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On Extremal Domains for Integral Inequalities in the Euclidean Space

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Abstract

On domains of the Euclidean space we consider Hardy and Rellich type inequalities with weight functions depending on the distance to the boundary of the domain. We show that extremal domains are not single for some known inequalities with sharp estimates of constants. We describe a family of extremal domains for Hardy type inequalities of a general form. On plane domains we study a new Rellich type inequality having a similar property.

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References

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Funding

This work was funded by the subsidy allocated to Kazan Federal University for the state assignment in the sphere of scientific activities (1.9773.2017/8.9), and by the RFBR, grant no. 17-01-00282-a.

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Authors and Affiliations

  1. Kazan Federal University, 18 Kremlyovskaya str., Kazan, 420008, Russia

    F. G. Avkhadiev

Authors
  1. F. G. Avkhadiev
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Correspondence to F. G. Avkhadiev.

Additional information

Russian Text © The Author(s), 2019, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2019, No. 6, pp. 80–84.

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Avkhadiev, F.G. On Extremal Domains for Integral Inequalities in the Euclidean Space. Russ Math. 63, 74–78 (2019). https://doi.org/10.3103/S1066369X19060082

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  • DOI: https://doi.org/10.3103/S1066369X19060082

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