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Nonlinear Analysis

  • Variational Methods
  • Perturbative Methods in Critical Point Theory
  • Elliptic Equations on R^n and Nonlinear Schroedinger Equation
  • H-Surfaces
  • Singularly Perturbed Problems
  • Geometric PDEs
  • Hamiltonian systems
  • Chaotic Dynamics and Arnold Diffusion
  • KAM Theory
  • Periodic Solutions of Infinite Dimensional Systems

Crystalline anisotropic mean curvature flow

  • Mean curvature flow of hypersurfaces. Introduction to the problem.
  • First variation of the perimeter.
  • Uniqueness of smooth flows. Short time existence.
  • Examples of singularities: Grayson example.
  • Fattening of the crossing.
  • Anisotropic mean curvature flow.
  • Finsler metrics and their duals.
  • The Wulff shape. Duality mappings.
  • Relations between the Minkowski content (or anisotropic perimeter) and the Hausdorff measure.

Nonlinear Analysis and Dynamical Systems

In the first part of this course, I will present the basic elements of calculus in infinite dimensional spaces, holomorphic functions, and the classical implicit function theorem in Banach spaces.

Nonlinear Analysis and Bifurcation Theory

The goal of the course is to cover the following topics:

  • Basic calculus in Banach spaces
  • Local inversion theorems and implicit function theory 
  • Properties of bifurcation points 
  • Bifurcation from the simple eigenvalue 
  • Bifurcation from multiple eigenvalues
  • Examples and applications 


  • Ambrosetti-Prodi, A primer in Nonlinear Analysis

Topics in Nonlinear Analysis

In this course I will present the Nash-Moser hard implicit function theorem in scales of Banach spaces with different applications, as, for example, the Siegel linear conjugacy theorem or the KAM (Kolmogorov-Arnold-Moser) theorem concerning quasi-periodic solutions of nearly integrable Hamiltonian systems.  


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