## Workshop on Variational Methods and Nash-Moser

• Start Date: 06/16/2008
• End Date: 06/22/2008
Speaker(s):

Patrick Bernard - Ceremade - Université Paris - Dauphine
Sergey V. Bolotin - University of Wisconsin, Madison WI
Luigi Chierchia - Dipartimento Di Matematica, Università Roma Tre
Helmut Hofer - Courant Institute - New York University, NY
Yiming Long - Nankai University, China
Ernesto Pérez Chavela - Universidad Autónoma Metropolitana - Iztapalapa, México, Mexico
Eric Sere - Ceremade, Universite Paris - Dauphine
Cristina Stoica -Wilfrid Laurier University, Waterloo, Ontario
Zhihong Jeff Xia - Northwestern University, Evanston, IL
Eduard Zehnder - ETH-Zentrum, Switzerland

Location:

University of British Columbia

Topic:
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

1. M. Berti, P. Bolle, Cantor families of periodic solutions for completely resonant nonlinear wave equations, Duke Math. J. 134 (2006) 359-419.
2. M. Berti, P. Bolle, Cantor families of periodic solutions for wave equations via a variational principle, Advances in Mathematics. 217 (2008) 1671-1727.
3. M. Berti, P. Bolle, Sobolev periodic solutions of nonlinear wave equations in higher spatial dimensions, preprint 2008.
4. J. Bourgain , Green's function estimates for lattice Schodinger operators and applications, Annals of Mathematics Studies 158, Princeton University Press, Princeton, 2005.
5. W. Craig, Problemes de petits diviseurs dans les equations aux derivees partielles, Panoramas et Syntheses, 9, Societe Mathematique de France, Paris, 2000.
6. 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

1. Symmetries and the variational formulation of the n-body problem.
2. Equivariant minimization
3. Planar symmetry groups
4. Collisions
5. McGehee coordinates and total collisions
6. Asymptotic estimates
7. Averaged variations
8. Local equivariant variations
9. Transitive decomposition of symmetry groups
10. Collisions and singularities.

Reference Material

1. D. L. Ferrario: Transitive decomposition of symmetry groups for the $n$-body problem: Adv. in Math. 213 (2007) 763-784.
2. 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.
3. 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.
4. 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)
5. 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..

Description:

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:
1. the spatial planetary three-body problem,
2. the planar planetary many-body problem,
3. periodic orbits approaching lower-dimensional elliptic KAM tori, and
4. 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 the
n-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.

Schedule:

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

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.

Lectures 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

1. M. Berti, P. Bolle, Cantor families of periodic solutions for completely resonant nonlinear wave equations, Duke Math. J. 134 (2006) 359-419.
2. M. Berti, P. Bolle, Cantor families of periodic solutions for wave equations via a variational principle, Advances in Mathematics. 217 (2008) 1671-1727.
3. M. Berti, P. Bolle, Sobolev periodic solutions of nonlinear wave equations in higher spatial dimensions, preprint 2008.
4. J. Bourgain , Green's function estimates for lattice Schodinger operators and applications, Annals of Mathematics Studies 158, Princeton University Press, Princeton, 2005.
5. W. Craig, Problemes de petits diviseurs dans les equations aux derivees partielles, Panoramas et Syntheses, 9, Societe Mathematique de France, Paris, 2000.
6. 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

1. Symmetries and the variational formulation of the n-body problem.
2. Equivariant minimization
3. Planar symmetry groups
4. Collisions
5. McGehee coordinates and total collisions
6. Asymptotic estimates
7. Averaged variations
8. Local equivariant variations
9. Transitive decomposition of symmetry groups
10. Collisions and singularities.

Reference Material

1. D. L. Ferrario: Transitive decomposition of symmetry groups for the $n$-body problem: Adv. in Math. 213 (2007) 763-784.
2. 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.
3. 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.
4. 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)
5. 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..

Other Information:

Scholarships

A restricted number of scholarships are available for graduate students and post-docs to attend the workshop.

Accommodation

1. Attachments below:

VanierInfo.pdf

GageInfo.pdf

Please print a copy of the attached document regarding your accommodations (it contains address & facilities & contact info)
**Please check the list below for your arrival & departure dates or you may need to arrange extra accommodations with me (kleung@pims.math.ca).
If I have not heard from your by Jun 9th, I will assume your acceptance for this arrangement.

2. Visitor info For directions to UBC Please print & visit this website http://www.ubc.ca/about/directions.html
UBC campus map:
http://www.maps.ubc.ca/PROD/images/pdf/ubcmap.pdf
**Note: West Coast Suite (Gage Residence) is in Box C5
If you take a taxi (cab) at the airport, it will cost you $25-30/trip If you take a bus, it will cost your$3.75/trip Please consult the tourist info desk at the airport for details.

3. Venue: West Mall Annex Room 110, PIMS-UBC - All participants will receive a campus map with further instructions how to get to WMAX 110 and PIMS-UBC at the Gage reception upon arrival. Please check the website http://www.pims.math.ca/science/2008/08nash/ regularly for latest updates.

4. Accommodation guest list for pims-funded participants at Tec De Monterrey Residence ordered by last name:
Last, First, Gender, Arrive, Depart
1 Biasco Luca M 15-Jun 22-Jun
2 Celli Martin M 15-Jun 21-Jun
3 Fokam Jean-Marcel M 15-Jun 22-Jun
4 Franco Perez Luis M 15-Jun 24-Jun
5 Niu Huawei M 15-Jun 22-Jun
6 Portalluri Alessandro M 14-Jun 21-Jun
7 Procesi Michela M 15-Jun 22-Jun
8 Santoprete Manuele M 15-Jun 22-Jun
9 Sanvito Cristina F 14-Jun 22-Jun
10 Stoica Cristina F 15-Jun 22-Jun
11 Su Feng M 15-Jun 21-Jun

5. Accommodation guest list for pims-funded participants at West Coast Suite ordered by last name:
1 Bernard Patrick M 7-Jun 23-Jun
2 Berti Massimiliano M 10-Jun 27-Jun
3 Bolle Philippe M 12-Jun 25-Jun
4 Bolotin Sergey M 15-Jun 22-Jun
5 Chierchia Luigi M 15-Jun 20-Jun
6 Craig Walter M 18-Jun 21-Jun
7 Diacu Florin M 15-Jun 22-Jun
8 Ferrario Davide M 14-Jun 21-Jun
9 Hofer Helmut M 18-Jun 21-Jun
10 Perez-Chavela Ernesto M 15-Jun 20-Jun
11 Sere Eric M 15-Jun 20-Jun
12 Terracini Susana F 14-Jun 21-Jun
13 Xia Jeff M 15-Jun 20-Jun
14 Zehnder Eduard M 14-Jun 21-Jun

Venue

Workshop will be held at West Mall Annex Room 110, PIMS-UBC

Registration

Registration is now closed