On Extremal Domains for Integral Inequalities in the Euclidean Space
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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.
KeywordsHardy inequality Rellich inequality
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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.