Theme  of the Conference

Variational methods in PDEs and in ODEs

The idea to use variational principles to prove existence or multiplicity results for solutions to PDEs (elliptic problems, minimal surfaces, Yamabe problem) or ODEs (periodic orbits in Hamiltonian Systems) goes back at least to Poincaré, and has been succesfully exploited throughout this century. Two classical methods are (1) the direct approach in the calculus of variations, in which existence of minimizers of functionals is proved by suitable compactness arguments, to obtain, e.g. existence results for minimal surfaces; and (2) Morse and Ljusternik-Schnirelman theory, in which topology and analysis are combined to deduce existence of  periodic orbits in mechanical systems.   The last few decades have seen a rapid growth in the number of problems which have been attacked, as well as the emergence of  new variational methods which have been brought to bear upon them, e.g.
 
Minimax arguments have been used for the  construction of closed orbits in Hamiltonian systems with starshaped energy surfaces

Floer introduced a new approach to Morse theory which allowed him to prove  the Arnol'd conjecture in many cases; the complete conjecture was proved by several groups of people. Further developments involve the variational study of closed characteristics on contact manifolds.

Variational methods have been used in varying degrees of abstractness and generality to construct so-called chaotic solutions or ìmultibump solutionsî in Hamiltonian systems, and in Lagrangian systems (monotone twist maps of an annulus being the simplest example.)

Bifurcation and Degree theory

The Brouwer and Leray - Schauder degrees have classically been used for existence proofs for PDEs, and to keep track of solutions to PDEs through bifurcations (e.g. in the Crandall - Rabinowitz bifurcation theorem). More recent developments concern the extension of the definition of the Brouwer degree to maps which are not continuous but close to continuous, and the use of the Leray - Schauder degree in singular perturbation problems.
 

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