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Syllabus 2012

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

Barcelona, May 28 - June 1, 2012



Course
Syllabus
Geometric methods for invariant manifolds in dynamical systems

M. Capinski (AGH Univ. of Science and Technology Al. Mickiewicza, Krakow, Poland)
  • Fixed points and periodic orbits: Interval Newton method, Poincare maps and periodic orbits from fixed points.
  • Stable/unstable manifolds for hyperbolic fixed points: Cone conditions, horizontal/vertical discs, geometric construction of the manifolds.
  • Covering relations: Propagation and transversal intersections of stable/unstable manifolds, symbolic dynamics and chaos.
  • Normally hyperbolic invariant manifolds: Geometric proof of existence based on cone conditions and covering relations.
  • Spectral gap condition: Construction of invariant fibres of manifolds, smoothness of manifolds.
Aubry Mather Theory from a Topological Viewpoint

M. Gidea (Northeastern Illinois University, Chicago, USA)
Schedule
  • Background on Aubry-Mather theory for twist maps and tilt maps. Monotone periodic orbits. Existence of Aubry-Mather sets and shadowing properties. The variational approach. The topological approach.
  • Monotone periodic orbits for area preserving homeomorphisms of the annulus, and for general homeomorphisms of the annulus. Thurston-Nielsen classification of surface homeomorphisms.
  • Monotone recurrence relations and generalizations of Aubry-Mather theory to higher dimensional twist maps.
  • Applications to Hamiltonian instability and to celestial mechanics.
Obstacle type free boundaries (Theory and applications)

H. Shahgholian (Royal institute of technology, Stockholm, Sweeden)
Schedule
  • Model problems.
  • Optimal regularity of solutions.
  • Preliminary analysis of the free boundary.
  • Regularity of the free boundary: first results.
  • Global solutions.
  • Regularity of the free boundary: uniform results.
References
  • Regularity of free boundaries in obstacle-type problems, Arshak Petrosyan, Henrik Shahgholian, Nina Uraltseva (Forthcoming: Graduate Studies in Mathematics series of the AMS).
(Non) local phase transition equations

E. Valdinoci ((Universita di Roma Tor Vergata, Roma, Italy)
  • Semilinear and quasilinear elliptic pde's, physical motivations, symmetry and rigidity properties, geometric Poincare inequalities, Harnack inequalities on level sets.
  • Fractional operators and nonlocal minimal surfaces, density estimates, regularity, asymptotics, open problems.
References
  • Alberti, G., Ambrosio, L., Cabre, X.: On a long-standing conjecture of E. De Giorgi: symmetry in 3D for general nonlinearities and a local minimality property. Acta Appl. Math. 65, no. 1-3, 9{33 (2001)
  • Ambrosio, L, Cabre, X.: Entire solutions of semilinear elliptic equations in R3 and a conjecture of De Giorgi, J. Amer. Math. Soc. 13, no. 4, 725{739 (2000)
  • Ambrosio, L., de Philippis, G., Martinazzi, L.: Gamma-convergence of nonlocal perimeter functionals, Manuscripta Math. 134, no. 3-4, 377{403 (2011)
  • Cabre, X., Cinti, E.: Energy estimates and 1-D symmetry for nonlinear equations involving the half-Laplacian. Discrete and Continuous Dynamical Systems 28, 1179-1206 (2010)
  • Cabre, X., Sola-Morales, J., Layer solutions in a half-space for boundary reactions, Commun. Pure Appl. Math. 5, no. 12, 1678-1732 (2005)
  • Cabre, X., Sire, Y.: Nonlinear equations for fractional Laplacians I: Regularity, maximum principles, and Hamiltonian estimates. Submitted paper. http://arxiv.org/abs/1012.0867 (2010)
  • Caffarelli, L. A., Roquejoffre, J.-M., Savin, O.: Nonlocal minimal surfaces, Comm. Pure Appl. Math. 63, no. 9, 1111--1144 (2010)
  • Caffarelli, L. A., Valdinoci, E.: Uniform estimates and limiting arguments for nonlocal minimal surfaces, Calc. Var. Partial Differential Equations 41, no. 1-2, 203-240 (2011)
  • Farina, A., Sciunzi, B., Valdinoci, E.: Bernstein and De Giorgi type problems: new results via a geometric approach, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 7, no. 4, 741-791 (2008)
  • Savin, O., Valdinoci, E.: Density estimates for a variational model driven by the Gagliardo norm. Submitted paper. http://arxiv.org/abs/1007.2114v3 (2011)
  • Savin, O., Valdinoci, E.: Gamma-convergence for nonlocal phase transitions. Submitted paper. http://arxiv.org/abs/1007.1725v3 (2011)
  • Silvestre, L.: Regularity of the obstacle problem for a fractional power of the Laplace operator. Ph.D. Thesis, Austin University. http://www.math.uchicago.edu/~luis/preprints/luisdissreadable.pdf (2005)
  • Sire, Y., Valdinoci, E.: Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result. J. Funct. Anal. 256, no. 6, 1842-1864 (2009)
  • Sternberg, P., Zumbrun, K.: A Poincare' inequality with applications to volume-constrained area-minimizing surfaces, J. Reine Angew. Math. 503, 63-85 (1998)