## Workshop on Variational Methods and Nash-Moser

- Start Date: 06/16/2008
- End Date: 06/22/2008

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

University of British Columbia

**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)**

*Davide L. Ferrario*

Lecture 1:

Lecture 1:

*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..

*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:

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.

GageInfo.pdf

VanierInfo.pdf

Sobolev periodic solutions of nonlinear wave equations in higher spatial dimensions.pdf

08_nash_poster3.pdf

**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..

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

**Scholarships
**

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

Please contact the organizers.

**Accommodation**

Please read the following items carefully.

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