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Tr. Mosk. Mat. Obs., 2014, Volume 75, Issue 2, Pages 245–276 (Mi mmo566)  

This article is cited in 1 scientific paper (total in 1 paper)

Uniform convexity and variational convergence

V. V. Zhikova, S. E. Pastukhovab

a Vladimir State University
b Moscow Institute of Radio-Engineering, Electronics and Automation

Abstract: Let $ \Omega $ be a domain in $ \mathbb{R}^d$. We establish the uniform convexity of the $ \Gamma $-limit of a sequence of Carathéodory integrands $ f(x,\xi )\colon \Omega { \times }\mathbb{R}^d\to \mathbb{R}$ subjected to a two-sided power-law estimate of coercivity and growth with respect to $ \xi $ with exponents $ \alpha $ and $ \beta $, $ 1<\alpha \le \beta <\infty $, and having a common modulus of convexity with respect to $ \xi $. In particular, the $ \Gamma $-limit of a sequence of power-law integrands of the form $ \vert\xi \vert^{p(x)}$, where the variable exponent $ p\colon \Omega \to [\alpha ,\beta ]$ is a measurable function, is uniformly convex.
We prove that one can assign a uniformly convex Orlicz space to the $ \Gamma $-limit of a sequence of power-law integrands. A natural $ \Gamma $-closed extension of the class of power-law integrands is found.
Applications to the homogenization theory for functionals of the calculus of variations and for monotone operators are given.

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English version:
Transactions of the Moscow Mathematical Society, 2014, 75, 205–231

UDC: 517.951, 517.956
MSC: 35J20, 35J60, 46B10, 46B20, 49J45, 49J50
Received: 29.03.2014

Citation: V. V. Zhikov, S. E. Pastukhova, “Uniform convexity and variational convergence”, Tr. Mosk. Mat. Obs., 75, no. 2, MCCME, M., 2014, 245–276; Trans. Moscow Math. Soc., 75 (2014), 205–231

Citation in format AMSBIB
\Bibitem{ZhiPas14}
\by V.~V.~Zhikov, S.~E.~Pastukhova
\paper Uniform convexity and variational convergence
\serial Tr. Mosk. Mat. Obs.
\yr 2014
\vol 75
\issue 2
\pages 245--276
\publ MCCME
\publaddr M.
\mathnet{http://mi.mathnet.ru/mmo566}
\elib{https://elibrary.ru/item.asp?id=23780165}
\transl
\jour Trans. Moscow Math. Soc.
\yr 2014
\vol 75
\pages 205--231
\crossref{https://doi.org/10.1090/S0077-1554-2014-00232-6}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84960110966}


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    This publication is cited in the following articles:
    1. A. Pankov, “Elliptic operators with nonstandard growth condition: some results and open problems”, Differential Equations, Mathematical Physics, and Applications: Selim Grigorievich Krein Centennial, Contemporary Mathematics, 734, eds. P. Kuchment, E. Semenov, Amer. Math. Soc., 2019, 277–292  crossref  mathscinet  zmath  isi
  • Trudy Moskovskogo Matematicheskogo Obshchestva
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