# JISD2012

### Tenth WORKSHOP ON INTERACTIONS BETWEEN DYNAMICAL SYSTEMS AND PARTIAL DIFFERENTIAL EQUATIONS (JISD2012)JORNADES D'INTERACCIÓ ENTRE SISTEMES DINÀMICS I EQUACIONS EN DERIVADES PARCIALS

#### Barcelona, May 28 - June 1, 2012

The tenth edition of the WORKSHOP ON INTERACTIONS BETWEEN DYNAMICAL SYSTEMS AND PARTIAL DIFFERENTIAL EQUATIONS (JISD2012) will be held in Barcelona, May 28 - June 1, 2012, at the Universitat Politècnica de Catalunya (UPC)

JISD former editions

There will be four main courses, some seminars, communications, and posters. The four courses will be taught by M. Capinski, M. Gidea, H. Shahgholian, and E. Valdinoci, within the Master of Science in Advanced Mathematics and Mathematical Engineering (MAMME) of the UPC Graduate School.

Supported by the MICINN, FME, UPC, SCM, RSME, SEMA.

New information: Participants / Abstracts

Organizers
- Xavier Cabré
- Maria del Mar González
- Tere M. Seara

Scientific Committee
- Massimiliano Berti (Univ. Federico II)
- Rafael de la Llave (Univ. Texas Austin)
- Jean-Michel Roquejoffre (Univ. Paul Sabatier. Toulouse)
- Alfonso Sorrentino (Univ. Of Cambridge)
- Marco Antonio Teixeira (Univ. Estatal de Campinas)
- Juan Luis Vázquez (UAM)

There will be some *financial support* available for this edition.
Deadline to apply for financial support: March 30, 2012.
Deadline to register: May 1st, 2012.

## Contents

Courses will be held in the room S02 of the FME building (Facultat de Matemàtiques i Estadística), at C/ Pau Gargallo, n. 5 Barcelona, 08028.

 Course Abstract Geometric methods for invariant manifolds in dynamical systems M. Capinski (AGH Univ. of Science and Technology Al. Mickiewicza, Kraków, Poland)(Syllabus) Prof. Capinski lecture notes:1, 2, 3, 4 NEW! Invariant manifolds can be used to determine global behaviour of dynamical systems. They can often be proved using purely geometric or topological arguments. In the course we demonstrate how to apply such techniques for proofs of fixed points, periodic orbits, stable/unstable manifolds and normally hyperbolic manifolds. We show how to combine geometric methods with rigorous-computer-assisted computations, obtaining proofs for problems inaccessible using standard techniques. Aubry Mather Theory from a Topological Viewpoint M. Gidea (Northeastern Illinois University, Chicago, USA) (Syllabus)Prof. Gidea lecture notes 1,2, 3 NEW! This course will review some basic facts on Aubry-Mather theory from a variational viewpoint, and then it will turn to a topological viewpoint. We will prove, using topological methods, the existence of Aubry-Mather sets, and their shadowing properties. We will also discuss some generalizations of the Aubry-Mather theory. In the end, we will provide some applications to Hamiltonian instability and celestial mechanics. Obstacle type free boundaries (Theory and applications)H. Shahgholian (Royal Institute of Technology, Stockholm, Sweeden) (Syllabus) In these lectures I shall treat the so-called Obstacle type free boundaries, which refers to problems of the type $$\Delta u = f(x,u, \nabla u),$$where there $f$ exhibits a jump discontinuity in its second and third variables. More precisely I shall consider three type of jump discontinuity for $f$: $$\chi_{\{u\neq 0\}}, \qquad \chi_{\{|\nabla u |>0\}}, \qquad \lambda_+ \chi_{\{u>0\}}- \lambda_- \chi_{\{u<0\}}$$ These problems are all in their "nature" close to the so-called obstacle problem, which minimizes the energy of a stretched membrane over given obstacle, and can be reformulated as $$\Delta u = \chi_{\{u>0\}}, \qquad u \geq 0.$$ These problems, and some variations of them, have been subject for intense studies in the last few decades. Today there is more or less a complete and comprehensive theory for these problems. (Non) local phase transition equationsE. Valdinoci (Università degli Studi di Milano, Italy) (Syllabus)Prof. Valdinoci references 1,2, 3 NEW! I would like to present some differential equations arising in phase cohexisence models. In the classical case, phase changes are regulated by a local interaction, which leads to an elliptic partial differential equations and which produces interfaces close to surfaces of minimal perimeter. In order to take into account long range effects, a new model has been recently considered, in which the interaction is non-local, for instance, modeled on the Gagliardo seminorm on fractional Sobolev spaces. The associated equation is driven by a fractional Laplace operator and the interfaces are now either local or non-local, according to the fractional parameter.