On passage to the~limit in nonlinear variational problems
Sbornik. Mathematics, Tome 76 (1993) no. 2, pp. 427-459
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A study is made of variational problems with convex Lagrangians $f(x,\xi)$ subordinate to a nonstandard estimate
\begin{gather*}
-c_0+c_1|\xi|^{\alpha_1}\leqslant f(x,\xi)\leqslant c_0+c_2|\xi|^{\alpha_2},
\\
c_0\geqslant 0, c_1>0, \quad c_2>0, \quad 1\alpha_1\leqslant\alpha_2.
\end{gather*}
The concepts of $\Gamma_1$-convergence and $\Gamma_2$-convergence are introduced for Lagrangians corresponding to boundary value problems of the first and second types. It is proved that the indicated class of Lagrangians is compact with respect to
$\Gamma_1$-convergence and with respect to $\Gamma_2$-convergence. Applications to compactness theorems and to various concrete averaging problems are given.
@article{SM_1993_76_2_a9,
author = {V. V. Zhikov},
title = {On passage to the~limit in nonlinear variational problems},
journal = {Sbornik. Mathematics},
pages = {427--459},
publisher = {mathdoc},
volume = {76},
number = {2},
year = {1993},
language = {en},
url = {http://geodesic.mathdoc.fr/item/SM_1993_76_2_a9/}
}
V. V. Zhikov. On passage to the~limit in nonlinear variational problems. Sbornik. Mathematics, Tome 76 (1993) no. 2, pp. 427-459. http://geodesic.mathdoc.fr/item/SM_1993_76_2_a9/