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Attainability of the best Sobolev constant in a ball

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Abstract

The best constant in the Sobolev inequality in the whole space is attained by the Aubin–Talenti function; however, this does not happen in bounded domains because of the break down of the dilation invariance. In this paper, we investigate a new scale invariant form of the Sobolev inequality in a ball and show that its best constant is attained by functions of the Aubin–Talenti type. Generalization to the Caffarelli–Kohn–Nirenberg inequality in a ball is also discussed.

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Acknowledgements

This work was partially funded by JSPS KAKENHI # 18K13441.

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Authors and Affiliations

  1. Graduate School of Science and Engineering, Ehime University, Matsuyama, Ehime, 790-8577, Japan

    Norisuke Ioku

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  1. Norisuke Ioku
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Correspondence to Norisuke Ioku.

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Communicated by Y. Giga.

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Ioku, N. Attainability of the best Sobolev constant in a ball. Math. Ann. 375, 1–16 (2019). https://doi.org/10.1007/s00208-018-1776-7

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  • DOI: https://doi.org/10.1007/s00208-018-1776-7

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