Uniform convexity and variational convergence
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V. V. Zhikov and S. E. Pastukhova
Translated by: V. E. Nazaikinskii - Trans. Moscow Math. Soc. 2014, 205-231
- DOI: https://doi.org/10.1090/S0077-1554-2014-00232-6
- Published electronically: November 5, 2014
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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 $|\xi |^{p(x)}$, 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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Bibliographic 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
- 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.
- © Copyright 2014 American Mathematical Society
- 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
- MathSciNet review: 3308610
Dedicated: Dedicated to the Centennial Anniversary of B. M. Levitan