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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
Research Group: 
Mathematical Analysis, Modelling, and Applications

Nash-Moser implicit function theorems and KAM theory

The main assumption of the classical implict function theorem in Banach spaces is that the linearized operator has a bounded inverse. This is sufficient for constructing bifurcation theory of periodic solutions of finite dimensional dynamical systems. On the other hand, there are several problems where this assumption is not satisfied, i.e. the linearized operator is unbounded, for example for the search of quasi-periodic solutions. To overcome this challenge, the Nash-Moser theory was developed. 
 

Topological and variational methods in critical point theory

  • Degree theory
  • Sard's Theorem
  • The Brouwer fixed point theorem with applications
  • The Schauder fixed-point theorem with applications
  • Critical points
  • Differential calculus and critical points; constrained critical points
  • Minimization problems
  • Linear eigenvalues and their variational characterization
  • Ekeland's variational principle
  • The Palais-Smale condition
  • Min-Max methods
  • Linking and Mountain-Pass  theorems

Topological Degree and Variational Methods, with Applications to the Problem of Bubbles with Prescribed Mean Curvarture

  • Degree theory:
    • Topological approach to finite-dimensional problems.
    • Sard's Theorem.
    • Finite dimensional degree theory and the Brouwer fixed point theorem.
    • Topological degree in infinite-dimensional Banach spaces.
    • The Schauder fixed-point theorem.
  • Application to the H-bubble problem:
    • Preliminaries on the H-bubble problem: the mean curvature of a radial graph in Rn over the unit sphere.

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