The Equivariant Spectral Flow and Bifurcation for Strongly Indefinite Functionals with Symmetries

• Applicant: Professor Dr. Nils Waterstraat
• Subject Area: Mathematics
• Term: from 2021 to 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 459826435

Let us consider a family of differential equations depending on a real parameter and let us assume that the constant function 0 is a solution of all these equations. Bifurcation theory is concerned with the question whether there are values of the parameter at which other solutions appear out of the zero solution. As solutions of differential equations are often the critical points of a functional on a Hilbert space, it is a common setting in bifurcation theory to study functionals with a parameter.A well-known theorem in bifurcation theory says that if the Morse indices of the functionals at 0 are finite, then a parameter value is a bifurcation point if the Morse index changes when the parameter passes that value. Often the functionals of interest in bifurcation theory do not have finite Morse indices (e.g. for Hamiltonian systems) and there have been numerous attempts to find suitable bifurcation invariants in such settings. A very general approach to this problem was introduced by Fitzpatrick, Pejsachowicz and Recht in 1999, when they showed that an integer-valued invariant from global analysis, called spectral flow, is a valuable substitute of the Morse indices in case that the latter are infinite. Their theorem covers the previously known results and it has been applied to various types of differential equations.Differential equations modelling phenomena in nature often have symmetries, which can be described by the effect of a Lie group action on the equations. Such symmetries can impose restrictions on Morse indices and so it may happen that they are forced to be constant in a family of functionals. Consequently, bifurcation cannot be studied in this case by looking for changes of the Morse index. Smoller and Wasserman showed in a seminal paper from 1990 that this problem can be overcome by using group representations. In recent years some progress has been made to use representation theory for finding bifurcation invariants for particular types of equations with symmetries that do not have finite Morse indices.The ultimate aim of this project is to prove an equivariant version of Fitzpatrick, Pejsachowicz and Recht’s bifurcation theorem that shows the existence of bifurcation points even if the spectral flow is taken out by symmetries. The new bifurcation invariant is an equivariant spectral flow which is an element of the representation ring of the acting Lie group. If the group action is trivial, our theorem covers Fitzpatrick, Pejsachowicz and Recht’s work. If the Morse indices are finite, we obtain Smoller and Wasserman’s theorem. We apply our bifurcation theorem to Hamiltonian systems and strongly indefinite elliptic systems with symmetries, which cannot be studied by already existing methods.

• DFG Programme: Research Grants