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For what functions does Nash inequality becomes an equality? Also any comment on the regularity of these functions (weak solutions to equality)?

Also same question about Poincare inequality.

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  • $\begingroup$ I searched google extensively and could not find anything about when equality is achieved. Hence I am posting this question. $\endgroup$
    – Rajesh D
    Commented Jul 28, 2018 at 1:41
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    $\begingroup$ You may consult the paper of Carlen and Loss academic.oup.com/imrn/article-abstract/1993/7/213/… $\endgroup$
    – nguyen0610
    Commented Jul 28, 2018 at 2:04
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    $\begingroup$ Check the papers "Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions" by Del Pino and Dolbeault and "A mass-transportation approach to sharp Sobolev and Gagliardo–Nirenberg inequalities" by Cordero, Nazaret, and Villani. $\endgroup$
    – Deane Yang
    Commented Jul 28, 2018 at 2:20

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