Geodesic
Lemmas on compensated compactness in elliptic and parabolic equations
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and dynamical systems, Tome 270 (2010), pp. 110-137 
V. V. Zhikov; S. E. Pastukhova. Lemmas on compensated compactness in elliptic and parabolic equations. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and dynamical systems, Tome 270 (2010), pp. 110-137. http://geodesic.mathdoc.fr/item/TM_2010_270_a7/
@article{TM_2010_270_a7,
     author = {V. V. Zhikov and S. E. Pastukhova},
     title = {Lemmas on compensated compactness in elliptic and parabolic equations},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {110--137},
     year = {2010},
     volume = {270},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2010_270_a7/}
}
TY  - JOUR
AU  - V. V. Zhikov
AU  - S. E. Pastukhova
TI  - Lemmas on compensated compactness in elliptic and parabolic equations
JO  - Trudy Matematicheskogo Instituta imeni V.A. Steklova
PY  - 2010
SP  - 110
EP  - 137
VL  - 270
UR  - http://geodesic.mathdoc.fr/item/TM_2010_270_a7/
LA  - ru
ID  - TM_2010_270_a7
ER  - 
%0 Journal Article
%A V. V. Zhikov
%A S. E. Pastukhova
%T Lemmas on compensated compactness in elliptic and parabolic equations
%J Trudy Matematicheskogo Instituta imeni V.A. Steklova
%D 2010
%P 110-137
%V 270
%U http://geodesic.mathdoc.fr/item/TM_2010_270_a7/
%G ru
%F TM_2010_270_a7

Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

We study the solvability of parabolic and elliptic equations of monotone type with nonstandard coercivity and boundedness conditions that do not fall within the scope of the classical method of monotone operators. To construct a solution, we apply a technique of passing to the limit in approximation schemes. A key element of this technique is a generalized lemma on compensated compactness. The parabolic version of this lemma is rather complicated and is proved for the first time in the present paper. The new technique applies to stationary and nonstationary problems of fast diffusion in an incompressible flow, to a parabolic equation with a $p(x,t)$-Laplacian and its generalization, and to a nonstationary thermistor system.

[1] Zhikov V.V., “K tekhnike predelnogo perekhoda v nelineinykh ellipticheskikh uravneniyakh”, DAN, 420:3 (2008), 300–305 | MR | Zbl

[2] Zhikov V.V., “K tekhnike predelnogo perekhoda v nelineinykh ellipticheskikh uravneniyakh”, Funkts. analiz i ego pril., 43:2 (2009), 19–38 | DOI | MR | Zbl

[3] Zhikov V.V., Pastukhova S.E., “Parabolicheskii printsip kompensirovannoi kompaktnosti i nekotorye ego prilozheniya”, DAN, 431:3 (2010), 306–312 | MR | Zbl

[4] Tartar L., Cours Peccot, Collège de France, Paris, 1977

[5] Murat F., “Compacité par compensation”, Ann. Scuola Norm. Super. Pisa. Cl. Sci. Ser. 4, 5 (1978), 489–507 | MR | Zbl

[6] Danford N., Shvarts Dzh., Lineinye operatory: Obschaya teoriya, Izd-vo inostr. lit., M., 1962

[7] Zhikov V.V., Pastukhova S.E., “O povyshennoi summiruemosti gradienta reshenii ellipticheskikh uravnenii s peremennym pokazatelem nelineinosti”, Mat. sb., 199:12 (2008), 19–52 | DOI | MR | Zbl

[8] Kinderlerer L., Stampakkya G., Vvedenie v variatsionnye neravenstva i ikh prilozheniya, Mir, M., 1983 | MR

[9] Lions Zh.-L., Nekotorye metody resheniya nelineinykh kraevykh zadach, Mir, M., 1972 | MR

[10] Billingsli P., Skhodimost veroyatnostnykh mer, Nauka, M., 1977 | MR

[11] Ladyzhenskaya O.A., Solonnikov V.A., Uraltseva N.N., Lineinye i kvazilineinye uravneniya parabolicheskogo tipa, Nauka, M., 1967

[12] DiBenedetto E., Degenerate parabolic equations, Springer, New York, 1993 | MR

[13] Zhikov V.V., “K probleme predelnogo perekhoda v divergentnykh neravnomerno ellipticheskikh uravneniyakh”, Funkts. analiz i ego pril., 35:1 (2001), 23–39 | DOI | MR | Zbl

[14] Alkhutov Yu.A., Zhikov V.V., “Teoremy suschestvovaniya i kachestvennye svoistva reshenii parabolicheskikh uravnenii s peremennym poryadkom nelineinosti”, DAN, 430:3 (2010), 295–299 | MR | Zbl

[15] Zhikov V.V., “Ob odnom podkhode k razreshimosti obobschennykh uravnenii Nave–Stoksa”, Funkts. analiz i ego pril., 43:3 (2009), 33–53 | DOI | MR | Zbl

[16] Simon J., “Compact sets in the space $L^p(0,T;B)$”, Ann. Mat. Pura ed Appl., 146 (1987), 65–96 | DOI | MR | Zbl

[17] Edmunds D.E., Rákosník J., “Sobolev embeddings with variable exponent”, Stud. math., 143 (2000), 267–293 | MR | Zbl

[18] Zhikov V.V., “Ob effekte Lavrenteva”, DAN, 345:1 (1995), 10–14 | MR | Zbl

[19] Zhikov V.V., “On Lavrentiev's phenomenon”, Russ. J. Math. Phys., 3:2 (1995), 249–269 | MR | Zbl

[20] Zhikov V.V., Pastukhova S.E., “O svoistve povyshennoi summiruemosti dlya parabolicheskikh sistem peremennogo poryadka nelineinosti”, Mat. zametki., 87:2 (2010), 179–200 | DOI | Zbl