# Continuity of nonlinear eigenvalues in $CD(K,\infty)$ spaces with respect to measured Gromov-Hausdorff convergence

## TL;DR

This paper establishes the stability of the nonlinear Krasnoselskii spectrum of the Laplace operator in $CD(K, ext{infinity})$ spaces under measured Gromov-Hausdorff convergence, ensuring spectral elements are actual eigenvalues.

## Contribution

It proves the spectral stability and eigenvalue characterization of the Laplace operator in $CD(K, ext{infinity})$ spaces under Gromov-Hausdorff convergence, extending known results to a nonlinear setting.

## Key findings

- Krasnoselskii spectrum is stable under measured Gromov-Hausdorff convergence.
- Every spectral element corresponds to an eigenvalue with a nontrivial eigenfunction.
- Stability holds under additional compactness assumptions, such as bounded diameter.

## Abstract

In this note we prove in the nonlinear setting of $CD(K,\infty)$ spaces the stability of the Krasnoselskii spectrum of the Laplace operator $-\Delta$ under measured Gromov-Hausdorff convergence, under an additional compactness assumption satisfied, for instance, by sequences of $CD^*(K,N)$ metric measure spaces with uniformly bounded diameter. Additionally, we show that every element $\lambda$ in the Krasnoselskii spectrum is indeed an eigenvalue, namely there exists a nontrivial $u$ satisfying the eigenvalue equation $- \Delta u = \lambda u$.