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Continuity of nonlinear eigenvalues in \({{\mathrm{CD}}}(K,\infty )\) spaces with respect to measured Gromov–Hausdorff convergence

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Abstract

In this note we prove in the nonlinear setting of \({{\mathrm{CD}}}(K,\infty )\) 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 \({{\mathrm{CD}}}^*(K,N)\) 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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Acknowledgements

The first author acknowledges the support of the MIUR PRIN 2015 grant. The second author acknowledges the support of the JSPS Program for Advancing Strategic International Networks to Accelerate the Circulation of Talented Researchers, the Grantin-Aid for Young Scientists (B) 16K17585 and the warm hospitality of SNS. The third author thanks Mark Peletier, Georg Prokert and Oliver Tse for helpful discussions and the SNS for its hospitality.

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Authors and Affiliations

  1. Scuola Normale Superiore, Pisa, Italy

    Luigi Ambrosio

  2. Tohoku University, Sendai, Japan

    Shouhei Honda

  3. Eindhoven University, Eindhoven, Netherlands

    Jacobus W. Portegies

Authors
  1. Luigi Ambrosio
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  2. Shouhei Honda
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  3. Jacobus W. Portegies
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Correspondence to Luigi Ambrosio.

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Communicated by M. Struwe.

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Ambrosio, L., Honda, S. & Portegies, J.W. Continuity of nonlinear eigenvalues in \({{\mathrm{CD}}}(K,\infty )\) spaces with respect to measured Gromov–Hausdorff convergence. Calc. Var. 57, 34 (2018). https://doi.org/10.1007/s00526-018-1315-0

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