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The \(\infty \)-capacity and Faber–Krahn inequality on Grushin spaces

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Abstract

In this paper we investigate the \(\infty \)-capacity and the Faber–Krahn inequality on the Grushin space \({\mathbb {G}}^n_\alpha \). On the one hand we show that the \(\infty \)-capacity equals the limit of the p-th root of the p-capacity as \(p\rightarrow \infty \) and give a simple geometric characterization in terms of the Carnot–Carathéodory distance, on the other hand we establish some basic properties for the \(\infty \)-capacity, \(\infty \)-eigenvalue and the Faber–Krahn inequality on the Grushin space \({\mathbb {G}}^n_\alpha \).

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Acknowledgements

The authors also would like to thank the referee for his valuable comments which improve the presentation of this article. Moreover, the second author is supported by the National Natural Science Foundation of China (No. 11671031, No. 11471018), the Fundamental Research Funds for the Central Universities (No. FRF-BR-17-004B), and Beijing Municipal Science and Technology Project (No. Z17111000220000).

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  1. School of Mathematics and Physics, University of Science and Technology Beijing, Beijing, 100083, People’s Republic of China

    Guoliang Li & Yu Liu

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  1. Guoliang Li
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  2. Yu Liu
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Correspondence to Yu Liu.

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Communicated by Adrian Constantin.

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Li, G., Liu, Y. The \(\infty \)-capacity and Faber–Krahn inequality on Grushin spaces. Monatsh Math 196, 135–162 (2021). https://doi.org/10.1007/s00605-021-01557-1

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