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 Tr. Mosk. Mat. Obs., 2019, Volume 80, Issue 2, Pages 221–246 (Mi mmo628)

An explicit form for extremal functions in the embedding constant problem for Sobolev spaces

I. A. Sheipak, T. A. Garmanova

Faculty of Mechanics and Mathematics, Moscow State University, Moscow, Russia

Abstract: The embedding constants for the Sobolev spaces $\mathring W^n_2[0;1]\hookrightarrow \mathring W^k_2[0;1]$ ($0\le k\le n-1$) are studied. A relationship between the embedding constants and the norms of the functionals $f\mapsto f^{(k)}(a)$ in the space $\mathring W^n_2[0;1]$ is given. An explicit form of the functions $g_{n,k}\in \mathring W^n_2[0;1]$ on which these functionals attain their norm is found. These functions are also extremals for the embedding constants. A connection between the embedding constants and the Legendre polynomials is put forward. A detailed study is made of the embedding constants for $k=3$ and $k=5$: explicit formulas for extreme points are obtained, global maximum points calculated, and the values of the sharp embedding constants is given. A link between the embedding constants and some class of spectral problems with distribution coefficients is established.

Key words and phrases: Sobolev spaces, embedding constants, Legendre polynomials.

 Funding Agency Grant Number Russian Science Foundation 17-11-01215 Russian Foundation for Basic Research 19-01-00240 The results of §§2 and 3 were obtained with the support of the Russian Foundation for Basic Research (grant no. 19-01-00240) and the results of §§4–6 were obtained with the support of the Russian Science Foundation (grant no. 17-11-01215).

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English version:
Transactions of the Moscow Mathematical Society, 2019, 80, 189–210

UDC: 517.518.23, 517.984
MSC: 46E35

Citation: I. A. Sheipak, T. A. Garmanova, “An explicit form for extremal functions in the embedding constant problem for Sobolev spaces”, Tr. Mosk. Mat. Obs., 80, no. 2, MCCME, M., 2019, 221–246; Trans. Moscow Math. Soc., 80 (2019), 189–210

Citation in format AMSBIB
\Bibitem{SheGar19} \by I.~A.~Sheipak, T.~A.~Garmanova \paper An explicit form for extremal functions in the embedding constant problem for Sobolev spaces \serial Tr. Mosk. Mat. Obs. \yr 2019 \vol 80 \issue 2 \pages 221--246 \publ MCCME \publaddr M. \mathnet{http://mi.mathnet.ru/mmo628} \transl \jour Trans. Moscow Math. Soc. \yr 2019 \vol 80 \pages 189--210 \crossref{https://doi.org/10.1090/mosc/292} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85083777138}