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Trudy Inst. Mat. i Mekh. UrO RAN, 2012, Volume 18, Number 2, Pages 48–61 (Mi timm807)  

This article is cited in 2 scientific papers (total in 2 papers)

On Friedrichs inequalities for a disk

R. R. Gadyl'shin, E. A. Shishkina

Bashkir State Pedagogical University

Abstract: We consider the Friedrichs inequality for functions defined on a disk of unit radius $\Omega$ and equal to zero on almost all boundary except for an arc $\gamma_\varepsilon$ of length $\varepsilon$, where $\varepsilon$ is a small parameter. Using the method of matched asymptotic expansions, we construct a two-term asymptotics for the Friedrichs constant $C(\Omega,\partial\Omega\backslash\overline\gamma_\varepsilon)$ for such functions and present a strict proof of its validity. We show that $C(\Omega,\partial\Omega\backslash\overline\gamma_\varepsilon)=C(\Omega,\partial\Omega)+\varepsilon^2C(\Omega,\partial\Omega)(1+O(\varepsilon^{2/7}))$. The calculation of the asymptotics for the Friedrichs constant is reduced to constructing an asymptotics for the minimum value of the operator $-\Delta$ in the disk with Neumann boundary condition on $\gamma_\varepsilon$ and Dirichlet boundary condition on the remaining part of the boundary.

Keywords: Friedrichs inequality, small parameter, eigenvalue, asymptotics.

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English version:
Proceedings of the Steklov Institute of Mathematics (Supplementary issues), 2013, 281, suppl. 1, 44–58

Bibliographic databases:

UDC: 517.956
Received: 29.09.2011

Citation: R. R. Gadyl'shin, E. A. Shishkina, “On Friedrichs inequalities for a disk”, Trudy Inst. Mat. i Mekh. UrO RAN, 18, no. 2, 2012, 48–61; Proc. Steklov Inst. Math. (Suppl.), 281, suppl. 1 (2013), 44–58

Citation in format AMSBIB
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\by R.~R.~Gadyl'shin, E.~A.~Shishkina
\paper On Friedrichs inequalities for a~disk
\serial Trudy Inst. Mat. i Mekh. UrO RAN
\yr 2012
\vol 18
\issue 2
\pages 48--61
\mathnet{http://mi.mathnet.ru/timm807}
\elib{https://elibrary.ru/item.asp?id=17736185}
\transl
\jour Proc. Steklov Inst. Math. (Suppl.)
\yr 2013
\vol 281
\issue , suppl. 1
\pages 44--58
\crossref{https://doi.org/10.1134/S0081543813050052}
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\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84879162667}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. R. R. Gadyl'shin, S. V. Repjevskij, E. A. Shishkina, “On an eigenvalue for the Laplace operator in a disk with Dirichlet boundary condition on a small part of the boundary in a critical case”, Proc. Steklov Inst. Math. (Suppl.), 292, suppl. 1 (2016), 76–90  mathnet  crossref  mathscinet  isi  elib
    2. R. R. Gadylshin, A. A. Ershov, S. V. Repyevsky, “On asymptotic formula for electric resistance of conductor with small contacts”, Ufa Math. J., 7:3 (2015), 15–27  mathnet  crossref  isi  elib
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