Geodesic
$\Gamma$-convergence of oscillating thin obstacles
Eurasian mathematical journal, Tome 4 (2013) no. 4, pp. 88-100.

Voir la notice de l'article provenant de la source Math-Net.Ru

We consider the minimization problems of obstacle type $$ \min\left\{\int_\Omega|Du|^2\,dx\colon u\ge\psi_\varepsilon\ \text{on}\ P,\ u=0\ \text{on}\ \partial\Omega\right\}, $$ as $\varepsilon\to0$. Here $\Omega$ is a bounded domain in $\mathbb R^n$, $\psi_\varepsilon$ is a periodic function of period $\varepsilon$, constructed from a fixed function $\psi$, and $P\subset\subset\Omega$ is a subset of the hyper-plane $\{x\in\mathbb R^n\colon x\cdot\eta=0\}$. We assume that $n\ge3$ and that the normal $\eta$ satisfies a generic condition that guarantees certain ergodic properties of the quantity $$ \#\left\{k\in\mathbb Z^n\colon P\cap\{x\colon|x-\varepsilon k|\varepsilon^{n/(n-1)}\}\right\}. $$ Under these hypotheses we compute explicitly the limit functional of the obstacle problem above, which is of the type $$ H^1_0(\Omega)\owns u\mapsto\int_\Omega|Du|^2\,dx+\int_PG(u)\,d\sigma. $$ 
@article{EMJ_2013_4_4_a5,
     author = {Yu. O. Koroleva and M. H. Str\"omqvist},
     title = {$\Gamma$-convergence of oscillating thin obstacles},
     journal = {Eurasian mathematical journal},
     pages = {88--100},
     publisher = {mathdoc},
     volume = {4},
     number = {4},
     year = {2013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EMJ_2013_4_4_a5/}
}
TY  - JOUR
AU  - Yu. O. Koroleva
AU  - M. H. Strömqvist
TI  - $\Gamma$-convergence of oscillating thin obstacles
JO  - Eurasian mathematical journal
PY  - 2013
SP  - 88
EP  - 100
VL  - 4
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/EMJ_2013_4_4_a5/
LA  - en
ID  - EMJ_2013_4_4_a5
ER  - 
%0 Journal Article
%A Yu. O. Koroleva
%A M. H. Strömqvist
%T $\Gamma$-convergence of oscillating thin obstacles
%J Eurasian mathematical journal
%D 2013
%P 88-100
%V 4
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/EMJ_2013_4_4_a5/
%G en
%F EMJ_2013_4_4_a5
Yu. O. Koroleva; M. H. Strömqvist. $\Gamma$-convergence of oscillating thin obstacles. Eurasian mathematical journal, Tome 4 (2013) no. 4, pp. 88-100. http://geodesic.mathdoc.fr/item/EMJ_2013_4_4_a5/

[1] H. Attouch, C. Picard, “Variational inequalities with varying obstacles: the general form of the problem”, J. Funct. Anal., 50:3 (1983), 329–386 | DOI | MR

[2] D. Cioranescu, F. Murat, “A strange term coming from nowhere”, Topics in the mathematical modelling of composite materials, Progr. Nonlinear Differential Equations Appl., 31, Birkhäuser Boston, Boston, MA, 1997, 45–93 | MR | Zbl

[3] G. Dal Maso, P. Trebeschi, “$\Gamma$-limit of periodic obstacles”, Acta Appl. Math., 65:1–3, Special issue dedicated to Antonio Avantaggiati on the occasion of his 70th birthday (2001), 207–215 | MR | Zbl

[4] E. De Giorgi, G. Dal Maso, P. Longo, “$\Gamma$-limits of obstacles”, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 68:6 (1980), 481–487 | MR | Zbl

[5] D. Kinderlehrer, G. Stampacchia, An introduction to variational inequalities and their applications, Classics in Applied Mathematics, 31, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000, Reprint of the 1980 original | MR | Zbl

[6] K.-A. Lee, M. Strömqvist, M. Yoo, Highly oscillating thin obstacles, 2012, arXiv: 1204.3462 | MR