RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Impact factor Subscription Journal history Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Tr. Mosk. Mat. Obs.: Year: Volume: Issue: Page: Find

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

Uniform convexity and variational convergence

V. V. Zhikova, S. E. Pastukhovab

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.

Full text: PDF file (377 kB)
References: PDF file   HTML file

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

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

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} 

• http://mi.mathnet.ru/eng/mmo566
• http://mi.mathnet.ru/eng/mmo/v75/i2/p245

 SHARE:

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. 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
•  Number of views: This page: 359 Full text: 124 References: 45