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Two Consequences of Davies’ Hardy Inequality

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Abstract

Davies’ version of the Hardy inequality gives a lower bound for the Dirichlet integral of a function vanishing on the boundary of a domain in terms of the integral of the squared function with a weight containing the averaged distance to the boundary. This inequality is applied to easily derive two classical results of spectral theory, E. Lieb’s inequality for the first eigenvalue of the Dirichlet Laplacian and G. Rozenblum’s estimate for the spectral counting function of the Laplacian in an unbounded domain in terms of the number of disjoint balls of preset size whose intersection with the domain is large enough.

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References

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Funding

R. L. F. acknowledges the support of U. S. National Science Foundation, grants DMS-1363432 and DMS-1954995. S. L. acknowledges the support of the Knut and Alice Wallenberg Foundation, grant KAW 2018.0281.

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Authors and Affiliations

  1. Mathematisches Institut, Ludwig-Maximilians-Universität München, München, Germany

    R. L. Frank

  2. Department of Mathematics, California Institute of Technology, Pasadena, USA

    R. L. Frank & S. Larson

Authors
  1. R. L. Frank
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  2. S. Larson
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Correspondence to R. L. Frank or S. Larson.

Additional information

Translated from Funktsional'nyi Analiz i ego Prilozheniya, 2021, Vol. 55, pp. 118–121 https://doi.org/10.4213/faa3863.

In memory of M. Z. Solomyak, on the occasion of his 90th birthday

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Frank, R.L., Larson, S. Two Consequences of Davies’ Hardy Inequality. Funct Anal Its Appl 55, 174–177 (2021). https://doi.org/10.1134/S0016266321020106

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  • DOI: https://doi.org/10.1134/S0016266321020106

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