The family index theorem and bifurcation of solutions of nonlinear elliptic bvp

TL;DR

This paper introduces new bifurcation criteria for nonlinear elliptic boundary value problems using the Atiyah-Singer family index theorem, providing explicit, stable conditions based solely on the coefficients of the differential operators.

Contribution

It develops a novel approach to bifurcation analysis for nonlinear elliptic systems by applying the Atiyah-Singer family index theorem, bypassing traditional reduction methods.

Findings

New bifurcation criteria derived from the Atiyah-Singer theorem

Explicit computation of bifurcation conditions from coefficients

Criteria are stable under lower order perturbations

Abstract

We obtain some new bifurcation criteria for solutions of general boundary value problems for nonlinear elliptic systems of partial differential equations. The results are of different nature from the ones that can be obtained via the traditional Lyapunov-Schmidt reduction. Our sufficient conditions for bifurcation are derived from the Atiyah-Singer family index theorem and therefore they depend only on the coefficients of derivatives of leading order of the linearized differential operators. They are computed explicitly from the coefficients without the need of solving the linearized equations. Moreover, they are stable under lower order perturbations.