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 Fundam. Prikl. Mat.: Year: Volume: Issue: Page: Find

 Fundam. Prikl. Mat., 2014, Volume 19, Issue 4, Pages 101–120 (Mi fpm1599)

On integral representation of $\Gamma$-limit functionals

V. V. Zhikova, S. E. Pastukhovab

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 $f_n(x,s,\xi)$.

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English version:
Journal of Mathematical Sciences (New York), 2016, 217:6, 736–750

Bibliographic databases:

UDC: 517.956.8

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

Citation in format AMSBIB
\Bibitem{ZhiPas14} \by V.~V.~Zhikov, S.~E.~Pastukhova \paper On integral representation of $\Gamma$-limit functionals \jour Fundam. Prikl. Mat. \yr 2014 \vol 19 \issue 4 \pages 101--120 \mathnet{http://mi.mathnet.ru/fpm1599} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3431886} \transl \jour J. Math. Sci. \yr 2016 \vol 217 \issue 6 \pages 736--750 \crossref{https://doi.org/10.1007/s10958-016-3002-z}