Geodesic
Degenerate equations of monotone type: Lavrent'ev phenomenon and attainability problems
Sbornik. Mathematics, Tome 198 (2007) no. 10, pp. 1465-1494 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

A non-linear monotone equation with degenerate weight function is considered. In the general case the smooth functions are not dense in the corresponding weighted Sobolev space $W$, which leads to a non-uniqueness of a particular kind. Taking for the energy space either $W$ itself or its subspace $H$ equal to the closure of the smooth functions one obtains at least two uniquely soluble problems. In addition, there exist infinitely many weak solutions distinct from the $W$- and $H$-solutions. The problem of approximability or attainability is considered: which solutions of the original equation can be obtained as limits of solutions of the equations with suitable non-degenerate weights? It is shown that the $W$- and the $H$-solutions are attainable; in both cases a regular approximation algorithm is described. Bibliography: 14 titles. 
@article{SM_2007_198_10_a4,
     author = {S. E. Pastukhova},
     title = {Degenerate equations of monotone type: {Lavrent'ev} phenomenon and attainability problems},
     journal = {Sbornik. Mathematics},
     pages = {1465--1494},
     year = {2007},
     volume = {198},
     number = {10},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2007_198_10_a4/}
}
TY  - JOUR
AU  - S. E. Pastukhova
TI  - Degenerate equations of monotone type: Lavrent'ev phenomenon and attainability problems
JO  - Sbornik. Mathematics
PY  - 2007
SP  - 1465
EP  - 1494
VL  - 198
IS  - 10
UR  - http://geodesic.mathdoc.fr/item/SM_2007_198_10_a4/
LA  - en
ID  - SM_2007_198_10_a4
ER  - 
%0 Journal Article
%A S. E. Pastukhova
%T Degenerate equations of monotone type: Lavrent'ev phenomenon and attainability problems
%J Sbornik. Mathematics
%D 2007
%P 1465-1494
%V 198
%N 10
%U http://geodesic.mathdoc.fr/item/SM_2007_198_10_a4/
%G en
%F SM_2007_198_10_a4
S. E. Pastukhova. Degenerate equations of monotone type: Lavrent'ev phenomenon and attainability problems. Sbornik. Mathematics, Tome 198 (2007) no. 10, pp. 1465-1494. http://geodesic.mathdoc.fr/item/SM_2007_198_10_a4/

[1] V. V. Zhikov, “Weighted Sobolev spaces”, Sb. Math., 189:8 (1998), 1139–1170 | DOI | MR | Zbl

[2] V. V. Zhikov, “To the problem of passage to the limit in divergent nonuniformly elliptic equations”, Funct. Anal. Appl., 35:1 (2001), 19–33 | DOI | MR | Zbl

[3] V. V. Zhikov, C. E. Pastukhova, “Usrednenie vyrozhdayuschikhsya ellipticheskikh uravnenii”, Sib. matem. zhurn. (to appear)

[4] V. V. Zhikov, “Homogenization of elasticity problems on singular structures”, Izv. Math., 66:2 (2002), 299–365 | DOI | MR | Zbl

[5] S. E. Pastukhova, “Homogenization of elasticity problems on periodic composite structures”, Sb. Math., 196:7 (2005), 1033–1073 | DOI | MR | Zbl

[6] V. V. Zhikov, A. L. Pyatnitskii, “Homogenization of random singular structures and random measures”, Izv. Math., 70:1 (2006), 19–67 | DOI | MR | Zbl

[7] I. Ekeland, R. Temam, Convex analysis and variational problems, North-Holland, Amsterdam–Oxford; Elsevier, New York, 1976 | MR | MR | Zbl | Zbl

[8] J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Paris; Gauthier, Paris, 1969 | MR | MR | Zbl | Zbl

[9] S. L. Sobolev, Some applications of functional analysis in mathematical physics, Transl. Math. Monogr., 90, Amer. Math. Soc., Providence, RI, 1991 | MR | MR | Zbl | Zbl

[10] V. V. Zhikov, “On two-scale convergence”, J. Math. Sci. (N.Y.), 120:3 (2004), 1328–1352 | DOI | MR | Zbl

[11] V. V. Zhikov, “A note on Sobolev spaces”, J. Math. Sci. (N.Y.), 129:1 (2005), 3593–3595 | DOI | MR | Zbl

[12] V. V. Zhikov, “Connectedness and homogenization. Examples of fractal conductivity”, Sb. Math., 187:8 (1996), 1109–1147 | DOI | MR | Zbl

[13] V. V. Zhikov, S. E. Pastukhova, “Homogenization for elasticity problems on periodic networks of critical thickness”, Sb. Math., 194:5 (2003), 697–732 | DOI | MR | Zbl

[14] S. E. Pastukhova, “Approximate conditions and passage to the limit in Sobolev spaces over thin and composite structures”, J. Math. Sci. (N. Y.), 133:1 (2006), 931–948 | DOI | MR | Zbl