PROGRAM:

• 2 courses of 10 hours each and 3 invited 1 hour talks

COURSES: (running June 18-22. See ABSTRACTS below)

• Changfeng Gui (Univ. of Connecticut):
  "Multi peak solutions and solutions of semilinear equations in phase transitions and a conjecture of De Giorgi"
• Regis Monneau (CERMICS-ENPC, France):
  "Introduction to dislocations dynamics"

INVITED 1-hour TALKS: (Sala d'Actes of the FME building)

• June 18 (16h - 17h) : Ioannis Athanasopoulos (University of Crete, Heraklion):
  "A free boundary problem of codimension two"
  
• June 18 (17h - 18h) : Juan Luis Vázquez (Universidad Autonoma de Madrid):
  "Stabilization with rates in fast diffusion and weighted Sobolev inequalities"
• June 19 (16h - 17h) : Henrik Shahgholian (Royal Institute of Technology, Stockholm):
  "The singular set of the free boundaries in two membrane problem"

REGISTRATION, PRACTICAL INFO, and SCHEDULE:

• Students attending the GLOBAL School should register at the web page JISD07.
• Note that there will be financial support, partially from GLOBAL, for students to attend the school.
• A REGISTRATION FORM (including financial support), PRACTICAL INFO, and the SCHEDULE for the GLOBAL school are posted in the JISD07.
• The "GLOBAL School on PDEs" is part of a larger event: JISD07. Students and researchers attending the GLOBAL school may be interested in attending other courses and events taking place within JISD07.
• "GLOBAL", Global and Geometric Aspects of Nonlinear PDE, is a European Science Foundation (ESF) Programme.

  For more info at this point, contact:
  xavier.cabre@upc.edu

ABSTRACTS for the COURSES:

• Changfeng Gui
  "Multi peak solutions and solutions of semilinear equations in phase transitions and a conjecture of De Giorgi"
  
  In this course, I will first introduce some of the fundamental methods in nonlinear partial differential equations such as variational methods, the maximum principle, etc.
  
  Then, I will discuss two types of interesting solutions: the spike-layer solutions which arise in the analysis of the shadow system of a biological pattern formation model (the Gierier-Meinhardt system); and the transition layer solutions which play an important role in the study of phase transition via the Allen-Cahn equation and its counterpart in system of equations.
  
  For the spike-layer solutions, I will emphasize the new variational methods for the existence of higher energy solutions with multiple concentration. Regarding transition layer solutions, I will talk about the symmetry of solutions in entire spaces including De Giorgi conjecture, the existence of triple, quadruple junction solutions, a new Hamiltonian type equality and its application, etc. The topics are related to current research interests.

• Regis Monneau
  "Introduction to dislocations dynamics"

  We will give an introduction to dislocations dynamics. Dislocations are curves defects in a crystal. When a stress is applied on the crystal, these curves can move with a dynamics given by the normal velocity depending on the whole shape of the curves and on the interactions with the other defects in the crystal.
  
  Mathematically, this dynamics is described by non-local Hamilton-Jacobi equations in the framework of viscosity solutions.
  
  After presenting classical results on homogenization, we will give in particular some results about the homogenization of the dynamics of self-interacting dislocations. In the limit we recover an effective plastic law which involves a fractional Levy operator. This will be an opportunity to present the introduction of recent tools on homogenization.
  

May 07 - RMC