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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

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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