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 Izv. Vyssh. Uchebn. Zaved. Mat., 2016, Number 1, Pages 69–73 (Mi ivm9070)

Brief communications

Rellich type inequalities in domains of the Euclidean space

Chair of Function Theory and Approximations, Kazan (Volga Region) Federal University, 18 Kremlyovskaya str., Kazan, 420008 Russia

Abstract: For smooth functions supported in a domain of the Euclidean space we investigate two Rellich type inequalities with weights which are powers of the distance function. We prove that for an arbitrary plane domain there exist positive Rellich constants in these inequalities if and only if the boundary of the domain is a uniformly perfect set. Moreover, we obtain explicit estimates of constants in function of geometric domain characteristics. Also, we find sharp constants in these Rellich type inequalities for all non-convex domains of dimension $d\geq2$ provided that the domains satisfy the exterior sphere condition with certain restriction on the radius of spheres.

Keywords: Rellich type inequality, uniformly perfect set, distance function, non-convex domain.

 Funding Agency Grant Number Russian Foundation for Basic Research 14-01-00351-a

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English version:
Russian Mathematics (Izvestiya VUZ. Matematika), 2016, 60:1, 60–63

Bibliographic databases:

Document Type: Article
UDC: 517.5+517.956.225

Citation: F. G. Avkhadiev, “Rellich type inequalities in domains of the Euclidean space”, Izv. Vyssh. Uchebn. Zaved. Mat., 2016, no. 1, 69–73; Russian Math. (Iz. VUZ), 60:1 (2016), 60–63

Citation in format AMSBIB
\Bibitem{Avk16} \by F.~G.~Avkhadiev \paper Rellich type inequalities in domains of the Euclidean space \jour Izv. Vyssh. Uchebn. Zaved. Mat. \yr 2016 \issue 1 \pages 69--73 \mathnet{http://mi.mathnet.ru/ivm9070} \transl \jour Russian Math. (Iz. VUZ) \yr 2016 \vol 60 \issue 1 \pages 60--63 \crossref{https://doi.org/10.3103/S1066369X16010060} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000409280700006} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84953311341} 

• http://mi.mathnet.ru/eng/ivm9070
• http://mi.mathnet.ru/eng/ivm/y2016/i1/p69

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. F. G. Avkhadiev, “Estimates of Hardy–Rellich constants for polyharmonic operators and their generalizations”, Ufa Math. Journal, 9:3 (2017), 8–17
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