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Fundamentalnaya i Prikladnaya Matematika, 2014, Volume 19, Issue 4, Pages 101–120
(Mi fpm1599)
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On integral representation of $\Gamma$-limit functionals
V. V. Zhikova, S. E. Pastukhovab a Vladimir State University
b Moscow State Institute of Radio-Engineering, Electronics and Automation (Technical University)
Abstract:
We consider the $\Gamma$-convergence of a sequence of integral functionals $F_n(u)$, defined on the functions $u$ from the Sobolev space $W^{1,\alpha}(\Omega)$ ($\alpha>1$), $\Omega$ is a bounded Lipschitz domain, where the integrand $f_n(x,u,\nabla u)$ depends on a function $u$ and its gradient $\nabla u$. As functions of $\xi$, the integrands $f_n(x,s,\xi)$ are convex and satisfy a two-sided power estimate on the coercivity and growth with different exponents $\alpha<\beta$. Besides, the integrands $f_n(x,s,\xi)$ are equi-continuous over $s$ in some sense with respect to $n$. We prove that for the functions from $L^\infty(\Omega)\cap W^{1,\beta}(\Omega)$ the $\Gamma$-limit functional coincides with an integral functional $F(u)$ for which the integrand $f(x,s,\xi)$ is of the same class as $f_n(x,s,\xi)$.
Citation:
V. V. Zhikov, S. E. Pastukhova, “On integral representation of $\Gamma$-limit functionals”, Fundam. Prikl. Mat., 19:4 (2014), 101–120; J. Math. Sci., 217:6 (2016), 736–750
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https://www.mathnet.ru/eng/fpm1599 https://www.mathnet.ru/eng/fpm/v19/i4/p101
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Abstract page: | 512 | Full-text PDF : | 189 | References: | 71 |
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