Department of Differential Equations

In the Department of Differential Equations, research is conducted in the following areas:

• Nonlinear Topological Analysis
  • Applications of the Brouwer and Leray–Schauder degrees and the Conley index to the study of solutions of abstract nonlinear problems.

• Invariant Topological Variational Methods
  • Applications:
    • degree of equivariant gradient mappings,
    • invariant Morse theory,
    • invariant Conley index
  • for the study of orbits of invariant abstract nonlinear problems possessing a variational structure.
  • Sufficient conditions for existence, continuation, global bifurcations, and symmetry breaking of critical orbits of invariant functionals.
  • Development of the theory of the degree of equivariant gradient mappings and its connections with the invariant Conley index and invariant Morse theory.
  • Classification of equivariant gradient mappings.

• Differential Equations in Mathematical Physics and Mechanics
  • Existence, continuation, global bifurcation, and symmetry-breaking problems for:
    • solutions of systems of elliptic equations with Dirichlet and Neumann boundary conditions,
    • periodic solutions of wave equations,
    • periodic solutions of autonomous Hamiltonian and Newtonian systems.
  • Hamiltonian equations with degenerate equilibrium positions and with resonance at equilibrium positions and at infinity.

• Celestial Mechanics
  • Existence, continuation, global bifurcation, and symmetry-breaking of periodic, homoclinic, and heteroclinic solutions of Hamiltonian systems in celestial mechanics.
  • Connected sets of periodic solutions.
  • The N-body problem and the restricted N-body problem.
  • The Hill problem.