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Uniform convexity and variational convergence


Authors: V. V. Zhikov and S. E. Pastukhova
Translated by: V. E. Nazaikinskii
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 75 (2014), vypusk 2.
Journal: Trans. Moscow Math. Soc. 2014, 205-231
MSC (2010): Primary 35J20; Secondary 35J60, 46B10, 46B20, 49J45, 49J50
DOI: https://doi.org/10.1090/S0077-1554-2014-00232-6
Published electronically: November 5, 2014
MathSciNet review: 3308610
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Abstract | References | Similar Articles | Additional Information

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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Additional Information

V. V. Zhikov
Affiliation: A. G. and N. G. Stoletov Vladimir State University, Vladimir, Russia
Email: zhikov@vlsu.ru

S. E. Pastukhova
Affiliation: Moscow State Technical University of Radio Engineering, Electronics, and Automation, Moscow, Russia
Email: pas-se@yandex.ru

DOI: https://doi.org/10.1090/S0077-1554-2014-00232-6
Keywords: Uniform convexity, $\Gamma$-convergence, Orlicz spaces, power-law integrand, nonstandard coercivity, and growth conditions
Published electronically: November 5, 2014
Additional Notes: Supported by RFBR grant no. 14-01-00192a, grant no. NSh-3685.2014.1 of the President of the Russian Federation, and Russian Scientific Foundation grant no. 14-11-00398.
Dedicated: Dedicated to the Centennial Anniversary of B. M. Levitan
Article copyright: © Copyright 2014 American Mathematical Society