Some sharp Sobolev inequalities on $ BV({\mathbb{R}}^n) $

Jin Dai; Shuang Mou; Jin Dai; Shuang Mou

doi:10.3934/math.2022925

AIMS Mathematics

2022, Volume 7, Issue 9: 16851-16868. doi: 10.3934/math.2022925

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Research article

Some sharp Sobolev inequalities on $ BV({\mathbb{R}}^n) $

Jin Dai ^,,
Shuang Mou

School of Mathematics and Statistics, Shaanxi Normal University, Xi'an, 710119, China

Received: 09 May 2022 Revised: 28 June 2022 Accepted: 04 July 2022 Published: 14 July 2022
MSC : 46B20, 46E35, 52A21

In this paper, some sharp Sobolev inequalities on $ BV({\mathbb{R}}^n) $, the space of functions of bounded variation on $ {\mathbb{R}}^n $, $ n\geq 2 $, are deduced through the $ L_p $ Brunn-Minkowski theory. We will prove that these inequalities can all imply the sharp Sobolev inequality on $ BV({\mathbb{R}}^n) $.
- Sobolev inequalities,
- functions of bounded variation,
- optimal constants
Citation: Jin Dai, Shuang Mou. Some sharp Sobolev inequalities on $ BV({\mathbb{R}}^n) $[J]. AIMS Mathematics, 2022, 7(9): 16851-16868. doi: 10.3934/math.2022925

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Abstract

In this paper, some sharp Sobolev inequalities on $ BV({\mathbb{R}}^n) $, the space of functions of bounded variation on $ {\mathbb{R}}^n $, $ n\geq 2 $, are deduced through the $ L_p $ Brunn-Minkowski theory. We will prove that these inequalities can all imply the sharp Sobolev inequality on $ BV({\mathbb{R}}^n) $.

References

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