2010/05/07 by Jacobo Pejsachowicz, Pejsachowicz, Jacobo · 2 citations
Mathematics · Physics and Astronomy · #47A53 #55N15 #58E07 (Primary) #58J20 (Secondary) #58J32 #58J55 #Advanced Differential Equations and Dynamical Systems #Algebraic Topology (math.AT) #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Nonlinear Differential Equations Analysis #Nonlinear Waves and Solitons #math.AP #math.AT #math.DG #msc:47A53 #msc:55N15 #msc:58E07 #msc:58J20 #msc:58J32 #msc:58J55
paper · pdf · doi:10.48550/arxiv.1005.2077
42 pages. Changes: added Lemma 2.31 and a reference + minor corrections. To appear on TMNA
openalex publication_date 2010/05/07 · arxiv created 2011/09/12 · arxiv updated 2011/09/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We associate to a parametrized family f of nonlinear Fredholm maps possessing a trivial branch of zeroes an \it index of bifurcation β(f) which provides an algebraic measure for the number of bifurcation points from the trivial branch. The index β(f) is derived from the index bundle of the linearization of the family along the trivial branch by means of the generalized J-homomorphism. Using the Agranovich reduction and a cohomological form of the Atiyah-Singer family index theorem, due to Fedosov, we compute the bifurcation index of a multiparameter family of nonlinear elliptic boundary value problems from the principal symbol of the linearization along the trivial branch. In this way we obtain criteria for bifurcation of solutions of nonlinear elliptic equations which cannot be achieved using the classical Lyapunov-Schmidt method.