Stability of variational eigenvalues for the fractional p-Laplacian

TL;DR

This paper studies how the eigenvalues and eigenfunctions of the fractional p-Laplacian behave as it approaches the classical p-Laplacian, using Gamma-convergence to establish stability and convergence results.

Contribution

It provides new insights into the stability and convergence of fractional p-Laplacian eigenvalues and eigenfunctions in the singular limit, extending the understanding of nonlocal to local operator transitions.

Findings

Eigenvalues are stable under Gamma-convergence as the operator approaches the p-Laplacian.

Normalized eigenfunctions converge in a suitable fractional norm.

Results apply to eigenvalues associated with a given topological index.

Abstract

By virtue of $\Gamma-$convergence arguments, we investigate the stability of variational eigenvalues associated with a given topological index for the fractional $p$-Laplacian operator, in the singular limit as the nonlocal operator converges to the $p$-Laplacian. We also obtain the convergence of the corresponding normalized eigenfunctions in a suitable fractional norm.