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Zap. Nauchn. Sem. POMI, 2010, Volume 385, Pages 224–233 (Mi znsl3907)  

This article is cited in 7 scientific papers (total in 7 papers)

Some Poincaré-type inequalities for functions of bounded deformation involving the deviatoric part of the symmetric gradient

M. Fuchsa, S. Repinb

a Universität des Saarlandes, Fachbereich 6.1 Mathematik, Saarbrücken, Germany
b St. Petersburg Department of Steklov Institute of Mathematics, St. Petersburg, Russia

Abstract: If $\Omega\subset\mathbb R^n$ is a bounded Lipschitz domain, we prove the inequality $\|u\|_1\le c(n)\operatorname{diam}(\Omega)\int_\Omega|\varepsilon^D(u)|$ being valid for functions of bounded deformation vanishing on $\partial\Omega$. Here $\varepsilon^D(u)$ denotes the deviatoric part of the symmetric gradient and $\int_\Omega|\varepsilon^D(u)|$ stands for the total variation of the tensor-valued measure $\varepsilon^D(u)$. Further results concern possible extensions of this Poincaré-type inequality. Bibl. 27 titles.

Key words and phrases: functions of bounded deformation, Poincaré' s inequality.

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English version:
Journal of Mathematical Sciences (New York), 2011, 178:3, 367–372

UDC: 517
Received: 30.05.2010
Language:

Citation: M. Fuchs, S. Repin, “Some Poincaré-type inequalities for functions of bounded deformation involving the deviatoric part of the symmetric gradient”, Boundary-value problems of mathematical physics and related problems of function theory. Part 41, Zap. Nauchn. Sem. POMI, 385, POMI, St. Petersburg, 2010, 224–233; J. Math. Sci. (N. Y.), 178:3 (2011), 367–372

Citation in format AMSBIB
\Bibitem{FucRep10}
\by M.~Fuchs, S.~Repin
\paper Some Poincar\'e-type inequalities for functions of bounded deformation involving the deviatoric part of the symmetric gradient
\inbook Boundary-value problems of mathematical physics and related problems of function theory. Part~41
\serial Zap. Nauchn. Sem. POMI
\yr 2010
\vol 385
\pages 224--233
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl3907}
\transl
\jour J. Math. Sci. (N. Y.)
\yr 2011
\vol 178
\issue 3
\pages 367--372
\crossref{https://doi.org/10.1007/s10958-011-0554-9}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-80053532861}


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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. Fuchs M., Repin S., “A posteriori error estimates for the approximations of the stresses in the Hencky plasticity problem”, Numer. Funct. Anal. Optim., 32:6 (2011), 610–640  crossref  mathscinet  zmath  isi  elib  scopus
    2. Fuchs M., “Computable upper bounds for the constants in Poincaré-type inequalities for fields of bounded deformation”, Math. Methods Appl. Sci., 34:15 (2011), 1920–1932  crossref  mathscinet  zmath  isi  elib  scopus
    3. Breit D., Schirra O.D., “Korn-Type Inequalities in Orlicz-Sobolev Spaces Involving the Trace-Free Part of the Symmetric Gradient and Applications to Regularity Theory”, Z. Anal. ihre. Anwend., 31:3 (2012), 335–356  crossref  mathscinet  zmath  isi  elib  scopus
    4. Neff P., Pauly D., Witsch K.-J., “Poincaré Meets Korn Via Maxwell: Extending Korn'S First Inequality To Incompatible Tensor Fields”, J. Differ. Equ., 258:4 (2015), 1267–1302  crossref  mathscinet  zmath  isi  elib  scopus
    5. Bauer S., Neff P., Pauly D., Starke G., “Dev-Div- and Devsym-Devcurl-Inequalities For Incompatible Square Tensor Fields With Mixed Boundary Conditions”, ESAIM-Control OPtim. Calc. Var., 22:1 (2016), 112–133  crossref  mathscinet  zmath  isi  scopus
    6. Breit D., Cianchi A., Diening L., “Trace-Free Korn Inequalities in Orlicz Spaces”, SIAM J. Math. Anal., 49:4 (2017), 2496–2526  crossref  mathscinet  zmath  isi  scopus
    7. Breit D., Diening L., Gmeineder F., “On the Trace Operator For Functions of Bounded a-Variation”, Anal. PDE, 13:2 (2020), 559–594  crossref  mathscinet  zmath  isi
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