Limiting behavior of principal eigenvalues and eigenfunctions for a class of elliptic operators with degenerate large advection

2025/08/22 by Cano-Casanova, S., López-Gómez, J., Molina-Meyer, M.
#34B09 #34D15 #34L15 #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2508.16108

Abstract

In this paper we study, both numerically and analytically, the asymptotic behavior of the principal eigenfunction of \eqref1.1, normalized by \eqref1.2, as s\uparrow +∞. Based on the numerical computations of this paper, we can prove that, under condition (Hm) bellow, φ_s approximates 1 and φ_s' approximates 0, uniformly in [-1,1], as s\uparrow +∞. As a byproduct of this result, we can derive the asymptotic behavior of the principal eigenvalue in a one-dimensional situation not previously covered by \citeChLo and \citePeZh, as we are working under minimal regularity assumptions on m(x). A recent result of \citeBWZ shows that the principal eigenvalue might oscillate as s\uparrow +∞ if m(x) is highly oscillatory. Thus, the oscillatory and regularity properties of m(x) might severely affect the asymptotic behavior of (λ_s,φ_s) as s\uparrow +∞.

Citations

• Non-convergence of the principal eigenvalue of elliptic operators for large advection
• Effects of large degenerate advection and boundary conditions on the principal eigenvalue and its eigenfunction of a linear second order elliptic operator
• Diffusion-mediated permanence problem for a heterogeneous Lotka–Volterra competition model

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