Geodesic
On the convergence of solutions of variational problems with bilateral obstacles in variable domains
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 22 (2016) no. 1, pp. 140-152 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

We establish sufficient conditions for the convergence of minimizers and minimum values of integral and more general functionals on sets of functions defined by bilateral obstacles in variable domains. The given obstacles are elements of the corresponding Sobolev space, and the degeneration on a set of measure zero is admitted for the difference of the upper and lower obstacles. We show that a weakening of the condition of positivity of this difference on a set of full measure may lead to a certain violation of the established convergence result.
Keywords: integral functional, minimizer, minimum value, bilateral obstacles, strong connected-ness.
Mots-clés : $\Gamma$-convergence 
@article{TIMM_2016_22_1_a13,
     author = {A. A. Kovalevsky},
     title = {On the convergence of solutions of variational problems with bilateral obstacles in variable domains},
     journal = {Trudy Instituta matematiki i mehaniki},
     pages = {140--152},
     year = {2016},
     volume = {22},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TIMM_2016_22_1_a13/}
}
TY  - JOUR
AU  - A. A. Kovalevsky
TI  - On the convergence of solutions of variational problems with bilateral obstacles in variable domains
JO  - Trudy Instituta matematiki i mehaniki
PY  - 2016
SP  - 140
EP  - 152
VL  - 22
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/TIMM_2016_22_1_a13/
LA  - ru
ID  - TIMM_2016_22_1_a13
ER  - 
%0 Journal Article
%A A. A. Kovalevsky
%T On the convergence of solutions of variational problems with bilateral obstacles in variable domains
%J Trudy Instituta matematiki i mehaniki
%D 2016
%P 140-152
%V 22
%N 1
%U http://geodesic.mathdoc.fr/item/TIMM_2016_22_1_a13/
%G ru
%F TIMM_2016_22_1_a13
A. A. Kovalevsky. On the convergence of solutions of variational problems with bilateral obstacles in variable domains. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 22 (2016) no. 1, pp. 140-152. http://geodesic.mathdoc.fr/item/TIMM_2016_22_1_a13/

[1] Kovalevskii A.A., “O nekotorykh voprosakh, svyazannykh s problemoi usredneniya variatsionnykh zadach dlya funktsionalov s peremennoi oblastyu opredeleniya”, Sovremennyi analiz i ego prilozheniya, Naukova dumka, Kiev, 1989, 62–70 | MR

[2] Kovalevsky A.A., “Obstacle problems in variable domains”, Complex Var. Elliptic Equ., 56:12 (2011), 1071–1083 | DOI | MR | Zbl

[3] Kovalevskii A.A., “$G$-skhodimost i usrednenie nelineinykh ellipticheskikh operatorov divergentnogo vida s peremennoi oblastyu opredeleniya”, Izv. RAN. Ser. matematicheskaya, 58:3 (1994), 3–35 | MR

[4] Murat F., Sur l'homogeneisation d'inequations elliptiques du 2`eme ordre, relatives au convexe $K(\psi_1,\psi_2)=\{v\in H^1_0(\Omega) | \psi_1\leqslant v\leqslant\psi_2\text{p.p. dans}\Omega\}$, Publ. Laboratoire d'Analyse Num'erique, no. 76013, Univ. Paris VI, 1976, 23 pp.

[5] Dal Maso G., “Asymptotic behaviour of minimum problems with bilateral obstacles”, Ann. Mat. Pura Appl. (4), 129:1 (1981), 327–366 | DOI | MR

[6] Khruslov E.Ya., “Asimptoticheskoe povedenie reshenii vtoroi kraevoi zadachi pri izmelchenii granitsy oblasti”, Mat. sb., 106:4 (1978), 604–621 | MR | Zbl

[7] Vainberg M.M., Variatsionnyi metod i metod monotonnykh operatorov v teorii nelineinykh uravnenii, Nauka, M., 1972, 416 pp. | MR

[8] Adams R.A., Sobolev spaces, Acad. Press, New York, 1975, 286 pp. | MR | Zbl

[9] Kovalevskii A.A., “Usloviya $\Gamma$-skhodimosti i usrednenie integralnykh funktsionalov s razlichnymi oblastyami opredeleniya”, Dokl. AN USSR, 1991, no. 4, 5–8 | MR

[10] Kovalevskii A.A., “O neobkhodimykh i dostatochnykh usloviyakh $\Gamma$-skhodimosti integralnykh funktsionalov s razlichnymi oblastyami opredeleniya”, Nelineinye granichnye zadachi, no. 4, 1992, 29–39 | MR

[11] Rudakova O.A., “O $\Gamma$-skhodimosti integralnykh funktsionalov, opredelennykh na razlichnykh vesovykh prostranstvakh Soboleva”, Ukr. mat. zhurn., 61:1 (2009), 99–115 | MR | Zbl

[12] Kovalevsky A.A., “On $L^1$-functions with a very singular behaviour”, Nonlinear Anal., 85 (2013), 66–77 | DOI | MR | Zbl

[13] Kovalevsky A.A., Rudakova O.A., “Variational problems with pointwise constraints and degeneration in variable domains”, Differ. Equ. Appl., 1:4 (2009), 517–559 | MR | Zbl