Singular analysis of the principal eigenvalue for cooperative elliptic systems with drifts

Abstract

We determine the small-diffusion limit of the principal eigenvalue for a one-dimensional cooperative elliptic system with component-dependent drifts and Neumann boundary conditions. For scalar equations, this limit is determined by local data at the drift equilibria and boundary points. We show that such localization fails for systems: switching between components can make some intervals spectrally relevant. We first prove that the logarithmic transforms of all component eigenfunctions converge to the same function, which solves a scalar Hamilton-Jacobi equation. The associated Aubry set determines the asymptotic behavior of the principal eigenvalue. Under nondegeneracy assumptions, this set decomposes into isolated points, regular intervals, and composite intervals containing common drift zeros. We then identify the effective value of each class through local Ornstein-Uhlenbeck spectra and interval transport problems, and establish that the limit of the principal eigenvalue is the minimum of these class values. A unique minimizing class also determines the common logarithmic eigenfunction profile.