Workshop on Variational Methods and Nash-Moser | PIMS - Pacific Institute for the Mathematical Sciences

Mini-course 1

The Nash-Moser method and applications (Massimiliano Berti and Philippe Bolle)

Lecture 1:
Periodic and quasiperiodic solutions near an elliptic equilibrium for Hamiltonian PDEs: presentation of the problem. We shall specially focus on periodic solutions of nonlinear wave equations. Lyapunov-Schmidt reduction: the range and the bifurcation equations. Small denominator problem and statement of a Nash Moser implicit function theorem for the range equation. Variational structure of the bifurcation equation.

Lecture 2:
Nash Moser-type iterative scheme. Convergence proof, under appropriate weak invert-ibility assumptions on the linearized problems.

Lectures 3-4:
Inversion of the linearized equations in presence of small divisors for periodic solutions in any spatial dimensions.

Reference Material

M. Berti, P. Bolle, Cantor families of periodic solutions for completely resonant nonlinear wave equations, Duke Math. J. 134 (2006) 359-419.
M. Berti, P. Bolle, Cantor families of periodic solutions for wave equations via a variational principle, Advances in Mathematics. 217 (2008) 1671-1727.
M. Berti, P. Bolle, Sobolev periodic solutions of nonlinear wave equations in higher spatial dimensions, preprint 2008.
J. Bourgain , Green's function estimates for lattice Schodinger operators and applications, Annals of Mathematics Studies 158, Princeton University Press, Princeton, 2005.
W. Craig, Problemes de petits diviseurs dans les equations aux derivees partielles, Panoramas et Syntheses, 9, Societe Mathematique de France, Paris, 2000.
S. Kuksin, Analysis of Hamiltonian PDEs, Oxford Lecture series in Mathematics and its applications 19, Oxford University Press, 2000.

Mini-course 2

Symmetries and collisions in the n-body problem 9 (Davide Ferrario and Susanna Terracini)

Lecture 1: Davide L. Ferrario
Lecture 2: Susanna Terracini
Lecture 3: Davide L. Ferrario
Lecture 4: Susanna Terracini

Symmetries and the variational formulation of the n-body problem.
Equivariant minimization
Planar symmetry groups
Collisions
McGehee coordinates and total collisions
Asymptotic estimates
Averaged variations
Local equivariant variations
Transitive decomposition of symmetry groups
Collisions and singularities.

Reference Material

D. L. Ferrario: Transitive decomposition of symmetry groups for the $n$-body problem: Adv. in Math. 213 (2007) 763-784.
D. L. Ferrario: Symmetry groups and non-planar collisionless action-minimizing solutions of the three-body problem in three-dimensional space. Arch. Rational Mech. Anal. 179 (2006), 389--412.
D. L. Ferrario and S. Terracini: On the existence of collisionless equivariant minimizers for the classical n-body problem. Inventiones Mathematicae, Vol. 155 N. 2 (2004), 305--362.
V.Barutello, D. L. Ferrario and S. Terracini: On the singularities of generalized solutions to $n$--body type problems: math.DS/0701174 (to appear in IMRN)
V. Barutello, D. L. Ferrario and S. Terracini: Symmetry groups of the planar 3-body problem and action-minimizing trajectories (to appear in Arch. Rational Mech. Anal..

All papers can be downloaded from http://www.matapp.unimib.it/~ferrario/papers/index.html

Both the Nash-Moser implicit function theorem and variational methods are well-established tools to study nonlinear differential equations. Both have met with great success in the past, and continue to be perfected. What is new, however, is the conjunction of theses methods. Roughly speaking, many nonlinear problems near resonance can be seen as bifurcation problems, which in turn can be solved by a Liapounov-Schmidt procedure. This means that one first has to "project" the problem on the image of the linearized operator (this is where Nash-Moser comes in, since there is loss of regularity), and then one has to solve the reduced problem (this is where the variational structure comes in). We refer to the survey paper of Biasco and Valdinocci for an excellent survey of this technique. They list as applications:

the spatial planetary three-body problem,
the planar planetary many-body problem,
periodic orbits approaching lower-dimensional elliptic KAM tori, and
long-time periodic orbits for the nonlinear wave equation.

As soon as one moves away from weak interaction, the picture changes and bifurcation methods can no longer be applied. In the domain of strong interaction, new progress has been made as well, with the discovery of new types of periodic solutions (choregraphies) in then-body problem. Variational methods have been essential in this progress. On the one hand, these solutions appear as critical points of some reduced problem, after quotienting by a finite group of symmetries. On the other, using the variational characterization, one has gained a much better understanding of collisions (or the absence thereof). Of course, these two approaches are complementary. We feel that there is much to be gained in the interplay between them, and this is the purpose of this workshop.

Abstracts etc.

Monday, June 16, 2008

9:00-10:30 Terracini-Ferrario
10:30-11:00 Coffee break
11:00-12:30 Berti-Bolle
12:30-2:00 Lunch break
2:00- 3:00 Zehnder
3:00- 3:30 Coffee break
3:30- 4:30 Stoica

Tuesday, June 17

9:00-10:30 Terracini-Ferrario
10:30-11:00 Coffee break
11:00-12:30 Berti-Bolle
12:30-2:00 Lunch break
2:00- 3:00 Chierchia
3:00- 3:30 Coffee break
3:30- 4:30 Santoprete

Wednesday, June 18

9:00-10:00 Xia
10:00-11:00 Bolotin
11:00-11:30 Coffee break
11:30-12:30 Perez-Chavela
12:30-2:00 Lunch break
2:00-3:00 Long
3:00- 4:00 Sere
4:00- 4:30 Coffee break
4:30- 5:30 Bernard
5:30- 6:30 Craig

Thursday, June 19

9:00-10:30 Terracini-Ferrario
10:30-11:00 Coffee break
11:00-12:30 Berti-Bolle
12:30-2:00 Lunch break
2:00- 3:00 Hofer
3:00- 3:30 Coffee break
3:30- 4:30 Diacu

Friday, June 19

9:00-10:30 Terracini-Ferrario
10:30-11:00 Coffee break
11:00-12:30 Berti-Bolle

Jump to Event Details

This is a Past Event

Event Type

Scientific, Workshop

Date

June 16–22, 2008

Time

9:00am - 5:00pm

Location

University of British Columbia

West Mall Annex Room 110
1933 West Mall, UBC