem: Study convexity and differentiability properties of the
unit ball
- (Victor Bangert)
Let
be a manifold with an almost complex structure
. A
pseudo-holomorphic line is a map
satisfying the PDE:
. Moser proved that for an almost complex structure
on
which is close to a standard complex structure
,
and any sufficiently irrational vector
, there exists
a foliation of
by pseudo-holomorphic curves which is
conjugate to a linear foliation by real 2-planes containing the
vector
.
Problem: Develop a global (non-perturbative) theory of such
foliations or laminations.
The theory is expected to be particularly rich for the case
due to the positivity of the intersection number of
pseudoholomorphic curves. This is an analog of the intersection
number for geodesics on
- the basis for Hedlund's results
on minimal geodesics.
- (Craig Evans)
Consider the Hamiltonian
, where
the potential
is
periodic. The corresponding
Hamiltonian operator in quantum mechanics is then
.
Much exciting research in ``semiclassical analysis'' concerns
studying the limit as
of solutions
of the
eigenvalue problem
and finding connections with classical Hamiltonian dynamics.
Problem: Do Mather sets play any role here? Or, conversely, can we
somehow ``quantize'' Mather sets? This would presumably mean to
build quasimodes (ie approximate solutions of (*)) corresponding
to Mather's sets and to prove good error bounds. Would some sort
of Diophantine condition be useful here?
- (Takis Souganidis)
Homogenization problem for random Hamilton-Jacobi equation
with non-convex stationary ergodic in
Hamiltonian
.
- (Elena Kosygina) Let
be a probability space with
probability measure
ergodic under a shift transformation group
,
. Consider the stochastic
Hamilton-Jacobi equation
where
is convex in
and satisfies some regularity
assumptions, for example
A homogenization result for this problem was obtained by Lions and
Souganidis. For the Hamiltonian above homogenization is equivalent
to a large deviation result for Brownian motion among random
obstacles (Snitzman).
Problem: Derive a large deviation result for general stationary
ergodic setting without any independence or mixing assumptions on
under the shifts.
- (Diogo Gomes) Investigate possible extensions of
the techniques used in the Aubry-Mather theory and
Monge-Kantorovich problems to study linear programming problems
in infinite dimensions. An example of such extensions are the
stochastic Mather measures which can be used to analyze second
order Hamilton-Jacobi equations.
- (Diogo Gomes) For a small perturbation of a completely
integrable Hamiltonian system a viscosity solution corresponding
to a non-resonant unperturbed torus can be uniformly approximated
by formal expansions. Similar expansions can be constructed for
the density of the Mather measures. However, it is not known how
well Mather measures themselves are approximated by these formal
expansions.
- (Massimiliano Berti) For a nonlinear wave equation
with Dirichlet boundary conditions and general nonlinearity there
exist a large number of small amplitude periodic orbit with fixed
period. Could one find small amplitude periodic orbits of large
period and quasi-periodic solutions for the wave equation?