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arXiv · 1706.08368

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

Luigi Ambrosio·Shouhei Honda·Jacobus W. Portegies
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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 $- \Delta u = \lambda u$.

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Luigi Ambrosio, Shouhei Honda, Jacobus W. Portegies. 2017-06-26. Continuity of nonlinear eigenvalues in $CD(K,\infty)$ spaces with respect to measured Gromov-Hausdorff convergence. https://arxiv.org/abs/1706.08368

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