Optimization of some eigenvalue problems with large drift

TL;DR

This paper investigates eigenvalue problems for non-symmetric elliptic operators with large drift terms, analyzing the asymptotic behavior of principal eigenvalues and eigenfunctions in bounded domains with Dirichlet conditions.

Contribution

It provides new insights into the asymptotic behavior of eigenvalues and eigenfunctions for large drifts, including their convergence and gradient profiles.

Findings

Max points of eigenfunctions converge to points maximizing distance to boundary

Eigenfunctions exhibit a uniform asymptotic profile near the boundary

Gradient directions align with the boundary in the asymptotic limit

Abstract

This paper is concerned with eigenvalue problems for non-symmetric elliptic operators with large drifts in bounded domains under Dirichlet boundary conditions. We consider the minimal principal eigenvalue and the related principal eigenfunction in the class of drifts having a given, but large, pointwise upper bound. We show that, in the asymptotic limit of large drifts, the maximal points of the optimal principal eigenfunctions converge to the set of points maximizing the distance to the boundary of the domain. We also show the uniform asymptotic profile of these principal eigenfunctions and the direction of their gradients in neighborhoods of the boundary.