Periodic homogenization of the principal eigenvalue of second-order elliptic operators

TL;DR

This paper studies how the principal eigenvalues and eigenfunctions of oscillatory second-order elliptic operators behave as the oscillations become small, providing convergence rates and extending results to nonlinear cases.

Contribution

It establishes homogenization results for the principal eigenvalue and eigenfunction of oscillatory elliptic operators, including nonlinear cases, with explicit convergence rates.

Findings

Eigenvalues converge to the effective operator eigenvalues.

Convergence rates are provided for linear and nonlinear problems.

Normalized eigenfunctions converge linearly under certain conditions.

Abstract

In this paper we investigate homogenization results for the principal eigenvalue problem associated to $1$-homogeneous, uniformly elliptic, second-order operators. Under rather general assumptions, we prove that the principal eigenpair associated to an oscillatory operator converges to the eigenpair associated to the effective one. This includes the case of fully nonlinear operators. Rates of convergence for the eigenvalues are provided for linear and nonlinear problems, under extra regularity/convexity assumptions. Finally, a linear rate of convergence (in terms of the oscillation parameter) of suitably normalized eigenfunctions is obtained for linear problems.